ISSi 5 1 G27 5 Un no. 4- MENT OF THE ARMY CORPS OF ENGINEERS BEACH EROSION BOARD OFFICE OF THE CHIEF OF ENGINEERS SHORE PROTECTION PLANNING AND DESIGN TECHNICAL REPORT NO. 4 [WM B mm nruu] t-tlH jni B \ 1961 Return this book on or before the Latest Date stamped below. Theft, mutilation, and underlining are reasons for disciplinary action and may result in dismissal from the University. University of Illinois Library oi'i...-; G.vjf w ' iiAti 6 - 'JUN 5 DEC 2 C 9/^1 373 . nv' L161—0-1096 /y / no. 4- !%( DEPARTMENT OF THE ARMY CORPS OF ENGINEERS SHORE PROTECTION PLANNING AND DESIGN TECHNICAL REPORT NO. 4 BEACH EROSION BOARD OFFICE OF THE CHIEF OF ENGINEERS Reprinted with revisions of February 1957 and May 1961 1961 \ For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. - Price $3.00 PRBFACS -ai* Vast soas are spent annually In the United States by States, municipali¬ ties and private owners for works designed to prevent erosion damage on sea- coasts and lake shores. The alarming number of such works which have failed due to structural inadequacy, or which have been allowed to deteriorate through lack of maintenance because they were functionally or economically unsound, testifies to the inadequacy of general technical knowledge in this f ie Id. The Beach Erosion Board and its staff have been engaged in the study of shore erosion problems since 1930 and since then have pursued an intensive program of research and development with a view to improving shore protection techniques. While the Board considers that this is a science still very much in the development stage, it is evident that many errors of the past may be avoided in the future by proper application of knowledge thus far gained. The need for providing a sound basis for shore protection planning and design is evident and it is with that objective that this report has been prepared. It is primarily designed to present in a single publication, techniques currently used in the solution of shore protection problems. It is expected that these techniques will be improved by further research and experience. This publication originally superseded Special Issue No. 2, Bulletin of the Beach Erosion Board, which was published in March 1953 and given prelimi¬ nary distribution. The report was prepared by the Engineering Division of the Beach Erosion Board under the direct supervision of J. V. Hall, Jr., and under general supervision of Colonel E, E, Gesler, President of the Board until 31 March 1953, and R. 0. Eaton, Chief Technical Assistant. The task group initially assigned to the preparation of the report was headed by K, P, Peel, temporarily assigned to the Board for this purpose from the South Pacific Division, Corps of Engineers, and Kenneth Kaplan of the Board*s staff. Other staff personnel tdio participated were R. H. Allen, C. T. Fray, R. L. Harris, W, J. Herron, T, Saville Jr., W, H, Vesper and L, L. Watkins. Revisions originating from many comments of interested engineers who received the pre¬ liminary issue (Special Issue No. 2 of the Bulletin) and those directed by the Board were made by R. A. Jachowski and G. M. Watts, and the report was first printed for sale to the public in June 1954. Members of the Board at the time this report was first approved for publication were: Colonel Leland H. Hewitt, President; Colonel Wendell P. Trower, Colonel Herman W. Schull, Jr., Colonel John U, Allen, Resident Member; Thorndike Saville, Morrough P. 0*Brien, and Lorenz G. Straub. Separate revisions to the report were issued in August 1957, The first printing of this report was sold out in 1958, and many requests for it have been received since that time. It is being reprinted at this time with revisions to make it current with the state of knowledge in the coastal engineering field through 1961, but in such form that the outstanding copies of the first printing can be brought up to date by insertion of separately printed pages. The additional material and revisions included in the current printing were prepared by the Engineering Division of the Beach Erosion Board under the supervision of J. V. Hall, Jr., and under general supervision of Colonel H. E. Sprague, Executive of the Board, and R. 0. Eaton, Chief Technical Advisor, Members of the staff who participated in the preparation included R. A. Jachowski, C. L. Bretschneider, G. M. Watts, T. Saville, Jr., J. M. Caldwell, R. P. Savage and L. L. Watkins. Editing for publication was accomplished by A. C. Rayner and R. L. Rector of the Project Development Division. Members of the Board at the time this printing of the report was approved for publication were: Major General Keith R. Barney, President; Brigadier General Robert G. MacDonnell, Brigadier General Thomas H. Lipscomb, Brigadier General Howard A. Morris, Thorndike Saville, Morrough P. O’Brien, and Lorenz G. Straub. This report is published under authority of Public Law 166- 79th Congress, approved July 31, 1945. May 1961 TABLE OP CX3NTENTS Page LIST OF TABLES. vii LIST OP FIGURES. viil INTRODUCTION . XV i PART 1 - FUNCTIONAL PLANNING CHAPTER 1 - PHYSICAL FACTORS Section 1.1 GENERAL. 1 1.2 WAVE ACTION. 1 1.21 GENERAL. 1 1.22 aOVriH AND DECAY OF WIND WAVES. 2b 1.23 WAVE FORECASTING FOR DEEP WATER AREAS. 4 1.231 Measurements for Ocean Areas . 5 1.232 Measurements on Lakes, Bays, Etc. 14 1.233 Forecasting Techniques for Deep Water Areas (S-M-B Method). 15 1.234 Decay Analysis for Deep Water Areas.. 17 1.235 Wave Height and Wave Period Variability. 17 1.24 WAVE FORECASTING FOR SHALLOW WATER AREAS. 25 1.25 WAVE SPECTRA METHOD FOR FORECASTING WIND-GENERATED WAVES . 28e 1.26 WAVES IN TRANSITIONAL AND SHALLOW WATER. 28q 1.261 Refraction of Waves. 29 1.262 Diffraction of Waves. 39 1.263 Refraction and Diffraction Combined . 49 1.264 Breaking Waves . 50 1.27 HURRICANE WAVES . 50 1.3 CHANGES IN WATER LEVEL. 51 1.31 TIDES . 51 1.32 WIND SET-UP. 52a 1.321 General. 52a 1.322 Determination of Wind Set-up and Storm Surge. 52f 1.33 SEICHES . 53 1.34 LAKE LEVELS. 54 1.4 CHARACTERISTICS OP BEACH MATERIALS .. 57 CHAPTER 2 - LITTORAL PROCESSES 2.1 GENERAL . 61 2.2 SOURCES AND CHARACTERISTICS OP MATERIALS. 61 Moy 1961 Section page 2.21 GENERAL. 61 2.22 CONTRIBUTIONS BY STREAMS. 62 2.23 CONTRIBUTIONS BY EROSION OF COASTAL FORMATIONS. 63 2.3 MODES OP LITTORAL TRANSPORT. 64 2.31 DEPTHS AT WHICH MATERIAL MOVES. 64 2.32 DETERMINATION OF DIRECTION AND DIRECTION VARIABILITY , . 67 2.321 Effects of Existing Structures . 67 2.322 Evidence at Headlands .... 67 2.323 Evidence at Tidal Inlets and Streams.. 70 2.324 Wave Analysis. 70 2.325 Variations in Material Characteristics . 76 2.326 Current Measurements ..... . 77 2.33 RATES OP LITTORAL TRANSPORT. 78 2.34 LOSSES OF LITTORAL MATERIALS. 83a 2.341 Losses by Longshore Transport. 83a 2.342 Movement Offshore . 83a 2.343 Losses in Submarine Canyons . 84 2.344 Losses by Deflation. 84 CHAPTER 3 - PLANNING ANALYSIS 3.1 GENERAL. 87 3.2 SEAWALLS, BULKHEADS AND REVETMENTS. 87 3.21 FUNCTIONS. 87 3.22 LIMITATIONS. 87 3.23 FUNCTIONAL PLANNING OF THE STRUCTURE.. . 87 3.24 USE OP THE STRUCTURE. 88 3.25 LOCATION OF STRUCTURE WITH RESPECT TO SHORE LINE .... 88 3.26 LENGTH OP STRUCTURE. 88 3.27 HEIGHT OP STRUCTURE.. 89 3.271 Wave Run-Up. 89 3.272 Wave Overtopping. 90a 3.28 DETERMINATION OP GROUND ELEVATION IN FRONT OP STRUCTURE . 92 3.3 PROTECTIVE BEACHES. 93 3.31 FUNCTIONS . 93 3.32 LIMITATIONS. 94 3.33 PLANNING CRITERIA. 94 3.331 Direction of Material Movement and Deficiency of Supply. 95 3.332 Selection of Borrow Material. 95 3.333 Berm Elevation and Width ,,,,, . 96 3.334 Slopes. 96a 3.335 Feeder Beach Location .. 96a Section SAND DUNES Page 96b 3.34 3.4 GROINS. 96b 3.41 DEFINITION. 96b 3.42 GROIN OPERATION. 97 3.43 PURPOSE. 99 3.44 LIMITATIONS ON TOE USE OF GROINS. 99 3.45 TYPES OP GROINS. 100 3.451 Perioeable Groins. 100 3.452 High and Low Groins. 100 3.453 Adjustable Groins . 100 3.46 DIMENSIONS OP (ROINS. 101 3.461 Horizontal Shore Section . 101 3.462 Intermediate Sloped Section . 101 3.463 The Outer Section. 101 3.464 Spacing of Groins. 102 3.465 Length of Groin. 102 3.47 ALIGNMENT OF GROINS. 106 3.48 ORDER OP GROIN CWNSTOUCTION. 106 3.5 JETTIES. 108 3.51 DEFINITION. 108 3.52 TYPES. 108 3.53 SITING. 108 3.54 EFFECrrS ON TOE SHORE LINE. 108 3.6 BREAKWATERS - SHORE CONNECTED. 109 3.61 DEFINITION. 109 3.62 TYPES. 109 3.63 SITING. 109 3.64 EFFECT ON TOE SHORE LINE. 110 3.7 KlEAKWATERS - OFFSHORE. 110 3.71 DEFINITION. 110 3.72 TYPE. 110 3.73 SITING. 110 3.74 EFFECTS ON TOE SHORE LINE. 110 3.75 OPERATION OP AN OPPSHCRE BREAKWATER. 112 3.76 OFFSHORE BREAKWATERS IN SERIES. 112 3.77 HEIGHT OF AN OFFSHORE BREAKWATER. 112 PART II - STRUCTURAL DESIGN CHAPTER 4 - PHYSICAL FACTORS Section Page 4.1 WAVE HEIGHT. 116 4.11 SELECTION OF DESIGN WAVE. 116 4.12 DETERMINATION OF SIGNIFICANT DESIGN WAVE AT STRUCTURE . . 116a 4.2 WAVE FORCES. 117 4.21 GENERAL. 117 4.22 DETERMINATION OF BREAKER DEPTH AND HEIGHT. 118 4.23 NON-BREAKING WAVES. 118 4.231 Sainflou Method; Forces Due to Non-breaking Waves , 118 4.232 Wall of Low Height. 120 4.24 WAVES BREAKING ON A STRUCTURE. 123 4,241 Minikin Method: Forces Due to Breaking Waves . , , 124 4.25 WAVES BREAKING SEAWARD OF A STRUCTURE. 125 4.26 EFFECT OF FACE SLOPE ON WAVE PRESSURES. 130 4.27 STABILITY OF RUBBLE-MOUND STRUCTURES. 131 4.271 Armor Units and Slope of Cover Layer . 131 4.272 Selection of Kp Factor. 132 4.273 Precast Concrete Armor Units. 133a 4.274 Crest Elevation and Width. 133b 4.275 Concrete Cap for Rubble<4(ound Structures. 133c 4.276 Additional Design Features. 133d 4.277 Plates and Tables... 134 4.28 WAVE FORCES ON PILES. 135 4.281 Drag and Inertial Coefficients. 138 4.282 Use of the Generalized Graphs. 138m 4.3 EARTH FORCES. 140 4.31 ACTIVE FORCES... 140 4.311 Unit Weights and Internal Friction Angles ..... 140 4.312 Application of Resultant of Earth Force ...... 143 4.32 PASSIVE FORCES. 143 4.33 SURCHARGE LOADS. 143 4.34 SUBMERGED MATERIALS. 143 4.35 UPLIFT. 144 4.4 ICE FORCES.. 144 4.5 VELOCITY FORCES. 145 4.6 MATERIALS. 146 4.61 CONCRETE. 146 4.62 STEEL. 146 4.63 TIMBER. 146 4.64 STONE. 146 May 1961 IV Section 5.1 5.11 5.12 5.121 5.122 5.123 5.124 5.13 5.131 5.132 5.133 5.134 5.2 5.21 5.22 5.221 5.222 5.223 5.224 5.225 5.226 5.227 5.228 5.229 5.23 5.24 5.241 5.242 5.3 5.31 5.32 5.33 5.331 5.34 5.341 5.35 5.4 5.41 5.42 5.421 5.422 5.423 5.424 CHAPTER 5 - STRUCTURAL ANALYSIS Pa^e SEAWALLS, BULKHEADS AND REVETMENTS. 147 TYPES. 147 SELECTION OP TYPE. 150 Foundation Conditions . 150 Exposure to Wave Action. 150 Availability of Materials. 150 Costs. 154 DESIGN. 154 High Semi-Gravity Type Concrete Wall .. 154 Steel Sheet Pile Cellular Seawall. 167 Steel Sheet Pile Bulkhead. 177 Revetment... 198 BREAKWATERS ;WD JETTIES. 200 GENERAL. 200 TYPES. 201 Rubble«Moiuid. 201 Precast Concrete Armor Units . 201 Composite Earth and Stone . 204 Composite Stone and Concrete. 204 Concrete Caissons... 204 Sheet Piling... 207 Crib Types. 207 Solid Fill. 210 Asphaltic Materials .... 210 SELECTION OF TYPE. 210 DESIGN PROBLEMS. 210b Caisson Type Breakwater . 210b Rubble«Mound Breakwaters .. 213 SAND bypassing. 216 GENERAL. 216 METHODS. 217 LAND-BASED WEDGING PLANTS. 217 Fixed Bypassing Plants ..... . 218 FLOATING DREDGES. 220b Bypassing Operations by Floating Plant . 220e MOBILE LAND-BASED VEHICLES. 220i SAND DUNES. 220^2 DUNE BUILDING. 221 DUNE STABILIZATION. 225c Types of Vegetation. 225c Climatic Regions of Dune Vegetation. 225d Available Plants in the Different Climatic Regions 225f Methods of Establishing Plantings .... . 225f May 1961 V Page Section 5.5 GROINS. 225j 5.51 GENERAL. 225 j 5.52 CONCRETE. 225j 5.521 Concrete Permeable Groins ..... . 225j 5.522 Concrete Impermeable Groins. 225j 5.53 STEEL SHEET PILE GROINS. 228 5.54 STONE GROINS. 230 5.55 TIMBER GROINS. 232 5.56 ASPHALT GROINS . .. 234 5.57 SELECTION OP TYPE. 235 5.58 DESIGN. 236 5.581 Concrete Block Groins. 236 5.582 Rubble-Mound Groins. 240 5.583 Vertical Sheet Pile Groins. 240 5.6 MISCELLANEOUS DESIGN PRACTICES. 241 APPENDICES A - GLOSSARY OF TERMS B - LIST OF COKMON SYMBOLS C - BIBLIOGRAPHY D - MISCELLANEOUS TABLES AND GRAPHS E - MISCELLANEOUS DERIVATIONS May 1961 VI LIST OP TABLES TABLE MO. title PAGE fO, 0-A WIND-SPEED ADJUSTMENTS NEARSHORE. 2c 0-B JOINT DISTRIBUTION OF H AND T FOR ZERO CORRELATION . « 18 1 FORECAST FOR SYNOPTIC CHARTS OP 26-27 OCTOBER 1950 . . 24 1-A SIGNIFICANT RANGE OF PERIOD FOR FULLY ARISEN SEA FOR DIFFERENT WIND VELOCITIES, V. 28h 1-B CHARACTERISTICS OF FULLY ARISEN SEA. 28h 1- C MINIMUM FETCH AND MINIMUM DURATION OF WIND ACTION NEEDED TO GENERATE A PRACTICALLY FULLY ARISEN SEA FOR DIFFERENT WIND VELOCITIES . 28i 2 COMPUTATIONS FOR VALUES OF Cj^/C2, T « 10 SEC. 33 2- A COMPUTATIONS FOR WIND WAVES OVER CONTINENTAL SHELF . . 50i 3 TIDAL RANGES ... 51 3- A MEAN RANGE AND HIGHEST AND LOWEST TIDES - ATLANTIC AND GULF COAST. 52d 3-B DIURNAL RANGE AND HIGHEST AND LOWEST TIDES - PACIFIC COAST. 52e 3-C CLASSIFICATION OF WIND SET-UP AND STORM SURGE PROBLEM 52f 3-D PARAMETER RELATIONS FOR WIND SET-UP IN RECTANGULAR CHANNEL OF CONSTANT DEPTH FOR NON-EXPOSED BOTTOM. . 52j 3-B PARAMETER RELATIONS FOR WIND SET-UP IN RECTANGULAR CHANNEL OP CONSTANT DEPTH FOR EXPOSED BOTTOM ... 52j 3-F WIND STRESS. 52m 3-G WIND SET-UP. 52o 3-H WIND SET-UP COMPUTATIONS. 52n 3-1 SUMMARY OP SURGE COMPUTATIONS. 52w 4 PLUCTATIONS IN WATER LEVELS - THE GREAT LAKES - PH(IOD 1860-1960 INCLUSIVE. 54 5 STANDARD SIZE CLASSIFICATION. 57 6 DETRITAL AND ASSOCIATED MINERALS. 60 7 & 8 NO TABLES. 9 DETERMINATION OF SIGNIFICANT DESIGN WAVE HEIGHTS ... 117 9-A PROVISIONAL Kd VALUES FOR USB IN DETERMINING ARMOR UNIT WEIGHT. 133a 9-B LAVER COEFFICIENT & POROSITY FOR VARIOUS ARMOR UNITS . 133e 10 UNIT WEIGHTS AND INTERNAL FRICTION ANGLES. 142 11 COEFFICIENTS AND ANGLES OP FRICTION. 142 12 SAFE LOAD PER PILE. 165 13 PLANTS OF THE EIGHT COASTAL REGIONS. 225g Moy 1961 VII LIST OF FIGURES FIGURE NO. TITLE PAGE NO, 1 WAVE DEVELOPMENT WITHIN A FETCH. 3 2 STATION MODEL. 6 3 GEOSTHOPHIC WIND SCALE. 8 4 SEA SURFACE TEMPERATURE SEPT. - OCT. - NOV. 9 5 SURFACE WIND SCALE. 10 6 ISOBAR IC PATTERNS AND CONDITIONS FOR FRONTAL LINES . . 13 7 DEEP WATER WAVE FORECASTING CURVES AS A FUNCTION OF WIND SPEED, FETCH LENGTH, AND WIND DURATION (FOR FETX:HES 1 to 1,000 MILES). 16a 7-A DEEP WATER WAVE FORECASTING CURVES AS A FUNCTION OF WIND SPEED, FETCH LENGTH, AND WIND DURATION (FOR FETCHES 100 to > 1,000 MILES). 16b 7_B H-t-F-T - DIAGRAM FOR FORECASTING WIND-GENERATED WAVES FOR FETCHES UP TO 1,000 MILES. 16d 7-C H-t-F-T - DIAGRAM FOR FORECASTING WIND-GENERATED WAVES FOR FETCHES UP TO 200 MILES. 16e 7-D COMPARISON OF COMPUTED AND GRAPHICALLY DETERMINED HEIGHTS AND TRAVEL DISTANCES OF WAVES GENERATED WITHIN A STATIONARY HURRICANE .. 16f 7-E SIGNIFICANT HEIGHTS AND PERIODS OF WAVES GENERATED IN A VARIABLY MOVING WIND SYSTEM OF VARIABLE WIND VELOCITY 16f 7-F RELATION OF EFFECTIVE FETCH TO WIDTH-LENGTH RATIO FOR RECTANGULAR FETCHES . 16i 8 DECAY CURVES . 19 9 TRAVEL TIME OF SWELL BASED ON tps: D/Cg. 20 10 SURFACE SYNOPTIC CHART FOR 1230Z 26 OCT., 1950 .... 21 11 SURFACE SYNOPTIC CHART FOR 0030Z 27 OCT., 1950 .... 22 12 SURFACE SYNOPTIC CHART FOR 0630Z 27 OCT., 1950 .... 23 13 PREDICTION LINES FOR WAVE HEIGHT DISTRIBUTION FUNCTIONS FOR VARIOUS MEAN WAVE HEIGHTS. ’ 26 14 GROWTH OF WAVES IN A LIMITED DEPTH. 27 15 GENERATION OF WIND WAVES OVER A BOTTOM OF CONSTANT SLOPE FOR UNLIMITED WIND DURATION AND f/m = 5.28: PRESENTED AS DIMENSIONLESS PARAMETERS. 28a 15-A GENERATION OF WIND WAVES OVER A BOTTOM OF CONSTANT SLOPE FOR UNLIMITED WIND DURATION AND f/m = 10.6: PRESENTED AS DIMENSIONLESS PARAMETERS. 28b 15-B GENERATION OP WIND WAVES OVER A BOTTOM OF CONSTANT SLOPE FOR UNLIMITED WIND DURATION AND f/m = 52.8: PRESENTED AS DIMENSIONLESS PARAMETERS. 28c 15-C GENERATION OF WIND WAVES OVER A BOTTOM OF CONSTANT DEPTH FOR UNLIMITED WIND DURATION PRESENTED AS DIMENSIONLESS PARAMETERS. 28d 15-D CO-CUMULATIVE POWER SPECTRA FOR OCEAN WAVES AT WIND VELOCITIES BETWEEN 20 KNOTS AND 36 KNOTS. 28g 15-E DURATION GRAPH - CO-CUMULATIVE SPECTRA FOR WIND SPEEDS FROM 10 to 20 KNOTS AS A FUNCTION OF DURATION .... 28k Moy 1961 VIII LIST OP FIGURES (ContM) FIGURE NO, TITUB PAGE NO. 15-P DURATION OlAPH - CX)-CUMULATIVE SPECTRA FOR WIND SPEEDS FROM 20 TO 36 KNOTS AS A FUNCTION OF DURATION. 281 15-G DURATION GRAPH - 00-CUMULATIVB SPECTRA FOR WIND SPEEDS FROM 36 TO 56 KNOTS AS A FUNCTION OP DURATION. 28m 15-H FETCH GRAPH - OO-CUMULATIVE SPECTRA FOR WIND SPEEDS PROM 10 TO 20 KNOTS AS A FUNCTION OF FETCH. 28n 15-1 FETCH GRAPH - OO-CUMULATIVE SPECTRA FOR WIND SPEEDS PROM 20 TO 36 KNOTS AS A FUNCTION OP FETCH. 28o 15-J FETCH GRAPH - OO-CUMULATIVE SPECTRA FOR WIND SPEEDS PROM 36 TO 56 KNOTS AS A FUNCTION OF FETCH. 28p 16 WAVE REmACTION AT WEST HAMPTON BEACH, L, I. , N.Y. DECENffiER, 1938. 30 17 REFRACTION TEMPLATE. 32 18 USE OP THE REDACTION TEMPLATE. 34 19 REDACTION DIAGRAM USING R/J METHOD. 36 20 USE OF FAN-TYPE REFRACTION DIAGRAM. 37 21 GENERALIZED DIFFRACTION DIAGRAM . 38 22 DIFFRACTION FOR A SINGLE BREAKWATER -NORMAL INCIDENCE . . 39 23 GENERALIZED DIFFRACTION DUGRAM FOR A BREAKWATER-GAP WIDTH OF TWO WAVE LENGTHS (BA *»2) . 40 24 CONTOURS OF EQUAL DIFFRACTION COEFFICIENT-GAP WIDTH a 0.5 WAVE LENGTH (BA « 0.5). 41 25 CONTOURS OP EQUAL DIPmAETION COEFFICIENT-GAP WIDTH * 1 WAVE LENGTH (BA *1). 41 26 CONTOURS OP EQUAL DIFHlACTION COEFFICIENT-GAP WIDTH * 1.41 WAVE LENGTHS (BA « 1.41). 42 27 CONTOURS OP EQUAL DIFFRACTION COEFFICIENT-GAP WIDTH * 1.64 WAVE LENGTHS (BA * 1.64). 42 28 CONTOURS OF EQUAL DIFFRACTION COEFFICIBNT-a4P WIDTH s 1.78 WAVE LENGTHS (BA * 1.78). 43 29 CONTOURS OP EQUAL DIFHIACTION COEFFICIENT-GAP WIDTH a 2 WAVE LENGTHS (BA * 2). 43 30 CONTOURS OF EQUAL DIPmACTION COEFFICIENT-GAP WIDTH a 2.50 WAVE LENGTHS (BA « 2.50). 44 31 CONTOURS OP EQUAL DIFFRACTION COEFFICIENT-GAP WIDTH a 2.95 WAVE LENGTHS (BA * 2.95). 44 32 CONTOURS OF EQUAL DIFFRACTION COEFFICIENT-GAP WIDTH a 3.82 WAVE LENGTHS (BA * 3.82). 45 33 CONTOURS OP EQUAL DIFFRACTION COEFFICIENT-GAP WIDTH a 5 WAVE LENGTHS (BA = 5). 45 34 DIFFRACTION FOR A BREAKWATER GAP OP WIDTH >5L (BA >5). . 46 35 WAVE INCIDENCE OBLIQUE TO BREAKWATER GAP. 46 35-A DIFFRACTION FOR A BREAKWATER GAP OF ONE WAVE LENGTH WIDTH (0 a 0 AND 15 DEGREES). 46a 35-B DIFFRACTION FOR A BREAKWATER GAP OF ONE WAVE LENGTH WIDTH (0 a 30 AND 45 DEGREES). 46b May 1961 iX LIST OF FIGURES (ContM) FIGURE NO, TITIJB PAGE NO. 35-C DIFFRACTION FOR A BREAKWATER GAP OF ONE WAVE LENGTH WIDTH (0 * 60 AND 75 DEGREES). 46c 35-D DIFFRACTION DIA®AM FOR A GAP OF TWO WAVE LENGTHS AND A 45-DH31EE APPROACH COMPARED WITH THAT FOR A GAP WIDTH V^WAVE LENGTHS WITH A 90-DEGREE APPROACH. 46d 36 SINGLE BREAKWATER - REmACTION - DIFFRACTION COMBINED . , 47 37 BREAKER HEIGHT INDICES. 48 38 BREAKER DEPTH INDICES. 48 38-A WAVE HEIGHT AND DEPTH OF WATER AT POINT OF BREAKING FROM SOLITARY WAVE RELATIONSHIPS. 50a 38-B ISOLINES OP RELATIVE SIGNIFICANT WAVE HBIOTT FOR SLOWLY MOVING HURRICANE. 50e 38-C Kg VERSUS tV dt. 50f 38-D RELATIONSHIP FOR FRICTION LOSS OVER A BOTTOM OF CONSTANT DEPTH. 50g 38-E WIND EFFECTS ON LAKE BRIE. 52b 38-P LAKE OKEECHOBEE, FLORIDA: HURRICANE OF AUG. 26-27, 1949 LAKE SURFACE CONTOURS. 52c 38-G WIND STRESS DIAGRAM. 52n 38‘-« EXAMPLE OF WIND SET-UP COMPUTED SURFACE PROFILE FOR AN INCLOSED LAKE (FIRST APPROXIMATION). 52q 38- 1 MEAN BOTTOM PROFILE OFF MOUTH OF CHESAPEAKE BAY. 52v 39- A HYWOGRAPH OP MONTHLY MEAN LEVELS OF THE GREAT LAKES (1860-1920) 55 39-B HYDROGRAPH OF MONTHLY MEAN LEVELS OF THE GREAT LAKES (1920-1960) 56 40 LATERAL MOVEMENT OF LITTORAL DRIFT. 65 41 LABORATORY RESULTS INDICATING TRENDS OF PROPORTIONATE BED LOAD AND SUSPENDED LOAD TRANSPORT RELATED TO WAVE ENERGY AND WAVE STEEPNESS. 65 42 SIZE DISTRIBUTION ACROSS A BEACH. 66 43 SEASONAL VARIATION OF SIZE DISTRIBUTION IN SEDIMENTS ALONG ”D*’ RANGE OFF LA JOLLA, CALIFORNIA. 66 44 TIME HISTORIES, PARTIAL VELOCITY DISTRIBUTION AND ORBITAL PATHS, OP WAVES IN DEEP WATER, IN SHALLOW WATER, AND AT BREAKING. 68 45 RELATIONSHIP BETWEEN GRAIN SIZE AND FORESHORE SLOPE ... 69 46 EFFECT OF SINGLE GROIN. 71 47 EFFECT OF A SERIES OP GROINS. 71 48 EFFECT OF OFFSHORE BREAKWATER. 72 49 EFFECT OF ENTRANCE JETTIES. 72 50 EFFECT OF SHORE-CONNECTED BREAKWATER. 73 51 EFFECT OF PILE CLUSTERS. 73 52 DIRECTION OF LITTORAL TRANSPORT INDICATED BY HEADLANDS . 74 53 TIDAL INLET TO BAY OR LAGOON. 75 Nloy 1961 X LIST OF FIGURES (Cont»d) FIGURE NO. TITLE PAGE 54 TIDAL INLET THROUGH A BARRIER BEACH. 75 55 REFRACTION DIAGRAM-WAVE CONDITIONS BY ORTHOGONALS ... 76 56 TYPICAL VANE FLOAT ASSEMBLY. 79 57 ESSENTIALLY COMPLETE LITTORAL BARRIER. 80 58 SUBSTANTIALLY COMPLETE LITTORAL BARRIER. 81 59 TEMPORARY LITTORAL BARRIER. 82 59-A RELATION BETWEEN ALONGSHORE COMPONENT OP WAVE ENERGY AND LITTORAL TRANSPORT RATE. 83b 60 WIM) VELOCITY GRADIENTS. 86 61 RELATION BETWEEN WIND VELOCITY AND RATE OF SAND MOVEMENT. 86 .61-A WAVE RUN-UP FOR SPECIFIC VALUES OF /T2 (dAlo*>3) . . 89d 61-B WAVE RUN-UP FOR SPECIFIC VALUES OP Ho'/Ta (l3. 90 62- A RELATION OF RATE OP OVERTOPPING TO ELEVATION OF WALL CREST (VERTICAL WALL). 90c 62-B RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CREST (SMOOTH 1 ON l| SLOPE). 90c 62-C RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CREST (STEP-FACE 1 ON l| SLOPE). 90d 62-D RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CaiBST (RIP-RAP COVER). 90d 62-E RELATION OP RATE OF OVERTOPPING TO ELEVATION OF WALL CREST ( VERTICAL WALL, DEPTH s 4.5*) . 90e 62-F RELATION OF RATE OF OVERTOPPING TO EUSVATION OF WALL CREST (CURVED WALL). 90e 62-G RELATION OP RATE OF OVERTOPPING TO ELEVATION OF WALL C31EST (CURVED WALL). 90f 62-H RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CREST (RECURVED WALL). 9Of May 1961 XI LIST OP FIGURES (ContM) FIGURE NO. TITLE PAGE WO. 62-1 RELATION OF RATE OP OVERTOPPING TO ELEVATION OF WALL CREST (SMOOTH 1 ON 3 SLOPE). 90g 62-J RELATION OP RATE OF OVERTOPPING TO ELEVATION OP WALL CREST (SMOOTH 1 ON SLOPE). 90h 62-K RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CREST (STEP-PACE 1 ON SLOPE). 90i 62-L RELATION OF RATE OF OVERTOPPING TO ELEVATION OP WALL CREST (RIPRAP (COVER). 90j 62-M RELATION OF RATE OF OVERTOPPING TO ELEVATION OP WALL CREST (VERTICAL WALL, DEPTH = 9* ). 901c 62-N RELATION OP RATE OF OVERTOPPING TO ELEVATION OP WALL CREST (SMOOTH SLOPE 1 ON 3). 90i 62-0 RELATION OP RATE OF OVERTOPPING TO ELEVATION OF WALL CREST (1 ON SMOOTH SLOPE). 90* 62-P RELATION OP RATE OP OVERTOPPING TO ELEVATION OF WALL CREST (STEP FACE). 90n 62-Q RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CREST (RIPRAP COVER). 91 63 EFFECrrS OP EROSION. 93 64 STABILIZED AND MIGRATING DUNES AT CAPE HATTERAS NATIONAL SEASHORE RECREATIONAL AREA, NORTH CAROLINA. 96c 65 EFFECT OF GROIN. 98 66 EFFECT OF GROIN FIELD. 98 67 GROIN FIELD OPERATION. 103 68 GROIN FIELD OPERATION WITH REVERSAL OF TRANSPORT. 103 69 TYPICAL GROIN PROFILE. 105 70 RECESSION AND ACCRETION BETWEEN GROINS. 105 71 PROFILE ON DOWNDRIPT SIDE OP INTERMEDIATE GROIN ..... . 107 72 RECESSION - DOWN DRIFT OF A SINGLE (ROIN. 107 73 EFFECT ON SHORE LINE OP OFFSHORE BREAKWATER. 109 74 OFFSHORE BREAKWATER; VENICE, CALIFORNIA. Ill 75 OFFSHORE BREAKWATER: WINIWOP BEACH, MASSACHUSETTS .... 113 76 AVERAGE EFFECT OP STRUCTURE ON WAVE HEIGHT RATIO WITH SUBMERGENCE. 114 77 RATIO OF ENERGY LANDWARD TO SEAWARD FOR VARIED SUBMERGENCE OF SIRUCTURES. 114 78 CLAPOTIS ON VERTICAL WALL. 119 79 DETERMINATION OP VALUE OF ho IN SAINPDOU'S FORMULA .... 121 80 DETERMINATION OF VALUE OF Pi IN SAINFLOU’S FORMULA .... 122 81 PRESSURE ON WALLS OF LOW HEIGHT. 123 82 MINIKIN WAVE PRESSURE DIAGRAM. 126 83 WAVE PRESSURES FROM BROKEN WAVES: WALL SEAWARD OP SHORE LINE. 129 84 WAVE PRESSURES FROM BROKEN WAVES: WALL LANDWARD OF SHORE LINE. 129 May 1961 XII LIST OP FIGURES (Cont'd) FIGURE NO. 85 85-A 85- B 86- A 86-B 86-C 86-D 86-E 86-F 86-G 86-H 86-1 86- J 87 87- A 87-B 88 89 89-A 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 May 1961 TITIJB WALL SHAPES . TOEORETICAL AND TYPICAL BREAKWATER SECTION FOR NON¬ BREAKING WAVE CONDITION . THEORETICAL AND TYPICAL BREAKWATER SECTION FOR BREAKING WAVE CONDITION. FORCES ON PIUSS - SCHEMATIC DIAGRAM OP PARTICLE MOTION . FORCES ON PILES - FORCE COMPONENTS . RATIO OF CREST ELEVATION ABOVE STILL WATER TO WAVE HEIGHT RELATIVE WAVE LENGTH AND PRESSURE FACTOR VS (AIRY WAVE THEORY). CORRECTION FACTOR FOR WAVE LENGTH DUE TO STEEPNESS . . . Kom VERSUS dA^. Kim VERSUS dA^. Pm/PDm VS Pim/FDm FOR THE BREAKING SOLITARY WAVE AND THE AIRY WAVE . RELATIVE LEVER ARM MEASURED PROM BOTTOM VS RELATIVE DEPTH CORRESPONDING TO MAXIMUM DRAG FORCE. RELATIVE LEVER ARM MEASURED FROM BOTTOM VS RELATIVE DEPTH CORRESPONDING TO MAXIMUM INERTIAL FORCE. RELATIONSHIPS OF ANGULAR POSITION OP MAXIMUM MOMENT PROM MODEL STUDIES. COEFFICIENT OF DRAG AS A FUNCTION OF REYNOIDS NUMBER OP WAVES HIGHER THAN 10 FEET. DRAG COEFFICIENT VS REYNOIDS NUMBER FOR CIRCULAR CYLINDER. EARTH PRESSURE ON AN INCLINED WALL. APPLICATION OF RESULTANT OF EARTH PRESSURE. VEIOCITY VS STONE WEIGHT AND EQUIVALENT STONE DIAMETER . COMBINATION STEPPED AND CURVED PACE WALL: CONCRETE . . . STEPPED PACE WALL: CONCRETE. CELLULAR SHEET PILE WALL: STEEL. STONE SEA WALL: STAGE PLACED RUBBLE-MOUND. SLAB AND KING PILE BULKHEAD: CONCRETE. SHEET PILE BULKHEAD: STEEL. TIMBER SHEET PILE BULKHEAD. STONE REVETMENT - PORT STORY, VIRGINIA. CONCRETE REVETMENT - PIONEER POINT, CHESAPEAKE BAY MARYLAND. BREAKING WAVE PRESSURE DIAGRAM. CROSS SECTION OP CONCRETE WALL. FORCE RESULTANT, OVERTURNING LANDWARD. BEARING PRESSURES, OVERTURNING LANDWARD . FORCE RESULTANT, OVERTURNING SEAWARD . BEARING PRESSURES, OVERTURNING SEAWARD . DISTRIBUTION OF PRESSURES, BROKEN WAVES. DISTRIBUTION OF PRESSURES: BREAKING WAVES. xiii PAGE NO. 130 133f 133g 136 136 138a 138b 138c 138d 138d 138e 138f 138g 138h 138i 138j 141 141 146 148 148 149 149 151 151 152 153 153 157 159 163 163 166 166 169 169 LIST OF FIGURES (Cont'd) FIGURE NO, TITLE PAGE NO. 107 DIAPHRAGM TYPE CELLULAR WALL. 173 108 COUPLE REPLACING OVERTURNING MOMENT. 175 109 RESULTANT FORCE . 176 110 DIAPHRAGM TYPE CELL. 177 111 BULKHEAD PLACEMENT. 179 112 LOADING DIAGRAM FOR MINIMUM PILE LENGTH. 179 113 LOADING, SHEAR, AND BENDING MOMENT DIA®AMS BETWEEN POINTS F AND Zj . 185 114 LOADING, SHEAR, AND BENDING MOMENT DIAGRAMS ABOVE DBADMAN TIE. 186 115 LOADING DIAGRAM FOR MINIMUM BENDING MOMENT. 188 116 LOADING, SHEAR, AND MOMENT DIAGRAMS BELOW POINT F ... . 189 117 LOADING DIAGRAM BELOW POINT G. 194 118 DEADMAN LOADING DIAGRAM . 195 119 DEADMAN DIMENSIONS. 195 120 SHEET PILE DEADMAN. 197 121 DEADMAN DISTANCE FROM WALL. 198 122 RUBBLE-MOUND BREAKWATER - TYPICAL SECTION - MORRO BAY HARBOR, CALIFORNIA. 202 123 LOS ANGELES AND LONG BEACH OUTER HARBOR BREAKWATER - COMPOSITE EARTH AND STONE. 202 124 TYPICAL SECTION OP CONCRETE CAPPED JETTY - LAKE WORTH INLET, FLORIDA. 203 125 TYPICAL SECTION - SOUTH BREAKWATER - MUSKEGON HARBOR, MICHIGAN. 203 126 PORT O’CONNOR DIKES - PORT O’CONNOR, TEXAS. 205 127 CIRCULAR TYPE OP CELLULAR BREAKWATER. 205 128 DETAILS OF TIMBER BREAKWATER.. . 206 129 TYPICAL SECTIONS - WEST PIER - MICHIGAN CITY HARBOR, INDIANA. 206 130 TIMBER CRIB TYPE BREAKWATERS. 208 131 BREAKWATER AT HARBOR BEACH, MICHIGAN - TIMBER CRIB TYPE . 208 132 TYPICAL SECTION - SOLID PILL BREAKWATER. 209 133 TYPICAL SECTION OF JETTY - LAKE WORTH INLET, FLORIDA AND GALVESTON, TEXAS. 209 134 TYPICAL TETRAPOD CROSS SECTION OP OUTER BREAKWATER - CRESCENT CITY, CALIFORNIA. 210a 135 PRESSURE DUGRAM FOR CAISSON TYPE BREAKWATER. 211 136 CROSS SECTION OF BREAKWATER. 215 137 SAND TRANSFER PLANT, SOUTH LAKE WORTH INLET. 219 137-A SOUTH LAKE WORTH INLET BYPASSING PLANT. 220 137-B FIXED BYPASSING PLANT AT LAKE WORTH INLET, FLORIDA ... 220 138 SHORE FEATURES AT DURBAN, SOUTH AFRICA. 220c 139 SANTA BARBARA HARBOR - SAND BYPASSING. 220e 139-A LOCATION MAP - PORT HUENEME, CALIFORNIA. 220g Moy 1961 Xiv LIST OP FIGURES (ContM) FIGURE NO. TITLE PAGE NO. 139-B PORT HUENEME, CALIFORNIA - APPROXIMATE SHORE LINE DURING DREDGING IN PHASE 1 . 220h 139-C GENERAL PLAN - VENTURA COUNTY HARBOR, CALIFORNIA , . . 220j 139-D DUNES READY FOR PIONEER PLANTING AT CAPE HATTER AS NATIONAL SEASHORE RECREATIONAL AREA, NORTH CAROLINA . 222 139-E WORK UNDER CURRENT PROGRAM AT CAPE HATTERAS NATIONAL SEASHORE RECREATIONAL AREA. 223 139- P FENCING DESIGNS USED ON OUTER BANKS, NORTH CAROLINA . . 224 140 DEVELOPMENT AROUND FENCE PANELS (V-TYPE) . 225 140- A DEVELOPMENT AROUND PENCE PANELS (SLANT TYPE). 225 141 DEVELOPMENT OF AN OILED DUNE. 225a 142 STABILIZED AND ERODED DUNES AT CAPE HATTERAS NATIONAL SEASHORE RECREATIONAL AREA, NORTH CAROLINA . 225b 143 COMBINED PRECAST CONCRETE AND ROCK PILL GROIN. 226 144 CONCRETE BLOCK GROIN (HARRISON WEBER). 227 145 CONCRETE BLOCK GROIN (C.L. LEEDS). 227 146 TYPICAL STEEL SHEET PILE OlOIN. 229 147 TYPICAL CELLUUR STEEL GROIN. 231 148 TYPICAL STONE BLOCK GROINS. 231 149 TYPICAL STONE GROIN. 232 150 TYPICAL TIMBER GROIN. 233 151 WAVE PRESSURE DIAGRAM. 239 152 Angle op wave approach. 239 May SHORE PROTECTION PLANNING AND DESIGN INTRODUCTION This report has been prepared with a view to assembling in a single volume, insofar as practicable, a manual of practice for shore protection. The term ’’shore protection”, as it is used herein, applies primarily to works designed to stabilize seacoasts and shores of large bodies of water where wave action is the principal cause of erosion. The nature and degree of protection required differ widely at dif¬ ferent localities and the proper solution of any specific problem requires a systematic and thorough study. The first requisite for such a study is a clear definition of the problem and the objectives sought; the first factor to be determined in the course of the study is the cause of the problem. Ordinarily there will be more than one method of obtaining the immediate objective. In the study, therefore, the long term effects of each method should be forecast and evaluated, beyond as well as within the problem area. All advantages and effects should be considered in comparing annual costs and benefits to determine the justification of remedial measures An attempt has been made to include herein a detailed summary of applicable methods, techniques and useful data pertinent to the solution of shore protection problems. Part I discusses the factors considered important in the analysis of such problems, beginning with the sources of energy and characteristics of material, proceeding with the interaction of these factors and ending with the development of functional plans. Part II presents generally accepted practice in structural design techniques for shore structures. Graphs and tables to facilitate computations and analysis are provided throughout the text and in appendices. Techniques presented herein are generally applicable to the broad scope of shore protection pro¬ blems but competent engineering judgment is required for determining their application to any specific problem. As the meanings of terms used in coastal engineering differ from place to place, the reader is advised to make full use of Appendix A (Glossary of Terms). XVI PART I FUNCTIONAL PLANNING CHAPTER 1 PHYSICAL FACTORS 1.1. GENERAL Beach and shore erosion consists of wearing away of the land by the application of energy to shore materials. Kinetic energy is available at the shore in the form of wind, which acts directly on shore materials. However, the principal method of application of the wind’s energy to the shore is through water waves generated by the wind. A considerable portion of the total energy of the wind acting over great expanses of water is thus enabled to reach the shore. Occasionally waves are also generated by other sources of energy, such as earthquakes. Other physical factors which establish the criteria for planning pro¬ tective measures are changes in water level, which determine the elevation at which the wave energy will act on the shore. Though ice may be a major factor, especially on the Great Lakes, it should primarily be considered in its structural aspects. Functionally only the fact that ice may reduce wave action at certain locations need be considered. 1.2 WAVE ACTION 1.21 GENERAL - Waves generated in deep water are usually of the type loiown as oscillatory waves, in which the particles of water making up the wave oscillate in a circular orbit about some mean position (see Figure A-3, Appendix A). An oscillatory wave is well defined if the wave length, L (see Appendix B List of Common Symbols), the horizontal distance between corresponding points on two successive crests; the wave height, H, the vertical distance to a crest from the preceding trough; the wave period, T, the time for two successive crests to pass a given point; and the depth, d, over which the wave moves, are known. The velocity, C, with which an oscillatory wave pro¬ gresses is related to period and length by L = CT (1) and to the depth and length by o' = tanh i-JLA (2) 2 TT L where g is the acceleration of gravity. As the water depth becomes large relative to the wave length, the hyperbolic tangent function (tanh ^ tt d ) May 1961 Part I Chapter 1 approaches unity, the wave velocity bt comes independent of depth, and C? - (2a) 2tt This condition (i.e. where the depth is great enough so that wave character¬ istics are independent of depth) is termed "deep water"; deep water conditions are generally indicated by the subscript "o", as Lq and Cq. Por deep water conditions, since Lq = CqT, then C 5.12 T (where T is in seconds and O ^TT ^ Cq is in feet per second) (2b) and 2 Lq = 5.12 T (where the deep water wave length Lo is in feet, and T is in seconds) (2c) Although "deep water" actually occurs only at an infinite depth, the tanh approaches unity closely at much smaller ratios of depth to wave length - called relative depth C-^). For a relative depth of 0.5 (that is, when the depth is one-half the wave length) the tanh is 0.9Q63, and this relative depth has by general usage been arbitrarily accepted as "deep water”. (However, for certain work requiring less precision, the deep water limit may be taken at some other relative depth. For example, the error in velocity computations is only 8% if a relative depth of 0.25 is considered "deep water"). When the water depth becomes quite shallow, the hyperbolic function tanh 2^ approaches and the wave velocity becomes: 2 C = gd (2d) Water of such a depth that equation 2d applies is termed "shallow water". The limiting depth for "shallow water" has by general usage been accepted as l/25th the wave length. Water having a depth between 1/2 and l/25th the wave length is termed "transitional". The total energv per unit crest length in one oscillatory v^fave is given by 2 E = 4- pgH^ L (1 - M \ ) ^8 (3) where p = w/g and is the mass density of water (saltwater = 2,0 lbs. sec^/ft^ or 2.0 slugs /ft^; freshwater = 1.96 lbs. sec^/ft*^ or 1.96 slugs/ft^) and M is an energy coefficient defined as Tr2 2 tanh (2TTd/L) May 1961 2 Part I Chapter 1 It should be recognized that the total energy in the wave is not forward- moving energy. Part of the energy is represented by the oscillatory motion of the water in the wave form and is referred to as the kinetic energy, E|q , in the wave form; this kinetic energy in effect remains on location without ad¬ vancing with the wave train. The remaining energy is represented by the fact that water has been moved out of the wave trough and appears above the mean water level as the wave crest. The latter energy is referred to as the poten¬ tial energy, Ep, in the wave form; this energy, in effect, moves forward with the wave form and generally expends itself on the shore. The relation holds that Et = Ek + E, (3a) The value of the potential energy, Ep, which is continually moving for¬ ward to the shore can be computed from the relationship, = n E. (4) where n = 1/2 1 + 4tt d/L s inh ) L (5) In water deeper than one-half the wave length, sinh (4Trd/L) becomes very large and n becomes approximately 1/2. Thus, in a wave train in deep water, one-half the energy, Et, appears as potential energy, Ep, and one-half as kinetic energy, Ej^. In a continuous wave train, i.e., one established from source to shore, the Ep remains essentially constant from deep-water to the breaker line, assuming no refraction or bottom friction loss. The E]^ value, however, gradually decreases from being equal to Ep in deep-water to being a rather insignificant quantity by the time the water depth is only l/25th of the wave length. Consideration of the above relationships shows that in effect, the kinetic energy, E^, is the mechanism which serves to transfer the potential energy Ep, to the shore. The mechanism requires a large value of Ek (equal to Ep) in deep water and less and less Ek as the water depth decreases. A wave train generated in the open ocean and moving toward the shore will in many cases have to travel through a region of calms to reach the shore. As the wave train intrudes into the calm area, it must use part of the forward- moving energy, Ep, to put the water into oscillation (as Ek) to transmit for¬ ward this potential energy, Ep, of the wave train. Thus the rate at which the Ep is transmitted is temporarily slowed until the required energy has been diverted into the calm waters as kinetic energy, Ek, thereby enabling the wave train to move forward. This results in waves disappearing from the front of the advancing wave train in such a manner that the wave group advances at a 2 a May 1961 Part I Chapter 1 slower velocity than the rate of advance of the wave crests within the wave train. This group velocity, Cg, is related to the individual wave crest velocity, C, by the relationship C = n C (5a) g where n has the value shown in equation 5, above. A study of the Cg and C relationships shows that in deep water the group velocity, Cg, is 1/2 the wave velocity, C. Also that in a depth of water l/25th of the wave length the value of Cg approaches C. An important point in this regard is to recognize that the C and Cg relationship is particularly useful while a wave train is advancing to establish itself in calm water, a use needed in predicting characteristics and arrival times of waves from distant storms. On the other hand, the energy relationship of equations 3, 4, and 5 are particularly useful when considering wave trains that have reached an equilibrium condition. For convenience, the coastal water over which a selected wave approaches the shore can be considered as divided into three zones, as follows; Deep water = water having a depth of more than 1/2 the wave length. Transitional water = water having a depth of from 1/2 to l/25th of the wave length. Shallow water = water having a depth of less than 1/25th of the wave length. 1.22 GROltfUi AND DECAY OF WI>JD WAVES - In an area of water over which the wind is blowing and generating waves, known as a fetch or fetch area, the growth of the waves is governed by at least three factors: The speed of the wind (U); the length of time the wind has been blowing, or duration (t^); and the length of the fetch (F) in the direction the wind is blowing. In cases (such as some reservoirs and bays) where the fetch area is relatively narrow compared to its length, an effective fetch length can be computed by averaging the effect of various directional segments, as described by Saville^^*^^^ With enclosed bodies of water, or in other areas in which the wind is blowing off the land, consideration of differing frictional effects of land and water may be necessary, and indicated wind speeds adjusted for these effects. Studies by the Weather Bureau (Myersfl98)^ and Graham and Nunnfi^Q) ) indicate recom¬ mended adjustments in wind speeds as shown in the table following, although the adjustment factor may vary considerably depending on the frictional characteristics of the shoreline. May 1961 2b Part I Chapter 1 TABLE 0-A Wind-speed adjustment, nearshore Wind direction Location Ratio* Onshore 2-3 miles offshore 1.0 Onshore At coast 0.9 Off shore At coast 0.7 Off shore 10 miles offshore 1.0 * Ratio of 30-foot wind speed at location to open-water wind speed. In relatively shallow water areas, in addition to these factors, the depth of water in the fetch limits wave growth. Both average wave heights and periods increase in a fetch, from the fetch rear where the wind is first significant to the fetch front, except when maximum height and period for the wind and duration are reached (see Figure 7). Diagreinatically wave growth may be illustrated as in Figure 1, Assum¬ ing that a wind of constant speed springs up, at an early stage of wave development, wave heights and periods will change from fetch rear to fetch front roughly as shown by the dashed line in the lower part of Figure 1, When the wind starts blowing, small waves develop throughout the fetch and move forward in the direction of the wind, growing in size as they move forward. No waves enter the fetch at the rear so at this point the wave height and period must be zero. At this stage the wind has been blow¬ ing only long enough to generate maximum heights or periods given by the 2 c Moy 1961 ERRATA SHEET 9 Tech, Rpt, No, 4 of the Beach Erosion Board "Shore Protection Planning and Design" On Page 28 - Replace 4th paragraph with - The wave period may be obtained directly from Figure 7 or 7A using the value of the corresponding equivalent deep water significant wave height (HJ,), An example of this use of the curves is given below. On Page 28e - Replace first 8 lines at top of page with — With the value of HJ, = 18,4 feet, a wind speed U = 50 tilph and fetch length F = 300 miles, use Figure 7 to determine (T* ) the equivalent deep water significant wave period. In this particular case, the limiting condition will occur, with U = 50 mph and Hjj = 18,4 feet (rather than with P = 300 miles : and HJ) = 18,4 feet). Thus 7 50 dAo = 50/5.12 (10,4)^ = = 0.09025 From Table D-1 H/HJj = 0.9419 H = 18.4 X 0,9419 = 17,3 feet, significant height at d_ = 50 feet for F = 300 miles, U = 50 mph and f/m = 10.6, Part I Chapter 1 horizontal dashed line in the figure. The waves are limited in height by the diiration of the wind. Any fetch length greater than that from 0 to Fm in Figure 1 will not be accompanied by greater wave heights or periods. The distance from 0 to Fm is known as the minimum fetch length Fm correspond¬ ing to the least duration of wind - called the minimum duration tj^ which will give rise to wave conditions illustrated by the dashed line. A later stage of wave development is illustrated by the solid line in Figure 1. The wind has been blowing long enough to generate maximum heights or periods given by the intersection of the solid curve with the ordinate representing the fetch front. After this stage is reached, an increase in duration of the wind gives no corresponding increase in wave height or period. Waves start from zero at the fetch rear, increase under the influence of the wind to the fetch front, then leave the generating area. In other words, wave growth is limited by the available fetch. In such a case the least length of time it takes a given wind blowing over a definite fetch length to cause such a steady state condition is known as the minimum duration (tj^). The fetch length involved, in this case the total available fetch, is still known as the minimum fetch corresponding to the minimum duration. Fetch / Front WAVE DEVELOPMENT WITHIN A FETCH FIGURE I 3 Part I Chapter 1 It should be particularly noted that for the early stage shown the measured fetch F is longer than the minimum fetch Fjj^ corresponding to t^ as given above. In the late stage, the measured d-uration may be longer than that minimum duration corresponding to the full measured fetch length F. The distributions shown in Figure 1 are actually simplified distributions, since a spectrum of wave heights and periods is generated in a fetch. These simplified distributions refer to what are known as significant waves , a statistical term which is used to describe the average of the heights and periods of the highest one-third of the waves in a group. After leaving a fetch, waves travel to a point some distance away - a coast for example - with speeds proportional to their periods (equations 2, 4 and 5). In this decay distance D, the longer period waves move faster than, and consequently arrive at the end of the decay distance before, those with shorter periods. An observer at the end of the decay distance sees a group of wave whose significant period T^j, is longer than the significant period T^ at the fetch front. Therefore, the significant period of waves seems to increase after they leave the fetch. The heights of waves actually decrease after leaving a fetch, and at the end of the decay distance, the observed significant wave height Hp will be smaller than Hp, the significant wave height at the front of the fetch. The time for the group of waves to travel over the decay distance is obtained approximately by dividing the decay distance by the deep water group velocity of waves with a period Tj^, and is known as the travel time /t --D. D \ \ ° . Cg 9 X 0 / 4 tt/ Most coastal areas of the United States are so situated that the major portion of the waves reaching them are generated in deep water, that is, in water deep enough so as to have no effect on wave generation. .In many of these areas, notably on the Pacific, South Atlantic and Great Lakes coasts, wave characteristics may be determined by first analyzing meteorological data to find offshore - deep water - wave conditions. Then, with refraction analysis procedures, the shallow water and shore line wave characteristics for these deep water waves may be found. In other areas, in particular along the North Atlantic coast, where the hydrography is complex, refraction procedure results are frequently extremely difficult to interpret, and the conversion of deep water wave data to shallow water and shore line data thus becomes laborious and sometimes inaccurate. Along the Gulf Coast and in many inland lakes, the generation of waves by wind is appreciably affected by the shallowness of the water. In addition the nature and extent of transitional and shallow water regions complicates ordinary refraction analysis by Introducing a bottom friction factor. 1.23 WAVE FORECASTING FOR DF.FP WATER AREAS - Wave forecasting procediires may be used to translate the meteorological data into wave data. Actually for both functional planning and structiaral design the term "hindcasting” might better be applied since it is historical wave data which is used. 4 Part I Chapter 1 However, the techniques are the same for both hindcasting and forecasting, and the latter term is usually used. To make a forecast it is necessary to delineate a fetch or generating area, to measure its length, and the decay distance, if any, and to know the wind speed and wind duration in the fetch. These determinations may be made in many ways depending on the forecasting area, and the type of meteorological data available. For relatively restricted bodies of water such as lakes, the fetch length is often the distance from the forecasting point to the opposite shore measured along the wind direction. In these cases there is no decay distance, and it is most often necessary to use ob¬ servational data to determine wind speeds and durations. When performing forecasts for shores of oceans or other large bodies of water, the most common form of meteorological data used for forecasting is the so-called synoptic surface weather chart. (’’Synoptic” refers to the fact that the charts are drawn by analysis of many individual items of data). On these charts are drawn lines of equal atmospheric pressure, called isobars. Pressures are recorded in millibars, a millibar being 1,000 dynes per square cm. One thousand millibars - a bar - equals 29.53 inches of mercury and is 98.7^ of normal atmospheric pressure. A simplified siirface chart for the Pacific Ocean is shown on Figure 11, which is drawn for 27 October, 1950, at 0030Z (0030 Greenwich Mean Time). Note in particular the area labelled L in the right center of the chart, and the area labelled H in the lower left corner of the chart. These areas designate low and high pressure areas respectively; the pressures increase moving out from L (isobars 972, 975, etc.) and decrease moving out from H (isobars 1026, 1023, etc.) Scattered about the chart are small arrow shafts with a varying number of feathers. The direction of a shaft shows the direction of the wind at that station and time, and the feathers are a code showing the wind force according to the following scheme: Each half feather represents one unit on the Beaufort Scale of wind force (Table D-3, Appendix D). Thus near the point 35° N latitude, 135° W longitude there are three such arrows, two of which show a Beaufort scale of 8(39-46 mph or 34-40 knots) and the third a Beaufort Scale of 7(32-38 mph or 28-33 knots). On an actual chart, much more meteorological data than wind speed and direction are shown for each station. This is accomplished by the use of coded symbols, letters, and numbers placed at definite points in relation to the station dot. A station model, showing the amount of informa¬ tion it is possible to report on a chart (not all is always reported or plotted) is shown in Figure 2. 1.231 Measurements for Ocean Areas (a) - Wind Speed and Direction - Certain relationships exist between wind and isobaric configurations which permit the use of these charts in wave forecasting for areas bordering oceans. If a line is drawn perpendicular to a set of Isobars, pressures at different points on the line would differ. 5 Part I Chapter 1 Feothtrs showing force of wind in Beaufort Scale.-_ (19 to 24 miles per hour.) Arrow shaft showing dir¬ ection of wind (Blowing from northwest.) - Figures showing temper¬ ature In degrees Fahrenheit. . Symbol showing amount of sky covered by clouds. (Completly covered.)_ Cods figure showing visibility. (1^ miles or more, but less than 22 miles.) Symbol showing present state of weother.* (Continuous light snow in flakes ) _ Figures showing dew point in degrees Fohrenheit. _ Symbol showing type of low cloud.'**' (Fractocumulus.) Height of lower clouds, except when lower clouds are only fragments below a layer of clouds whose base is below 8,200 feet'*' (328 to 655 feet.) NOTE The letter symbols for eoch weather element ore shown above. "^Omitted when doto ore not observed or ore not recorded. Symbol showing type of middle cloud.*(Altostratus.) Figures showing barometric pressure at seo level. Initial 9 or 10 for "hundreds' of millibars, ond also deci¬ mal point, omitted (995.3 millibars.) Figures showing net amount of barometric change In post 3 hours, (in tenths of millibars.) Symbol showing barometric tendency in past 3 hours. (Falling or steady, then rising, or rising, then rising more quickly.)**" Plus or minus sign showing whether pressure Is high¬ er or lower than three hours ago. Code figure showing time precipitation began or ended.**" (Began three to four hours ago.) Past weather during six hours preceding observation (Rain) Figures showing amount of precipitation in last 6 hours '*' (In hundredths of an inch.) Courfesy United States Weather Bureau STATION MODEL FIGURE 2 (Stewart, 1945) since each isobar crossed represents a different pressiare. The rate of this pressure change toward lower pressures is called the gradient. If no other factors were present, this gradient would cause wind to blow normal to the isobaric pattern. However, other forces are present, in particular the Coriolis force (caused by the rotation of the Earth on its axis, which acts normal to the wind direction), centrifugal force, and frictional force. Without friction, equilibrium between these forces is attained when wind blows .along the isobars. If the isobars are straight, only Coriolis force and the force due to the pressiire gradient are acting, and the resultant computed wind is called the geostrophic wind. If the isobars are curved, centrifugal forces also act; the resultant wind is called the gradient wind. Frictional forces, however, cause the wind to blow, not parallel to the isobars, but at a slant across them toward the lower pressure region. Near the surface, the angle between the gradient or geostrophic winds and the actual wind is 10 to 15 degrees over sea areas and about 40 degrees over land areas. The spiraling wind pattern thus created is called a cyclonic or anticycIonic pattern depending on whether the wind is blowing about a low pressure area as center, or a high 6 Part I Chapter 1 pressure area as center. In the northern hemisphere, cyclonic winds (about a low) rotate counterclockwise. The opposite is true in the southern hemisphere. Friction causes actual wind speeds to be lower than computed gradient or geostrophic wind speeds. Two graphs are used to facilitate the computation of actual wind speed. The first is used to calculate a geostrophic wind speed assuming the isobaric configuration under consideration is straight. The second is used to find the ratio of the actual to the geostrophic wind speed taking accoTint of both isobar curvature (to convert geostrophic to gradient wind) and friction. To utilize the first of these (Figure 3) it is necessary to have a measure of the pressvire gradient over the area in question. Since most charts are drawn with either a 3-millibar or a 5-millibar spacing, for either of these standard spacings the geographical distance between isobars is adequate as such a measure. To use Figure 3, the distance between isobars on a chart is measured in degrees of latitude (an average spacing over a fetch is ordinarily used), and the latitude position of the fetch is determined. Using the spacing as ordinate and location as abscissa, the plotted or interpolated slant line at the intersection of these two values gives the geostrophic wind speed. For example, on Figure 11, a chart with 3-millibar isobar spacing, the average isobar spacing over F2, located at 37 degrees N latitude, is 0.70 degrees of latitude. Using the scales on the bottom and right of Figure 3, a geostrophic wind of 67 knots is found. The second graph (Figure 5) relates the ratio speed” geostrophic wind speed to the difference between the sea and air temperatiire for various radii of isobar curvature measured in degrees latitude. The difference of sea and air temperature here acts as a measure of the friction effect, and the isobaric curvature as a measure of the relationship between gradient and geostrophic wind. Thus, on this graph, two corrections are applied. As an example of its use, reference is made again to fetch F2, Figure 11, in which the average isobar radius of curvature was found to be about 10 degrees of latitude (mild cyclonic curvature). There was found to be no difference of sea-air temperatures, and the curves of Figure 5 give 0.59 as the ratio of actual surface to geostrophic wind speed. The actual wind speed in this fetch, found by multiplying the 67-knot geostrophic wind by the 0.59 ratio above was about 40 knots. Note that for this particular storm the sea temperature used (65°) was that given on Figure 4, from the World Atlas of Sea Surface Temjjeratures (130)^ Sea temperatures are also given in the Climatic Atlas(137), The air temperature was taken from the station reports in the area (these reports are not plotted). The procedures illustrated above are those ordinarily used to deter¬ mine wind speeds. However, if the speeds so derived differ from the average value of ship reported wind speed in the fetch area by more than one * Numbers shown in this manner refer to references listed in Appendix C. 7 5mb ISOBAR SPACING - DEGREES LATITUDE GEOSTROPHIC WIND SCALE V=_!_^ V An FOR Ap=5mb8 3mb An: Otgrees Lotitud* P>IOI3.3mb T » IO*C ^ = 1.2 X 10"3 qm/cm? (Br*ttchn«i(l«r, 1952) FIGURE 3 February 1957 8 3mb ISOBAR SPACING - DEGREES LATITUDE FIGURE 4 9 surface wind scale FIGURE 5 (Bretschneider, 1952) 10 Part I Chapter 1 Beaufort Scale division the forecaster must judge for himself which of the two values is more likely to represent actual conditions. For example, if there are many ship reports in the immediate fetch area, all in approximate agreement the wind speeds from these may be more reliable than that derived from the isobaric pattern. (b) Delineating a Fetch - The problem of determining on a surface chart that area which may properly be called a generating area when fore¬ casting for a specific locality, is quite complex. The following factors must be given consideration. (1) Variability of wave direction: It is possible, with a fair degree of assurance, to obtain the general direction of the wind for a given ocean area. However, it has been shown^^' that waves in a generating area move not only in the direction of the wind, but at various other angles to the wind. For a fetch with relatively straight Isobars, those waves moving at 30° to the average wind direction will have heights of 80 to 90 percent of the waves moving in the direction of the average wind. When Isobars are curved, it may be expected that this 80 to 90 percent semi-sector will be larger than 30°. It has been common practice to account for this variability by assming that undiminished waves may move at 30° to the wind direction for a relatively straight isobaric pattern, and at 45° for a curved isobaric pattern.(132) (2) Isobar spreading: The greater the distance between Isobars, the smaller is the wind speed. When the wind speed drops below a certain value, and the decay distance is relatively large, waves generated by this wind will not be significant at the point for which forecasts are being made. The commonly adopted limiting criterion (132) is that for decay distances of 500 miles or more, wind speeds of 20 knots or less may be ignored. Noticeable spread¬ ing of isobars is often a sign useful in locating both a fetch front and rear. (3) Frontal lines: The continually changing isobar patterns represent constantly moving air masses, the characteristics of any two of which usually differ though they may be contiguous. Boundary areas of different air masses are plotted on surface charts, four different symbols being used depending on the type of air mass movement. The symbol A A , called a cold front, is used to indicate the line on the earth's surface along which a mass of cold air is advancing into a mass of warm air. The symbol .AA., called a warm front, is used to indicate the line along which a mass of warm air is advancing into a mass of cold air. Both symbols on a surface chart represent only the line of intersection of the surface of separation of the two air masses with the earth's surface. The actual surface of separation,called a frontal surface, will be one in which the cold air forms a tongue intruding at low levels into the warm air mass. The symbol AAJLJk. , called an occluded front, represents a line along which a cold front having overtaken a warm front and lifted Part I Chapter 1 the warm air from contact with the ground, meets the cold air mass which had been ahead of the now lifted warm air mass. The symbol is called a stationary front, and represents a front not moving at the time of the chart. Pictorial representations of these conditions are shown in Figure 6. It may be seen from the typical isobaric configura¬ tions also shown on Figure 6 that at a front line the direction of the isobars often abruptly changes. Since wind direction depends on isobar direction, at a front wind direction often changes abruptly, often enough to limit a fetch at or near a front. (4) Accuracy: One of the most important considerations in locating and limiting a fetch is the accuracy with which charts are drawn. Especially over ocean areas, the information from which a meteorologist constructs, a synoptic chart is usually very limited. Few ships sail outside normal shipping lanes, thus presenting the analyst with wide expanses of ocean for which little data exist. Many isobaric configurations on synoptic charts represent estimates, limited in accuracy by the extent of detailed data available. (5) Land Masses: It should be noted that land masses often limit the water area available for a fetch. This is true for lakes, bays and land-locked or partly land-locked sea areas. With these considerations in mind certain generalized rules may be adopted to aid in delineating fetch areas. (l) Depending on the relative locations of the point for which forecasts are to be made and a storm area, waves which are moving at angles of from 15° to 50° with the isobar directions will be significant. For simplicity the somewhat inaccurate rule (see follow¬ ing rule 4) that for the general isobaric pattern - without fronts or significant isobar spreading - directions of approach of 30° with the isobars for relatively straight isobars, and 45° for curved isobars are allowed. To locate either a fetch front or rear, therefore, a straight edge may be rotated about the forecasting area until it cuts the various isobars of a particular possible fetch at about 30° or at about 45° depending on isobar curvature (See Figure lO). (2) If isobars spread noticeably in either the front or rear of a possible fetch area, there is a good possibility that the fetch front or rear will lie in this area of spreading. With decay distances of more than 500 miles, areas of 20-knot winds can be considered to be fetch boundaries of an area with greater wind speeds. (See Figure 10, isobars 990 and 993). (3) Wind directions often change so abruptly at front lines that the front line itself may limit the fetch. (4) Accuracy of the chart limits accuracy of the forecast. 12 C old oir—»■ ^Front COLD FRONT Front Cold air WARM FRONT Cold front Occluded front Warm front Worm air Cold air-► i _——Front OCCLUDED FRONT Cold air Warm air Worm air Cold air STATIONARY FRONTS ISOBARIC PATTERNS AND CONDITIONS FOR FRONTAL LINES FIGURE 6 13 Part I Chapter 1 (c) Decay - Waves, after leaving a generating area, will follow generally a great circle path to a coast. However, for most of the ordinary forecasting situations, adequate accuracy is obtained by assuming wave travel in a straight line on the synoptic chart (see rule 1 above). Decay distances may, therefore, be found by measuring the straight line distance between the front of a fetch and a coastal point under consideration. 1*232 Measurements On Lakes. Bays. Etc. (a) Wind Speed ; The relationships mentioned previously for the determination of wind speeds and directions from isobaric patterns, in general, apply to land areas as well as sea areas. However, the friction factor which causes winds to spiral, crossing isobars, and to be smaller than geostrophlc or gradient winds is very much more variable over land areas. When a fetch under consideration is in close proximity to land, this vari¬ ability will manifest itself in altering anticipated wind directions and velocities. In enclosed or semi-enclosed bodies of water, such as lakes, bays, etc., rather than analyzing isobaric patterns to deduce wind speeds and directions, these should be taken wherever possible from actual station reports. This procedure is not ordinarily resorted to when analyzing fetches over ocean areas, for two reasons: Most often not enough ship reports are located in the fetch area for reliability to be assumed over the entire fetch; and in any case, the directions and magnitudes of reported winds re¬ present the recorded or average values at the particular time for which the chart is drawn. These values are sensitive to possibly transitory, local variations. However, on lakes, especially the Great Lakes, there are usually enough station reports so that objections to their use are satisfied. Often, especially over relatively small or well defined fetch areas, it is not convenient or even possible to utilize surface charts for fore¬ casting parajneter determinations. Estimates of surface wind speeds and durations must still be made, even though available data are limited. Where wind records exist in or near the fetch area, these may be utilized, in which case the accuracy of the forecast will depend on both the completeness of these records, and the extent of fetch over which they are to be applied. Where wind records do not exist, no duration estimates can be made, though local wind speed reports may still be utilized to perform forecasts assuming unlimited durations (i.e. wave growth limited by the available fetch). It should be recognized that wave characteristics deduced by this technique can be no more than qualitative. (b) Fetches and Decay Distances :- All that has been said previously about locating and limiting fetches, and determining decay distances, applies to water areas contiguous with land as well as those in the open ocean. Certain points may be emphasized, however. Since wind directions are determined from actual station reports, it is permissible in general to use the 30° semi-sector rule, where necessary, to limit a possible fetch. That is, waves may be assumed to vary in direction by as much as 30° with direction and still be undiminished in size. It may be expected, though, that most fetches will be limited either at the front or at the rear by a land mass. It should also be kept in mind that decay distances will be most 14 Part I Chapter 1 often, relatively small or non-existent. 1,233 Forecasting Techniques for Deep Water Areas (BretSchneider revised , Sverdrup-Munk Method) - Figures 7 and 7A are plots of the forecasting curves presently in use. With these, one may determine the significant wave height (Up) and significant wave period (Tp) at the front of a fetch, knowing the wind speed in the fetch, and either the duration of the wind or the fetch length. In using the plots, the actual wind velocity (U), the fetch length (P), and the estimated duration (t) of the wind, when a fetch first appears on a chart, are tabulated, (When a fetch first appears on a chart the dura¬ tion of the wind at the time of the chart may be taken as one-half the time (Z) between charts, ) Figure 7 or 7A is then entered with the known value of U, on the left if U is in knots, or on the right if in statute miles per hour. This **U'' line is then followed across to its intersection with the fetch length (P) line, or the duration (t) line, whichever comes first from the left side of the graph. For exan^^le, with a wind of 35 knots or 40 mph, a duration of 10 hours comes before a fetch length of 200 nautical miles. Similarly an P of 80 nautical miles comes before a t of 10 hours. At this point Hp and Tp, the wave height and period at the head of the fetch, may be read off and tabulated. For the case given above with U = 35 knots, t » 10 hours, and P » 200, Hp » 14,7 feet, Tp « 9,6 seconds, tg| of course equals 10 hours, and ^m » 115 nautical miles. If P had been 80 nautical miles with t still 10 hours, the heights, periods, minimum duration and fetch would be Hp = 12,7 feet, Tp « 8,7 seconds, ^m * ^ ,6 hours and Pm = 80 nautical miles. The wave pattern in the first case where the duration is limiting, would correspond to the dashed line in Figure 1, that of the second case, where, the fetch is limiting, to the solid line in Figure 1. For the same fetch on a later chart drawn for a time Z after the first chart, U, F, and t are again tabulated. Using the subscript ’'2'' to refer to forecasting parameters of this second chart, and subscript "I” to refer to those of the first chart, if U 2 * U^ the above procedures should be followed using either t 2 ■ tj^j ♦ Z or P 2 , If» however, U 2 1 ^ Uj certain additional assumptions must be made before using the forecasting curves. It has been suggested (14), that for purposes of forecasting, a change in wind speed from Ui to U 2 in a time Z between charts, may be assumed to take place instantaneously at a time Z/2, Waves due toU^may then be cal¬ culated by assuming that the first chart’s minimum duration time has been lengthened by an amount Z/2 or that its minimum fetch has been changed by A F/2, where s F represents the change in fetch length between weather charts. Since at the assumed abrupt change in wind speed, the energy im¬ parted to the waves by Ui, with a minimum duration tj^j ♦ Z/2 for a minimum fetch Fgjj + fi F/2, does not change, U 2 will begin imparting energy to waves which already contain energy due to Uj, Plotted on Figures 7 and 7A are dotted lines of constant (Hp^ Tp^) which may be thought of as lines of constant wave energy, (To a first May 1961 15 Part I Chapter 1 approxifflation. deep water wave energy is given by E „ w H^L . 5.12 pg(HT)2 8 8 compare with equation 3), If the energy had been imparted to the waves by U 2 acting alone, these waves would be of length and height given on Figure 7 or 7A by the intersection of the U 2 ordinate with the constant energy line (plotted or interpolated) corresponding to energy imparted by Ui with a minimum duration of tj^j^ + Z/2 or a minimum fetch Pj^l ”*■ AF/2, By increasing the minimum duration at this point by an amount Z/2 or by changing the mini¬ mum fetch by an amount aF/ 2 , wave conditions under U 2 at the time of the second chart may be approximated. For example, with Uj = 35 knots, t^i = 10 hours, Fj^i = 115 nautical miles, t^i -f Z/2 = 13 hours; an interpolated (by eye) dotted line would be followed up to the U 2 * 40-knot line where the duration * 8.8 hours. To this value Z/2 or 3 hours is added and Hp 2 » Tp 2 , t^2 ^m2 read off as Hp 2 » 19.4 feet, Tp 2 = 11.0 seconds, ti „2 * 11.8 hours, and Fj „2 * 155 nauti¬ cal miles. If the measured fetch F 2 had been less than 155 nautical miles. this length of fetch would limit the growth of waves. Though the preceding discussion would indicate that AF should be calculated, in practice this need not be done; the results gotten through calculation of aF would be obtained by reading off wave heights at the intersection of U 2 and F 2 if F 2 is limiting. Therefore, if P 2 had equalled 150 nautical miles in this case, (less thah 155 miles and therefore limiting) at the intersection of U 2 = 40 knots, and P 2 ** 150 nautical miles, Hp 2 * 19.2 feet, Tp 2 = 11.0 seconds, tjj 2 “ 11.6 hours, and Fni 2 = 150 nautical miles. Note this important distinc¬ tion, tj, Pj^ and F 2 are measured or determined from the synoptic chart, t„jj, tm 2 t Pflil 2 ind Pjg 2 calculated by use of Figure 7, Some of the measured and calculated values will be the same, but not all of them. If the wind velocity U 2 is less than Uj, essentially the same procedures arc followed though there are some differences. Prom the intersection of Uj and -f Z/2 a constant energy line is followed to its intersection, if there is one, with either U 2 or P 2 whichever comes first from the left side of the figure. If comes first Z/2 is <»dded to the duration at this point, and the U 2 ordinate is followed to either this new duration or to F 2 which¬ ever is first from the left side of the chart. (Compare with the preceding paragraph). At this point, Hp 2 , Tp 2 , t^2» ^m2 inclusive are read off. If the constant energy line had intersected F 2 before U 2 , it is only neces¬ sary to drop down along the F 2 abscissa to its intersection with U 2 , and at this ^oint read Hp 2 , Tp 2 , t m?. and Fni2* (This lipocedure could be used for many cases in which U 2 is greater than Ui), For examples of this procedure see notes 8 and 11 at the end of the typical forecast given later. The major differences in technique which must be used when U 2 is less than Uj, occurs when the constant energy from the intersection of Uj and Vl ♦ z /2 does not intersect either U 2 or P 2 , Forecasting theory used here predicts that the waves due to any wind blowing over an unlimited fetch for an unlimited duration will eventually attain a limiting height and period May 1961 16 0001 006009 OOZ 009 OOS 00¥ OOC 002 0^ jnoH JSd S8|iw ui paads pui/v\ SiOux uj paads puiM I6 a May I96I Fetch Length in Nouticol Miles Re* FIGURE 7. DEEP WATER WAVE FORECASTING CURVES ASA FUNCTION OF WIND SPEED, FETCH LENGTH, AND WIND DURATION Fetch Length in Statute Miles 150 200 _ 300 400 500 600 700 600 1.000 _ 1,500 2000 2500 ^000 4000 SiOOO 6^000 8000 10.000 jnoH J3d ui paads pui* May 1961 16 b FIGURE 7-A.DEEP WATER WAVE FORECASTING CURVES AS A FUNCTION OF WIND SPEED, FETCH LENGTH, AND WIND DURATION (for Fetches 100 to > 1,000 miles ) Part I Chapter 1 and growth will not increase. On Figure 7A the lower limit of this state is delineated by the line labelled ^maximum conditions”. To the right of this line, no wave growth takes place. Generation of Deep Water Waves by Variable Wind Speed and Direction and Moving Storms - If the wind field is not too irregular and the move- of the storm it fairly slow, then wave forecasting relationships given herein (either the Sverdrup*44unk*Bretschneider significant wave method or the Pierson-Neumann-Janies wave spectrum method) are satisfactory. For moving wind systems and changing fetches methods and techniques used are given by Kaplan However, when the variables are ill-defined, the graphical method proposed by Wilson^^ll) must be used, the method of which also applies for wi^s of constant speed and direction. The forecasting curves of Wilson ^211) called an HtFT diagram. Figures 7B and 7C are the HtFT diagrams for forecasting waves by this method. These diagrams are not the original diagrams used by Wilson but are based on the more recent revision of the wave forecasting relationships by Bretschneider (154), and are therefore consistent with the forecasting curves given in Figures 7 and 7A. Figures7D and 7E, taken from reference 211, are respectively a typical pro¬ file of a variable wind field and a typical example in the application of the graphical technique. In explaining Figure 7E the following is quoted from reference (211): **Wind of Variable Velocity in a Variably Moving Wind System o f Finite Fetch "Dispensing now with the restriction of a uniform wind velocity, U, but retaining the concept of uniformity along closed contours of a space-tiiae wind field, it becomes possible to represent a wind system that has both variable wind velocity and variable speed of forward (or rearward) progression by a wind-field of closed contour lines whose intervals apart repre¬ sent equal increments of wind velocity U. "Figure 7fi shows such a wind-field with contours of wind velocity at 5-knot intervals from 20 to 40 knots. Superimposed thereon at an arbitrarily selected point 0 in space and time is the HtFT diagram with H(F), F(t(i) and Tftd) curves drawn in for the same 5-knot intervals of wind velocity U from 20 to 40 knots. "The problem now is one of determining the history of the height and period growth of the waves originating at the point 0 . "In relation to the wind-field the origin 0 is seen to be at a point where the wind velocity would be of the order of 21 knots. Waves originating at 0 would be obliged to follow a space-time path somewhere along the belt of propagation lines forming the relationships F^ft^). It is clear that the actual May 1961 I6C Wind Velocity - U-( Knots ) FIGURE 7-B. Ht-FT - DIAGRAM FOR FORECASTING WIND-GENERATED WAVES (for Fetches up to 1000 miles) Moy I 1961 I6d Wind Velocity - U- ( Knots) Wind Velocity - U-( Knots ) H FIGURE 7-C. H t-FT- DIAGRAM FOR FORECASTING WIND - GENERATED WAVES (for Fetches up to 200 miles) May 1961 I6e Wind Velocity - U - ( Knots ) WindVelocity-U-( Knots) u. CO 0 > d > x» d o *d • g >» (0 +5 XJ bi) •H o C! O •H •H O U f-H © O 0 Cl^ 6 > xJ d (—4 d d rO •rj d CO •H -P U 0 s: d rH hD > rO •rH d 0 d •H d U d -P •H > d d xJ 4h o © o •H -P Ch d S •H U 0 d 0 -p W) d CO •H © >> CO bO CO UJ 1 1 • bO r n-(s»ou>() AijDOiaA pu!M S ? H- CM) IM6!9 h a^OM in O »o O — — C\J (sjnog) uoiiOJHQ O iuo0|^iu6|$ t U (T -P > d d o d • S xJ o o o d d (0 -P . 'S) d :d -P o .H X3 CO ® •H ^ d dS t3 d &( ® 0 6 fl d O •H 0 o e bD a • fciO ■£! Wilson 1955 May 1961 I6f Part I Chapter 1 path of the waves must initially be along a line intermediate between the propagation lines for U = 20 and U * 25 knots as far as a, the intersection point with the 25-knot wind-field contour. Along the path Oa the waves would be under the in¬ fluence of winds ranging from 21 to 25 knots so that, to all intents and purposes»0a can be regarded as the propagation line for U « 23 knots. •'Over the same interval of tine the growth in significant period of the waves will follow the line Ob (Figure 7E), equiva¬ lent to the curve Tyft^j) for U = 23 knots, ••Having arrived at a, the waves pass into the next incre¬ mental wind zone over which wind velocity rises from 25 to 30 knots. Their further space-time path from a to e must be at a rate (or group velocity) appropriate to the average wind of U ss 2l\ knots, but the propagation rate must start off from a at the same slope as the line Oa has at a. •To ensure that the group velocity shall remain the same at the transition, it becomes necessary to trace a line be at constant period and locate a point c intermediate between ^25^*d^ and T 3 Q(t^), The condition of constant period ensures constant wave group velocity since group velocity is directly proportional to wave period under deep water conditions. By drawing the abscissa cd, the point d is found intermediate between the curves P25^^d^ ^30^^d^* imaginary propaga¬ tion line P274^^d^» drawn through d would now have the same slope as the ciurve Oa at a. To find ae, therefore, it is only necessary to transcribe, as it were, a piece of the P27^^*d^ curve from d, parallel to itself, and add it to the curve Oa at a. By this means the point e is established, "In the same sense, by transcribing a portion of the curve T27^(td) from c, parallel to itself, and adding it to the curve Ob at b, the point f can then be located (via ef), marking the further growth in period of the waves, bf, under the influence of the 25 to 30-knot wind, ••This procedure may be followed consistently to trace the the actual space-time path of the waves, Oaekosw, through the wind-field and to give the history of the period growth of the waves, Obflptx, It will be noted that the same method applies in the zone of declining wind velocities as in the zone of increasing velocities. Thus the portion os of the wave propa¬ gation curve is drawn parallel to P32|(‘td^ at r, the wave group velocity at 0 and r being different in the declining 35 to 30- knot wind zone from that at h and k in the increasing 30 to 35- knot wind zone. I6g May 1961 Part I Chapter 1 **The graphical charting of the corresponding growth in significant height of the waves follows essentially the same procedure as described above. The curve Ob* follows the H 23 (P) isoline as far as b*, which is the intersection point with the ordinate drawn through a. Further increase in height of the waves in the next incremental wind zone (U s 25 to 30 knots) must continue at a rate appropriate to H 27 ^(F), start¬ ing, however, at the same height as at b*. Accordingly b*f*is drawn, parallel to H27|(P) st c*, to give the intersection point f* , with the ordinate drarwn from e on the propagation line. The final curve of significant wave height follows the line 0b*f*l*p* and tapers off to a maximum value which is main¬ tained to the end of the wind field. In the same way the curve of significant wave period, Obflptx, is found to taper off to a maximum value of wave period.*' The above quote from Wilson should be sufficient to understand the practical applications of the graphical technique, but if more details are required the reader is referred to the paper by Wilson^211)^ Experience and practice of coturse are necessary to perfect one*s techniques in the above method. The Effect of Fetch Width on Wave Generation - The effect of fetch width in limiting wave growth in a generating area has long been recog¬ nized but has generally been neglected since, for the generation of waves in the ocean, the vast majority of fetches have widths of the same order of magnitude as the lengths; in such cases the limiting effect of the fetch width has been considered as being very minor. However, in consider¬ ing wave generation in inland waters (lakes and reservoirs) the fetches are limited not by the (generally) large bounds of the meteorological disturb¬ ances (as in the ocean) but by the land forms surrounding the body of water; in these cases, fetches of rather great length in comparison to their width are frequently observed, and the width effect of the fetch may become quite important, resulting in the generation of waves materially lower than those that would be expected from the same generating conditions over more open waters. A method of determining the effect of fetch width on wave generation has been proposed by Saville^^^'^^ Figure 7F is reproduced from this report. For example, consider a channel 20 miles long and 5 miles wide, with a 50-mph wind blowing lengthwise. Compute W/P * 5/20 » 0,25. From Figiure 7F read Fg/F = 0,45, using wind effective over t 45° of the wind direction. May 1961 I6h 0» d 00 6 N d u> d lO d d ro d CV) d a> a> q: I > o CO t- ■o $ sz u a> o o o CC H O 2 LU _J I X I- Q O I- o H UJ liJ > I— o LiJ lJ_ U. UJ ^ O o < O) llJ o h- UJ cr < CD < o UJ cr ir o u. UJ QC I N UJ cr Z) CD H < tt: d/ ssauaAi|oa^^3 M^iSd 16 i May 1961 Part I Chapter 1 Compute Fg = 0,45 X 20 » 9 miles Using the forecasting relations for a fetch of 9 miles and wind speed of 50 mph determine significant wave height and period, H = 6.5 feet and T « 5.2 seconds. Part I Chapter 1 1,234 Decay Analysis for Deep Water Areas - Figures 8 and 9 are the curves used to find wave characteristics after the waves have left the fetch but are still travelling in deep water. With Figure 8 it is possible to compute .. « Decayed wave height Hp Decayed wave period "^D Fetch wave height Hp Fetch wave period Tp HPf Tp, F^ and D (the decay distance). An example of its use is shown on the figure. With Figure 9, it is possible to compute wave travel time between a fetch and a coast, knowing the decayed wave period Tp, and the decay distance D, This information enables the determination of wave times of arrival at the end of the decay distance. To illustrate the foregoing methods, essentially those of Bretschneider (14) the following short forecast is made (Table 1), Refinements for the analysis of moving fetches are given by Kaplan (67), Synoptic conditions are shown on Figures 10, 11 and 12, 1,235 Wave Height and Wave Period Variability - It was noted in section 1,22 that the wave height and period determined from the above forecasting or hindcasting procedures are significant waves, or the average values of the one-third highest waves. The significant waves might be considered more- or-less as conservative quantities around which are distributed the individual heights and periods comprising the groups of waves or the wave spectrum. It has been found by Putz(165) from statistical analysis of a number of 20-minute wave records that the mean wave height, the average of the highest 10 percent and the maximum probable heights are given essentially by the following equations: Hpj a 0,624 Hg (6) Hio = 1.29 Hs (6a) Hj^ax * 1.87 Hg (average of 25, 20-minute records) (6b) Similar relationships have also been verified by theory^l®^\ = 0,625 Hg (6c) Hio * 1.27 Hg (6d) Hjnax ® 1.78 Hg (derived for 300 waves) (6e) Insofar as the most probable maximum wave height is concerned, the duration of the storm or length of the wave record is important. That is, the most probable maximum wave is a function of the number of waves as well as the significant wave duri ng a p eriod of steady state and is given theoretically by = 0,707 H^vAoge^» where N is the number of waves dur¬ ing the steady state condition. For example if N = 300 consecutive waves then the most probable maximum wave will be given by H^ax = 1.99 Hs; if N » 1000, Hjaax *® 1.86 Hg, Figure 13 is reproduced from (106) and shows the relationships between the mean wave height^and the statistical distribution of the other waves. These relationships are not too much different from those one might obtain from theoretical deductions of Longuet - Higgins^lSS). When using Figure 13, the mean wave height of a group of waves should first be found from the significant height by use of equation 6, The diagonal line on the figure corresponding to Hj^ is May 1961 17 Part I Chapter 1 followed to its intersection with the vertical line labelled (say) 95 percent. The height which will not be exceeded by 95 percent of the waves in a group whose mean height is H is read on the vertical scale. In the example on the graph this height is 5 feet where is 2.5 feet, (i.e. 5 percent of the waves will be equal to or greater than 5 feet). Just as the wave height has variability or statistical distribution about the mean or significant wave height, the wave period also has a variability or statistical distribution about the mean. An empirical relationship for wave oeriod variability based on the analy¬ sis of wave records is given by Putz^^^^\ Based on a reanalysis of his data together with a great number of additional records a more general relationship is given by Bretschneider (189) This relationship combined with wave height variability relationships results in a joint distribution relationship for wave height and wave period. Table 0-B reproduced from (189) gives the prob¬ ability that a wave will have a particular height and period once the mean wave height and period are obtained. For all practical purposes the mean wave period can be taken equal to the significant period and the mean height equal to 0.625 times the significant height. For example if a significant height and period of 16 feet^ and 10 seconds respectively ar_e obtained from wave forecastin^g curves, then K = 16 x 0.625 = 10 feet. Since T is almost equal to T 1/3 then T = 10 seconds may be used. From Table 0-B it will be seen that for HAl = 1.6 about 1 wave (0.90) per 1000 waves will have a height between 14 and 16 feet and period between 16 and 18 seconds; that about 1 wave (1.02) per 1000 waves will have a height between 24 and 26 feet and period between 12 and 14 seconds; that about 35 waves (34.97) will have a height between 10 and 12 feet and period between 10 and 12 seconds. IA.BLE 0“B JOINT DI-S miBQ TT ON OF H AND T FOR ZERO CORREIA HON Number of waves per 1,000 consecutive waves for various ranges in height and period. Range in Relative Height R A NOE IN RE L A T I V E PERI 0 D ^/t H/h 0-.2 .2-.U .6-.8 .8-1.0 1.0-1.2 1.2-1.a i.a-1.6 1.6-1.0 1.0-2.0 0-2.0 tccuBuIatlve 0- .2 .03 .50 2.05 a.86 7.66 8,09 5.31 1.92 .3a .03 30.81 30.61 .2- .U .10 l.Ul 5.61 13.78 21.76 23.92 15.05 5.aa .90 .07 88.32 U9.13 .U- .6 .Ui 2.06 8.5I1 20.23 31.95 33.65 22.10 7.99 i.ua .11 120.21 2a7.3a .6- .6 .16 2.U0 9.91 23. U8 37.08 39.06 25.65 9.27 1.67 .12 las.eo 396.la .8-1.0 .16 2.U0 9.92 23.51 37.13 39.11 25.69 9.26 1.67 .12 ia8.99 5a5.13 1.0-1.2 .15 2.Ill 8.87 21.02 33.19 3a.97 22.96 0.30 i.a9 .11 133.20 67B.33 I.2-I.I1 .12 I.7I1 7.21 17.07 26.96 26.ao 16.65 6.7a 1.21 .09 108.19 786.52 l.b-1.6 .09 1.30 5.37 12.72 20.09 21.16 13.90 5.02 .90 .07 60.62 867. Hi 1.6-1.6 .06 .90 3.72 8.82 13.93 ia.67 9.6a 3.ae .63 .05 55.90 923.06 i.e-2.0 .03 .U0 1.99 U.72 7.a5 7.05 5.15 1.86 .33 .03 29.09 952.93 2.0-2.2 .03 .U2 1.72 U.09 6.U5 6.60 a.a? 1.61 .29 .02 25.90 978.63 2.2-2.(i .01 .16 .76 1.80 2.8a 2,99 1.97 .71 .13 .01 u.ao 990.23 2.U-2.6 .01 .09 .39 .93 i.a? 1.55 1.02 .37 .07 5.90 996.13 2.6-2.6 .ou .18 .U3 .67 .71 .a7 .17 .03 2.70 998.83 2.8-3.0 0-3.0 1.09 16.06 66.U1 157.U6 2it6.65 262.93 172.03 62.16 11.18 .83 Accuvulative 1.09 17.15 63.59 21a .05 a89.70 752.63 92a.66 986.82 990.00 998.83 18 May 1961 , SECONDS, SIGNIFICANT WAVE PERIOD AT END OF F„,. T^/Tp, RELATIVE WAVE PERIOD AT END OF DECAY DISTANCE OOiaad 3AVM 3AliVT3b iHDI3H 3AVM 3AliVT3a ' ^h/°H in LlI > cr 3 O >- < o UJ Q 00 UJ or O 19 DECAY DISTANCE D,(STATUTE MILES) WAVE PERIOD Tg , ot end of decay (seconds) FIGURE 9 (Bretschneider, i952) 20 DECAY 0ISTANCED,(NAUTICAL MILES) 21 SURFACE SYNOPTIC CHART FOR I230Z 26 OCT. 1950 FIGURE 10 22 Pi Argutllo 23 Part I Chapter 1 TABLE I FORECAST FOR SYNOPTIC CHARTS OF 26-27 OCTOBER 1950 Chart date (Oct 1950) Chart time (Greenwich) Fetch No, Isobar spacing (degrees latitude) Latitude (degrees) Geostrophic wind speed Vg (knots) Longitude (degrees) Sea Temperature 0g (degrees F) Air Temperature 6 _ (degrees F) a Difference, sea-air temperature (degrees F) Isobar curvature U/Vg Actual wind speed U (Knots) Fetch length (nautical miles) Duration t (hours) Decay distance (nautical miles) Tp (seconds) Hp (feet) Minimum duration tj„ (hours) Minimum fetch Pj„ (nautical miles) Td/Tp V «P Tp (T^) (seconds) (Hq) (feet) Travel time (Hours) Arrival time - Greenwich (Z) time - Pacific Std. time * Por foot notes see following page. 26th 27th 27th 1230 0030 0630 2 2 2 0,65 0.70 1.0 38 37 37 70(1)* 67(^) 47 ( 1 ) 137 133 132 66^2) 65^2) 64<2) 65 65 64 + 1 0 0 Mild to Great Cyclonic Mild Cyclonic Straight 0,59^3) 0 . 59 ( 3 ) 0.62(3) 41 40 29 360 6(4> 470 460 660 430 420(11) 8 . 9 ( 5 ) 12 . 7 ( 8 ) 12 . 2 ( 11 ) 14 . 4 ( 5 ) 23 . 5 ( 8 ) 17.2(11) 6 18.7(^^ 33 65 270 460 1 . 07 ( 6 ) 1 . 19 ( 9 ) 1.18 0 . 22 ( 6 ) 0 . 52 ( 9 ) 0 . 51 ( 9 ) 9.5 15.1 14.4 3.2 12.2 8.8 45('7^ 19 ( 10 ) 2o(10) 28 Oct 27 Oct 28 Oct 0930 1930 0230 28 Oct 27 Oct 27 Oct 0130 1130 1830 May 1961 24 Part I Chapter 1 (1) From Figure 3 (2) From Figure 4 (3) From Figure 5 (4) Estimated (5) From Figure 7 with U = 41 knots and tmin =6 hours (6) From Figure 8 with Tp = 8,9 sec, and D = 660 nautical miles (7) From Figure 9 with Tp = 9,5 sec, and D = 660 nautical miles, (8) From Figure 7 in the following manner. Follow the line U = 41 knots to the duration line of t Z/2 = 6 + 6 = 12 hours; follow an imaginary dotted line from that point to its intersection with the U = 40 knots line which occurs at a point where the duration = 12,7 hours. Add Z/2 to this duration (12,7 + 6 = 18,7), and follow along U = 40 knots to the point where tj^ = 18,7 hours. Read off Tp and Hp, (9) From Figure 8 (10) From Figure 9 (11) From Figure 7, in the following manner. Follow the line U = 40 knots to the duration line of tj^ + Z/2 = 18,7 + 3 = 21,7; follow an imaginary dotted line from that point to its intersection with the fetch line F = 460 miles and drop down along this line to its intersection with the wind velocity line of 29 knots (which occurs at a Point where the duration = 33 hours). Read off Tp and Hp, 1,24 WAVE FORECASTING FOR SHALLOW WATER AREAS - It has already been noted that in relatively shallow water areas, wave generation is affected by the water depth. For a given set of wind and fetch conditions, wave heights will be smaller and wave periods shorter if generation takes place in transitional or shallow water areas. To date, two separate approaches to the problem have been made. Both have shown worth in predicting wave generation, but as yet they have not been adequately verified. This being the case, forecasting curves like those of Figure 7, have not been prepared. Both these methods permit determinations very like those of the Sverdrup, Munk, Bretschneider deep water methods presented previously. For the first method (that of Thysse and Schijf) the empirically determined relationships between forecasting parameters are presented as two sheaves of curves, both shown on Figure 14, To utilize them, wind direction and speed (U) must be determined by any means available, (wind records, etc,), and the available fetch (F) in the wind direction, measured. The relation¬ ships gF/U^ and gd/U^ are then calculated (g is the acceleration of gravity and d is the mean depth over the fetch) after which a gF/U^ abscissa is followed to its intersection with the computed gd/U^ curves, either plotted or interpolated. Values of g^I/U^ and gL/2nU^ are read off as ordinates. Once these have been determined, simple multiplication enables determina¬ tion of the fetch wave height (H) and wave length (L), Wave period may be determined from a combination of equations 1 and 2 , vdiich gives: T^ s _ IJLJ; _ (7) g Tanh ^ L For example, given a fetch of 10 miles, a wind speed of 45 miles per hour and a mean depth in the fetch of 20 feet. May 1961 25 (21.5) (I9.5)(I7.9) (16.3) (14.7) (13.1) (12.3) (11.5) (lO. 3) (9 50) (8.72) (7.95) (7.18) 13.0 12.0 n o 10.0 9.0 8 0 7.50 7.0 6.50 6.0 5.50 5.0 4.50 o>a>^ — o !2 Ll) (T o o 26 Part I Chapter 1 ^ = 0.148, ^ = 3.90 X 10^ U Then from the curves of Figure 14, ( the solid curve is used to read the ^ values of gH/U^, and the dashed curve is used to read the values of Lg/2TTU ) = 3.5 X 10“^ and = 7 x 10“^ 2nU'^ from which H = 4,7 feet and L = 59.6 feet. Equation 7 gives the period as 3.5 seconds. The second method (tha^ of Bretschneider) takes bottom friction and percolation in the permeable sea bottom into account. To date as far as is known there has beer no single theoretical development for determin¬ ing the actual growth of waves generated by winds blowing over relatively shallow water. The numerical method presented is essentially that of successive approximations wherein wave energy is added due to wind stress and subtracted due to bottom friction and percolation. This is done by makiriguse of the deep water forecasting relationships orieinally develop¬ ed by Sverdrup and Munk^^^^^ and revised by Bretschneider^^^^' for determining the energy added due to wind stress. The amount of wave energy los February 1957 27 Part I Chapter 1 due to bottom friction and percolation is determined from the relationships developed by Bretschneider and Reid^^^^\ The resultant wave heights and periods axe obtained by combining the above relationships by numerical methods. The basic assumptions applicable to the development of the deep water wave generation relationships^as well as the development of the relationships for bottom friction loss^^^^' and percolation loss^^^^^ apply to the development. Although there are insufficient wave data to date to verify completely all relationships derived, it is believed that the method will prove sat¬ isfactory once sufficient wave data become available to enable accurate evaluation of the constants. 2 Figures 15, 15A and 15B are dimensionless plots of gH^ /U versus gdp/u2 with gFAJ^ as a parameter for f/m = 5.28, 10.6 and 5 2.8 respectively where f is the friction factor (usually chosen as 0.01) and m is the bottom slope. Figure 15C is a similar plot for a bottom of constant depth. The wave period may be determined directly from the gT/U curve in Figure 7A as the value corresponding to the equivalent deep water signifi¬ cant wave height. An example of this use of the curves is given below. Example Given; F = 300 miles or 15.84 x 10^ feet U = 50 mph = 73.4 ft/sec m = 5 ft/mile or 9,5 x 10 feet/foot f = 0.01 dp = 50 feet Solution: f/m = 10,6 gF/U^ = 32.2 X 15.84 X 10^ (73.4)^ = 9467 32.2 X 50 (73.4)^ 0.2988 Say 0.30 0.11 Figure 15A H • = 18.4 feet o Februory 1957 28 OOoOO o ooooo o ooooo o O O O o o o o o o o o o o o o o o o o o o o O O O Tavi the period of the band in the wave spec¬ trum in which most of the spectral energy is concentrated, Tmax* the wave height data are given in Table 1-B. b. The non-fully developed state - In the fully arisen sea the domina¬ ting wave pattern covers a relatively wide portion of the significant range of periods in the spectrum. The width and range of this portion of the spectrum depend upon the wind velocity. For non-fully arisen sea it also depends on the fetch length and the duration of wind action. In the growing stage of the wave pattern it is assumed that the waves pass through three principal stages, before becoming fully arisen sea. The short period, steep wave components develop first. These remain in a quasi-steady state, always being regenerated after breaking by energy supply from the wind. Once these waves have developed, longer wave trains may arise which cover a spectral band around an average "wave" which travels with the wind speed. If the wind continues to blow, and the fetch and dura¬ tion are long enough, still larger waves can develop. 28 f May 1961 28 0 20.0 16.0 14.0 12.0 10.0 9.0 8.0 7.0 6.0 5.0 4 0T f='/T FIGURE 15-D. CO-CUMULATIVE POWER SPECTRA FOR OCEAN WAVES AT WIND VELOCITIES BETWEEN 20 KNOTS AND 36 KNOTS. THE ORDINATE VALUES Ef, ARE PROPORTIONAL TO THE TOTAL WAVE PATTERN AND DETERMINE THE HEIGHT CHARACTERISTICS OF THE SEA. 28 g Februory 1957 Port I Chapter I TABLE 1-A SIGNIFICANT RANGE OF PERIODS FOl FULLY ARISEN SEA FOR DIFFERENT WIND VELOCITIES. V. (T^ = lower limit, T^ = upper limit of significant periods) V T u V •^L T u (Knots) (Sec.) (Sec.) (Knots) (Sec. ) (Sec.) 10 1.0 6.0 34 5.5 18.5 12 1.0 7.0 36 5.8 19.5 14 1.5 7.8 38 6.2 20.8 16 2.0 8.8 40 6.5 21.7 18 2.5 10.0 42 6.8 23.0 20 3.0 11.1 44 7.0 24.0 22 3.4 12.2 46 7.2 25.0 24 3.7 13.5 48 7.5 26.0 26 4.1 14.5 50 7.7 27.0 28 4.5 15.5 52 8.0 28.5 30 4.7 16.7 54 8.2 29.5 32 5.0 17.5 TABLE 1-B 56 8.5 31.0 CHARACTERISTICS OF FULLY ARISEN SEA V T av T max f mjoc H av »l/3 ” 1/10 (Knots) (Sec.) (Sec.) (cycles/sec.) (ft.) (ft.) (ft.) 10 2.8 4.0 0.250 0.88 1.41 1.79 12 3.4 4.8 0.208 1.39 2.22 2.82 14 4.0 5.6 0.179 2.04 3.26 4.15 16 4.6 6.5 0.154 2.85 4.56 5.8 18 5.1 7.2 0.139 3.8 6.1 7.8 20 5.7 8.1 0.123 5.0 8.0 10.2 22 6.3 8.9 0.112 6.4 10.2 12.8 24 6.8 9.7 0.103 7.9 12.5 16.0 26 7.4 10.5 0.0952 9.6 15.4 19.6 28 8.0 11.3 0.0885 11.3 18.2 23.1 30 8.5 12.1 0.0826 13.5 21.6 27.6 32 9.1 12.9 0.0775 16.1 25.8 32.8 34 9.7 13.6 0.0735 18.6 29.8 38.0 36 IP.2 14.5 0.0690 21.6 34.5 43.8 38 10.8 15.4 0.0649 24.7 39.5 50.2 40 11.4 16. 1 0.0621 28.2 45.2 57.5 42 12.0 17.0 0.0588 31.4 50. 1 63.9 44 12.5 17.7 0.0565 36.0 57.7 73.4 46 13.1 18.6 0.0538 39.7 63.5 80. 6 48 13.7 19.4 0.0515 44.4 71.0 90.4 50 14.2 20.2 0.0495 48.9 78.0 99.4 52 14.8 21.0 0.0476 54 87 110 54 15.4 21.8 0.0459 59 95 121 56 15.9 22.6 0.0442 64 103 130 T av. : average "peri<’d’’ (seconds) T max : period of most energetic wave in the spectrum \v’ ”l/3* ” 1/10 Heights of average wave, average height of 1/3 highest waves and average of 1/10 highest waves, (see Table 1-A) Februory 1957 28 h Part I Chapter 1 By the assumption ot three characteristic "waves'* which are equivalent to these three broad spectral bands in the actual wave spectrum, an ap¬ proximation is given for calculating and describing the state of the sea during wave generation and growth. Relationships between wind and wave characteristics have been established on the basis of energy considerations, which allow calculation of the characteristics of the complex sea for different fetch lengths and wind durations, such as the range of time intervals between succeeding crests at a fixed position, the average "period", the range of wave heights, and the average wave height. Observations have been compared with theoretical results and they support the ideas involved in the theoretical approximation. The minimum fetch length and minimum wind duration required to main¬ tain a fully arisen sea are shown in Table 1”C for various wind velocities. In the practical case, if either or both of these values are not exceeded by the actual fetch and duration a fully arisen sea will not have developed, and other methods (to follow) must be used to describe the state of the sea. TABLE 1-C MINIMUM FETCH AND MINIMUM DURATION OF WIND ACTION NEEDED TO GENERATE A PRACTICALLY FULLY ARISEN SEA FOR DIFFERENT WIND VELOCITIES V F t V F t m ra m m (Knots) (n. miles) (hours) (Knots) (n. miles) (hours) 10 10 2.4 34 420 30 12 18 3.8 36 500 34 14 28 5.2 38 600 38 16 40 6.6 40 710 42 18 55 8.3 42 830 47 20 75 10 44 960 52 22 100 12 46 1100 57 24 130 14 48 1250 63 26 180 17 50 1420 69 28 230 20 52 1610 75 30 280 23 54 1800 81 32 340 27 56 2100 88 1957 28 i February Part I Chapter 1 Case A: Wave Characteristics Limited by the Wifid Duration . A constant wind with mean velocity V (knots) blows over an unlimited fetch P (nautical miles). The energy added to the composite wave motion is the same everywhere so that the waves grow at all localities at the same rate with time, t, (duration, hours). The stage of development of the sea depends only upon the duration of wind action, and is given by the "duration-graphs" in Figures 15E, 15F, and 15G. Case B; Wave Characteristics Limited by the Length of the Fetch , The wind duration, t, is long enough to produce a steady state but the fetch, F, is limited; the state of the sea then depends only upon the length of the water surface over which the wind has blown. This case is given by the fetch-graphs in Figures15H, 151, and 15J, The intersection points of the CCS curves with the duration or fetch lines, (Figures 15E-J respectively)sliow the limit of the development of the composite wave motion at the given duration or fetch. Physically, it means that the state of development is limited by a certain maximum amount of total energy which the wave motion can absorb from the wind within the given conditions. The E^ value of the ordinate of each intersection point is a practical measure of the total energy accumulated in the wave motion of the non-fully arisen state, limited either by the fetch or duration. Under actual conditions, both fetch and duration may be limited, and the E^ value for any given situation, in most cases, will be different for the fetch and duration. In such cases the smaller of the E^ values is taken. As in the case of a fully arisen sea the wave height characteristics can be computed from the E^ value. The upper limit of the significant range of periods for a sea not fully arisen is approximately determined from the wave frequency value (f.) at the intersection of a CCS curve for a given wind speed with the given ^ fetch or duration line, respectively. The wave spectrum may be considered as cut off abruptly at this given minimum frequency, f. , or maximum period, T.. The wave components with periods a little longer than T., and which are just in the beginning stage of development,have a small amplitude and contribute so little to the total wave energy, that they may be neglected. When the value f., is larger than or equal to the frequency, f , of the optimum band as given in Table 1-B where T = 1/f , a smaf^ correction may be applied in order to make allowance for tRe presence of some longer waves. This correction , has been determined empirically from wave observations on limited fetches which indicate that = 0.15 f^ 28 j Februory 1957 Ef(feet2) 4.50 - 6.0 -5.5 —5.0 -40 -3.5 -2.5 -20 -1.5 - 1.0 Z_ 5 >^0 FIGURE 15 E Februory 1957 28 k SIGNIFICANT WAVE HEIGHT (feet^) 28.0 20.0 16.0 14.0 12.0 10.0 9.0 ao 7.0 6.0 FIGURE 15 F 5.0 -30 -28 -26 -24 -22 -20 -18 -16 -12 -10 -8 -6 4 .2 -0 4.0 T February 1957 28 i SIGNIFICANT WAVE HEIGHT(feef) —100 a> Oj LlI 02 03 -95 -90 I 30 .04 I 24 .05 I 20 06 I 18 .07 I 14 .08 09 .10 10 12 .13 I 8 .14 i5f=!4 ; T T -80 —75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 = 0 FIGURE 15 G Februory 1957 28 m SIGNIFICANT WAVE HEIGHT (feet) Ef (feet2) -6jO -55 - 5.0 — 4.0 — 3.5 —ZO FIGURE 15 H Februory 1957 28 n SIGNIFICANT WAVE HEIGHT (fttt) 28.0 20.0 16.0 14.0 120 100 9.0 ao 70 6.0 FIGURE 15 I 5.0 .26 4.0 T -30 -28 -26 -24 -22 -20 -18 -IQ -14 -12 -10 -8 -6 -4 -2 0 February 1957 28 0 SIGNIFICANT WAVE HEIGHT(feet) Ef(feet^) FIGURE 15 J 28 p Februory 1957 SIGNIFICANT WAVE HEIGHT (f*et) Part I Chapter 1 In most practical cases* the wave-generating winds will be complicated, and the determination of the parameters ’’fetch*' and "duration" is probably the most subjective factor in the process of wave forecasting. It is difficult to set up strict rules for determining these important parameters of wave forecasting. In practice, innumerable possibilities of different fetch and duration conditions may be encountered by the wave forecaster. It must be left to his experience and good judgment in the interpretation of consecutive synoptic weather ms^s to use the given graphs and tables in the most effective way. The corrected upper limit of the significant range of wave periods for a non~fully arisen sea, where f. > f , is then given by the equation: 1 max Example: With a wind velocity of 20 knots and a duration of 6 hours, E^ = 2,99 and f^^ = 0,165 (T^ = 6,1 seconds) from Figure 15E, Table 1-B shows that f = 0,124; thus f. > f and the correction may be applied, = max 1 max t 0,025, and f^ = 0,165 - 0,025 = 0,14; thus T^ = 7,1 seconds and is used in place of T. =6,1 seconds. The lower limit of the significant range of wave periods, t|^, can be determined in the same manner as for a fully arisen sea (i,e,, the rrequency corresponding to 3 percent of E^), For this case 0,4 and 2,5, Thus the significant range of wave periods for this examp le i s 2,5 to 7,1 seconds. The wave height characteristics are computed fromV B- = 1,73, It follows: H ='2,1 feet, H . = 4,9 feet, H . = 6,2 feet, 1,26 WAVES IN TOANSITIONAL AND SHALLOW WATER , - From equation 2, it may be seen that wave velocity varies with the depth of water in which the wave is moving; as the depth decreases, the velocity also decreases. In addition, assuming constant wave period, equation 1 indicates that wave length also varies with water depth. The variation in wave velocity along the crest of a wave moving at an angle to underwater contours, (i,e,, that portion of the wave in deeper water is moving faster than the portion in shallow water) causes the crest to bend toward alignment with the con¬ tours (see Figure 16), This effect, called refraction, is more or less significant depending on the relation of the water depth to the wave length. In water deeper than one-half the wave length, the hyperbolic tangent function in the formula f,2 gL 2TTd ' It *>">>—. <2) IS nearly equal to unity, and equation 2 reduces to February 1957 28 q H*« TU-u-’ir*'''^ f* «? -'• "riu.a»2^#.4 »/4 fci*' ■#<<» •»M#a -»*<•»•: f,r :lff >,iifi 'tu^, t^r^inr .«h V » ■ * • M <*i 1^. M r^tlK 1ft- 3 f.fi ylfj . P f ■ '"- «.*t 4«» t i«i i J,/ % .'JT 3 •liq' ln i - -- e- *<.-.:vi v«r .>4 ^TT*^ ^ ;..VT ^ r^- " >j f. ti,- V.. v:i/i>i(>*iil^-^v'‘'-- # O «' ’ ..ttiU ’^.^'''*^'1- "^3.3*^i<» -^1 ,«t. t.c^* ^-.^^ 3 ;. i . r ■>' ,; ' , ''■•■"‘'•&V t' iT.j ■‘■,r fJ-iq-TMt)'■*;«•,I i^tig -Ms-jikiiM' Mil-' *. . . -^ Aj» »*<*■■ ni’ 1 ■' vvr,^ v-..> 'ncilj ^.,lq;g:v *?v;' .i+je.'tii )»>«n# «|^iif * a*,’* tUi • : ;.nO •'•. tt^ #4 I, * ‘>^1 2 ^’? ’ ' •'» fc> 4C 4[ji liv*'. J ) ;»» ^'r;i^^1n'i Ttsijt ■r... VU. >3:%««. v; C-.7i»»r ;r% ^ j.i- .21 ..,^r m f -..A;,^XAW .*' ♦ .■» *’ 1. • k<3A/ilt: .'■»11J.itg *• ^ .' ■■' : * '■ 7 ■;. ’ - v'T '■■■< ■•-Xa^, -:^£: fl-. ■• ♦ mf **9^ A-jSruft&t 1^ «M . ,>•3 *1 zjr>p « . Part I Chapter 1 In this equation, the velocity (C) is not dependent on depth; therefore in those regions deeper than one-half the wave length (deep water) refraction is not significant. In the region where water depth is between 1/2 and 1/25 the wave length (transitional water), and in the region where water depth is less than 1/25 the wave length (shallow water) refraction effects are significant. In the first of these, wave velocity must be computed from equation 2; in the second, tanh 2Trd/L becomes nearly equal to 2Trd/L and equation 2 reduces to = gd (0) Both equations 2 and 9 show a dependence of wave velocity on depth. To a first approximation, equation 3 for total energy in a wave per unit crest width may be written as Et w H^L 8 ( 10 ) It has already been noted (equations 4 and 5) that the entire wave energy, Et, is not transmitted forward with the wave, only 1/2 being trans¬ mitted forward in deep water as Ep. The value of this Ep for a given wave remains essentially constant as it moves from deep water on in to the breaker line, however both Et and Ej^, the kinetic energy, decrease progressively across the transitional zone with E;^ becoming relatively insignificant when the shallow-water zone is reached. The value of Et of course decreases to approximately the value of Ep when the shallow-water zone is reached. 1,261 Refraction of Waves . In performing refraction analyses, it is assumed that for a wave advancing toward shore, no energy flows laterally along a wave crest. That is, the transmitted energy remains constant between two lines (called orthogonals) drawn perpendicular to a wave crest as it passes over changing hydrography. The wave energy transmitted forward between any two adjacent orthogonals in deep water is Eq = (1/2) bo Eto» where bo is the spacing of the orthogonals in deep water. (The subscript "o" always refers to deep water conditions). This energy may be equated to the energy trans¬ mitted forward between the same two orthogonals in shallow water (E = n b Et), where b is the spacing between the orthogonals in shallow water. Therefore (1/2) bo Eto = n b E^ or Et/^to = ^i/2) (1/n) (bo/b) (12) From equation 10, H/Hq = V (Et/Eto^ JLq/l) therefore equation 12 may be written / \ H/Ho = VC 1/2) (1/n) (Lq/L) ) (13) May 1961 29 •JT- Of; WAVE REFRACTION AT WEST HAMPTON BEACH, L.I.,N.Y. FIGURE 16 30 Part I Chapter 1 ) The term Aj(l/ 2 )(l/n)(L q/l), is known as the shoaling coefficient (H/Hq’)* This shoaling coefficient is a function of wave length and depth. It. and various other functions of d/L, eg^ 2 7 rd/L, 47 r d/L , tanh (2 7 rd/L), sinh (477’d/L), etc. are tabulated in Appendix D (Table D-1 for even increments of d/liQ and Table D-2 for even Increments of d/b). For any de^. and deep water wave length, the ratio d/L may be determined by 4 method of successive approximations from the ratio d/Lg . Equation 13 shows that wave heights in transitional or shallow water may be found, knowing deep water wave heights, if the relative spacing between lines drawn perpendicular to wave crests can be determined. The square root of this relative spacing (bo/b) is known as the refraction coefficient, usually designated K. It should also be noted that these per¬ pendicular lines, when constructed, will show the direction of movement of the waves to which they are drawn perpendicular. The lines drawn perpendicular to the wave crests are known as ortho- gonals . Various methods have been proposed for constructing these lines. The earlier approaches required drawing positions of wave crests, ( 64 ) then erecting perpendiculars to them; later approaches eliminate the inter¬ mediate wave crest step, permitting the immediate construction of ortho- gonals themselves (3) (67)(ll6). It can be shown (see Appendix E) that the change of direction of an orthogonal as it passes over relatively simple hydrography is sinag = (C 2 /C 1 ) sinai ( 14 ) where a, is the angle a normal to an orthogonal makes with a contour the orthogonal is passing over, 02 is a similar angle measured as the orthogonal passes over the next contour, Cq is the wave velocity (equation l) at the depth of the first contoiir, and C 2 is the wave velocity at the depth of the second contour. From this equation, a template may be constructed with which the angular change in a as an orthogonal passes over a definite contour interval may be found, and the changed-direction-orthogonal may be constructed (see Appendix E). Such a template is shown on Figure 17. Procedures in Refraction Diagram Construction - For a chosen shore location, charts showing bottom topography of the area are obtained. Two or more charts may be necessary, of differing scale, but the procedures are Identical for charts of any scale. Next, underwater contours are drawn on the chart, or on a tracing paper overlay, for various depths, depending on the diagram accuracy desired. If overlays are used the shore line should be traced in for reference. In tracing the contours. Judgment must be used I 31 Februory 1957 32 REFRACTION TEMPLATE FIGURE 17 Part I Chapter 1 in "smoothing out" small irregularities, since bottom features which are comparatively small in respect to the wave length do not affect the wave appreciably. The range of wave periods to be used is determined by a hindcasting study of historical weather charts or from other historical data relating to wave periods. With the wave period so determined, Cp/C2 values for each contour interval should be marked between the contoiars. The method of computing Cp/C2 is illustrated by Table 2. A tabulation of C2/C2 for various contours intervals and wave periods is given in Table D-9 of Appendix D. TABLE 2 COMPUTATIONS FOR VALUES OF Ct/C? T = 10 seconds d ^ , 27rd d L Cd/Cs Cs/Cd (1) (Feet) —t2l- -nr (5) 6 0.0117 0.268 1.40 0.72 12 0.0234 0.374 1.21 0.83 18 0.0352 0,453 1.14 0.88 24 0.0469 0.516 Column 1 gives depths corresponding to chart contoiirs. These would extend from 6 feet to a depth equal to Lq/2. Column 2 is coliimn 1 divided by Lq corresponding to the determined period. Column 3; these values may be fo\ind in Table D-1 of Appendix D, as a function of d/L^. This term is also C/Cq. Column 4 is the quotient of successive terms in column 3 . Column 5 is the reciprocal of column 4. To construct an orthogonal from deep to shallow waiter, a deep water direction of wave approach is first selected by a hindcasting study of historical weather charts, by fan diagram analysis or by direct observation. A deep water wave front (crest) is drawn as a straight line perpendicular to this wave direction and suitably spaced parallel lines (orthogonals) are drawn from this wave front in the chosen direction of wave approach. These lines are extended to the first depth contour shoaler than Lo/2 where (in feet) = 5.12 t2. 33 34 Part I Chapter 1 Procedures for a less than 80° - Starting with any one orthogonal, the following steps should be taken: (a) Sketch in a mid-contour between the first two contours to be crossed, extend the orthogonal to this mid-contour, and construct a tangent to the mid-contoxir at this point; (b ) Lay the line labelled ”orthogonal” along the incoming orthogonal with the point marked 1.0 at the intersection of orthogonal and mid-contour (see Figure I8a); ( c) Rotate the template about the ”t\irning point" until the C1/C2 value (C(3^/Cg) corresponding to the contour interval being crossed intersects the tangent to the mid-contour. The "orthogonal" line now lies in the direction pf the turned orthogonal (see Figure I8b); (d ) Place a triangle along the base of the plotter and erect a perpendicular to it so that the intersection of this perpendicular with the incoming orthogonal is equidistant from the two contours measured along the incoming orthogonal and this perpendicular. This line represents the turned orthogonal; (e) Repeat the process for successive contour intervals. If the orthogonal is being constructed from shallow to deep water, the same procedure is followed, except that Cg/C^i values are used for Cq/C2 instead of Cd/Cg . Procedures for CL greater than 80° - In any depth of water, when a becomes greater than 80° , the above procediires may not be used. The orthogonal no longer appears to cross the contours, but tends to run parallel to them. In this case the contour interval is crossed in a series of steps. In essence, the whole interval is divided into a series of smaller intervals, at the mid-points of which, orthogonal angle turnings are made. Referring to Figure 19, an interval to be crossed is divided into segments or boxes, by transverse lines. The spacing, R , of the lines is arbitrarily set as a ratio of the distance, J , between the contours. For the complete interval to be crossed, C2/C1 is computed or foiind from Table D-9 of Appendix D (Note: G2/C1 not G^G2)» On the template (Figure 17) is a graph showing orthogonal angle turnings ( AC! ) at the center of a box, plotted as a function of the G2/C1 value of any contour interval for various values of the R/J ratio, which may be chosen. The orthogonal is brought into the middle of the box, A a is read from the graph, and the orthogonal turned by that angle. The procedure is repeated for every box until a at a plotted or interpolated contour becomes 35 Part I Chapter 1 J = Distance between contours at turning points,* R = Distance along orthogonal T =12 seconds Lq= 737 ft. For contour interval from 40 fm to 30ftt» C/C = 1.045, C /C =0.957 12 2 1 less than 80°. At this point this method of orthogonal construction must be stopped or error will result. Refraction Fan Diagrams - It is often convenient, especially where a coastal area is shielded by land features from waves approaching in certain directions, to construct refraction diagrams from shallow toward deep water. In such cases, a sheaf or fan of orthogonals may be projected seaward in direction some 5 or 10 degrees apart. (See Figure 20a) With the deep water directions detemined by the individual orthogonals, companion orthogonals may be projected shoreward on either side of the seaward projected ones in order to determine the refraction coefficient for the various directions of wave approach. (See Figure 20b). Refraction Diagram Limitations - In many cases refraction diagrams provide a reasonably accurate measure of the changes waves undergo on ap¬ proaching a coast. Quite often they provide the only measure of these changes available. However, the accuracy of data determined from refraction diagrams is limited by the validity of the theory of their construction and the accuracy of depth data on which they are based. The orthogonal direction change equation 14 is derived for the simplest case of straight parallel contours, and although little error is introduced by bringing orthogonals over relatively simple hydrography, it is difficult (94) to carry an orthogonal accurately into shore over complex bottom features. Moreover, the equation is derived for small waves moving over relatively flat slopes. Although May 1961 36 Azimuths meosured clockwise from true north. O H Z Z < a. Z o o z < q; o < o z g 1- q: w u. UJ iLl a: tr o UJ g 0. li. >- I z < < (T C9 < O UJ cr. UJ tr UJ C) I < u. u. o UJ (O 3 37 1 Februory 1957 38 Part I Chapter 1 no strict limits have been set, strict accuracy as far as height changes are concerned cannot be expected where bottom slopes are steeper than 1 on 10, although model tests have indicated that direction changes occur nearly as predicted even over a vertical discontinuity, A third limitation is inherent in the assumption that no energy travels laterally along a wave crest. No strict limits have been set, but the accuracy of wave heights derived from orthogonals which bend sharply, is questionable, 1,262 Diffraction of Waves , - Diffraction in water waves is that phenomenon whereby energy is transferred laterally along a wave crest. It is most notice¬ able where an otherwise regular train of waves is interrupted by a barrier such as a breakwater, Putnam and Arthur (105) presented experimental data verifying a method of solution proposed by Penny and Price (98) for the behavior of waves after passing a single breakwater. Blue and Johnson (11) have dealt with the problem of the behavior of waves after passing through a gap, as between two breakwater arms, (a) - Waves Passing a Single Breakwater , The solution to this problem is presented in Appendix E, From this solution an overlay has been prepared (Figure 21) which, for the case of uniform depth shoreward of the breakwater, shows positions of diffracted wave crests and lines of equal wave height reduction. Plotting data used in the construction of this figure is given on Table E-1, Appendix E, k Template Overlay Breakwater is negative ^ v«l ; '/// - -- -- Hi fl ilUl l ( 11 f 11( l ( l(R Wove Crests DIFFRACTION FOR A SINGLE BREAKWATER-NORMAL INCIDENCE FIGURE 22 The diagram of Figure 21^^^^ is presented in dimensionless form and can therefore be used for any condition of wave period and water depth by scaling the entire figure up or down. The manner of use of the overlay is illustrated in Figure 22, The wave length, L, at the depth d at the break¬ water tip must be found by computing the ratio d/Lo (L© = the deep water 39 4.0 fit -0,4-—-.10 -.16-- 17 -.14-T-;-.OB-r-.06 -v-.09 40 GENERALIZED DIFFRACTION DIAGRAM FOR A BREAKWATER GAP WIDTH OF TWO WAVE LENGTHS (B/L:2) FIGURE 23 y/L Fig.25 CONTOURS OF EQUAL DIFFRACTION COEFFICIENT GAP WIDTH = 1 WAVE LENGTH (B/L = I) DIFFRACTION OF WAVES AT A BREAKWATER GAP (Johnson, 1952) 4 \ y/L (Johnson, 1952) Fig.27 CONTOURS OF EQUAL DIFFRACTION COEFFICIENT GAP WIDTH =1.64 WAVE LENGTHS (B/L=l.64) DIFFRACTION OF WAVES AT A BREAKWATER GAP (Johnson, 1952) 42 J/l 10 iz “T" GAP J=l.78 L l\ BREAKWA TER K'=0.4 DOW'V V9C.UIVIC. 1 niu OHM 14 16 16 K=0.6 n DIRECTION OF ^ INCIDENT WAVF Fig.28 CONTOURS OF EQUAL DIFFRACTION COEFFICIENT GAP WIDTH = 1.78 WAVE LENGTHS (B/L=l.78) (Johnson, 1952) (C Id 20 Fig. 29 CONTOURS OF EQUAL DIFFRACTION COEFFICIENT GAP WIDTH = 2 WAVE LENGTHS (B/L = 2) DIFFRACTION OF WAVES AT A BREAKWATER GAP (Johnson, 1952) 43 DIRECTION OF INCIDENT WAVE Diffrooted Wove Height Incident Wove Height Fig. 31 CONTOURS OF EQUAL DIFFRACTION COEFFICIENT GAP WIDTH = 2.95 WAVE LENGTHS (B/L = 2.95) DIFFRACTION OF WAVES AT A BREAKWATER GAP (Johnson, 1952) 44 Rg. 33 CONTOURS OF EQUAL DIFFRACTION COEFFICIENT GAP WIDTH = 5 WAVE LENGTHS (B/L=5) DIFFRACTION OF WAVES AT A BREAKWATER GAP (Johnson, 1952) 45 46 FIGURE 35-A. DIFFRACTION FOR A BREAKWATER GAP OF ONE WAVE LENGTH WIDTH (0=0 AND 15 DEGREES). February 1957 46-a FIGURE 35-B. DIFFRACTION FOR A BREAKWATER GAP OF ONE WAVE LENGTH WIDTH (0=30 AND 45 DEGREES). February 1957 46 b 6 FIGURE 35-C. DIFFRACTION FOR A BREAKWATER GAP OF ONE WAVE LENGTH WIDTH (0=60 AND 75 DEGREES). February 1957 46-c TWO WAVE LENGTHS AND A 45-DE6REE APPROACH COMPARED WITH THAT FOR A GAP WIDTH \/I WAVE LENGTHS WITH A 90-DEGREE APPROACH. Februory 1957 46 d Ovtr-«ll refroction—diffraction coefficient is given by ( K|) ( K') V b|/ ba . -Where: K|= Re f r 0 c t i 0 n coefficient to breakwater. K'= Diffraction coefficient at point on diffracted wave crest from which orthogonal is drawn. bisOrthogonol spacing at diffracted wove crest. b2’:Orthogonal spacing nearer shore. Wave Orthogo na Is Lines of Equal Diffraction Coefficient (K'] Breakwater 2ZZZZZZZZZZZZ22ZZZZZZZZZZZ1 Wave Crests SINGLE BREAKWATER-REFRACTION-DIFFRACTION COMBINED FIGURE 36 4T BREAKER HEIGHT INDICES FIGURE 37 FIGURE 38 May 1961 48 Part I Chapter 1 wave length), by referring to Table D-1, Appendix D for the corresponding value of d/L and dividing, d, by this ratio. The diagram itself must then be scaled up or down so that the distance from yA. = 0, to yA = 1 corresponds to one wave length on the scale of the chart on which the diagram is to be drawn. The diffraction diagram is then placed so that the xA = 0 ordinate lies in the direction of wave approach, with xA positive values along the breakwater. The lines labelled "wave crests" then represent positions of successive diffracted wave crests. The lines labelled K* are lines of equal decreased wave height. For example, along the K* = 0,20 line wave heights are 0,2 of their incident value, (b) - Waves Passing a Gap of Width Less Than Five Wave Lengths , - The solution for this problem (Appendix E) is more complex, and it is not possible to construct one diagram for all conditions, A new diagram must be drawn for each different ratio of gap width to wave length (BA). One for a BA~ratio of 2 (see Appendix E) is shown in Figure 23, which also illustrates its use. Figures 24 through 33^65) show lines of equal diffraction coefficient for BA'iatios of 0,50, 1, 1,41, 1,64, 1,78, 2, 2,50, 2,95, 3,82 and 5, drawn for a somewhat more complex solution of the diffraction problem than that in Appendix E, In all but Figure 29 for BA = 2, the wave crest lines have been omitted. Wave crest lines are usually only of pictorial use. They are, however, required for accurate estimation of the combined effects of refraction and diffraction. In such cases, estimation of wave crest lines of sufficient accuracy may be made by use of circular arcs. For a single breakwater, the arc will be centered on the breakwater end; the portion of the wave crest extending to unprotected water (from the K-line of 0,5) may be approximated as a straight line. For a breakwater gap, crests beyond about 8 wave lengths behind the breakwater may be approximated by an arc centered on the middle of the gap; crests to about 6 wave lengths may be approximated by two arcs, cen¬ tered on the two ends of the breakwater; these two arcs are connected by a smooth curve - sufficiently approximated by a circular arc centered on the middle of the gap. One-half the diffraction coefficient lines have been eliminated from these figures but the diffraction coefficients are symmetrical about the xA = 0 ordinate, thus the diagrams may be completed by folding the diagram about that ordinate, (c) - Waves Passing a Gap of Width Greater Than Five Wave Lengths , - Where the breakwater gap width is greater than five wave lengths, the diffraction effects of each wing are essentially independent, and the diagram (Figure 21) for a single breakwater may be used to define the diffraction characteristics in the lee of both wings, (See Figure 34), (d) - Diffraction at a Gap - Oblique Incidence .- When waves approach at an angle to the centerline of the breakwater, the diffracted wave characteristics differ from those resulting from wave approach in a direction normal to the center- line, Figures 35A, 35B and 35C show the diffraction diagrams for the angles 0, 15, 30, 45, 60 and 75°, Though by using these diagrams somewhat more accurate results may be obtained, an approximate appraisal of diffracted wave characteristics may be made by considering the gap to be as wide as its projection in the direction of incident wave travel as indicated in Figure 35, A comparison of a 45° incident wave using the approximate method and the more exact diagram is shown on Figure 35D, 1,263 Refraction and Diffraction Combined . In the usual case, the bottom seaward and shoreward of a breakwater is not flat, therefore refraction as well as diffraction occurs. Though a unified theory of the two has not yet been devised, an approximate picture of wave changes may be drawn by: (1) constructing a refraction diagram to the breakwater, (2) at this point, constructing a diffraction diagram carrying successive crests 3 or 4 wave lengths shoreward, if possible, and (3) with the wave crest and wave Moy 1961 49 Part I Chapter 1 direction indicated by the last shoreward wave crest determined from the diffraction diagram, constructing a new refraction diagram to the breaker line, A typical refraction-diffraction diagram is shown on Figure 36, Note the method of determining combined refraction-diffraction coefficients. 1,264 Breaking Waves - At a certain point in its advance toward shore a wave becomes unstable, peaks up and breaks. The water motion changes from a laminar orbital motion to turbulent, white-water conditions. The determi¬ nation of point of breaking and breaking wave heights is of major importance in the planning and designing of shore protection measures. Three different sets of curves are currently used to determine break¬ ing depths and wave heights on breaking; one theoretical, and two empirical, (61) (93) The theoretical curve is derived from the analysis of a so-called solitary wave (93) and results in the following equations: d./H b D b o « H • s o _ 1 _ 3.3 A, s 1,28 1,28 3,3 3/2 1,837 (d. ) D 0,667 (H 'T) o 2/3 (15) (16) (17) (17a) (17b) Graphs drawn from these relations as well as those from the two other studies (60) are presented in Figures 37 and 38, All three curve sets relate the breaker depths d^ and the breaker wave height Hb to deep water wave length Lq and deep water wave height, Hq*, which would exist if refraction were ignored, (Knowing the deep water wave height Hq and the refraction coeffici¬ ent K, Ho* may be determined from Ho* =K Ho; i.e., H = K (H/Ho*) Ho*(H/Mo*) Hq*; see section 1,261. The solid line curves include the additional para¬ meter of bottom slope, the effect of which was adequately verified under con¬ trolled laboratory conditions. To use the curves on Figures 37 and 38, Ho* is determined as above, Lo from the known relationship Lo (in feet) « 5,12T^, and the beach slope from hydrographic charts. By computing the ratio Ho*/Lo the ratios Hb/^o* and db/Ho* may be picked off the graphs and, from these, Hb and db determined by multiplication by Ho*. Figure 38A is a graphical solution of equation 17a. The figure shows the wave height and depth of water at point of breaking as a function of wave period for the solitary wave theory relationship. 1,27 HURRICANE WAVES . This section is devoted to the generation of wind waves in deep water and over the continental shelf due to onshore winds. May 1961 50 001 (489*) 0^ ‘J8*DM daSQ U! *m6i8H 8ADM 5Qa Depth of Breaking d(,(feet) FIGURE 38-A WAVE HEIGHT AND DEPTH OF WATER AT POINT OF BREAKING FROM SOLITARY WAVE RELATIONSHIPS Part I Chapter 1 with particular interest given to hurricane waves, A number of generalized graphs were presented in the section on wave forecasting, which could be used to predict wind waves in deep water and also in shallow water for certain conditions. These conditions were for winds of constant speed and direction, and those for shallow water were for the special case of a flat bottom of constant depth or bottom of constant slope. In many cases the above graphs are inadequate for forecasting waves, particularly when the bottom is of vari¬ able slope and depth. The procedures used in this section for hurricane wave generation can also be applied to the normal case of constant wind speed and direction. In fact the procedure used for hurricane waves is first to deter¬ mine the deep water significant waves and then propagate them shoreward under the continued influence of the maximum wind as the hurricane moves shoreward. In this respect the wind acting on the waves being propagated shoreward, being the maximum wind, will be of constant direction and will remain at constant speed unless the intensity of the hurricane changes. In so far as the design or standard project hurricane is concerned it can be assumed that the intensity of the part of the hurricane associated with maximum waves will remain unchanged. If one is interested in forecasting deep water waves of some particular hurricane which might occur, it is possible to use the graphical method of forecasting as originally proposed by Wilson^^^^^ and which has been revised in the section on wave forecasting (see section 1.233, Generation of Deep Water Waves by Variable Wind Speed and Direction and Moving Storms.) However, when considering a standard project or a design hurricane, the method pre¬ sented by Bretschneider(212) ^as simplified the use of the graphical method. Both methods will result in essentially the same deep water waves. For a slowly moving standard hurricane the following formulas can be used to obtain the deep water significant wave height and period H o « 16.5 e RAP Too 1 ♦ 0.208aVp (17c) and T *s 8.6 e Rap 200 1 -»• O.lO^qVP (17d) s where » deep water significant wave height in feet T * the corresponding significant wave period in seconds s R s radius of maximum wind in nautical miles AP = P - P # where P is the asymptotic or sometimes the normal a o a pressure of 29.92 inches of mercury, and P is the central pressure of the hurricane in inches of mercury. May 1961 50 b Part I Chapter 1 = the forward speed of the hurricane in knots, is the maximum sustained wind speed for the stationary hurricane in knots, measured at 30 feet above the mean sea surface. e (U - I V_) for the moving hurricane max ^ P ^ a is a coefficient depending on the forward speed of the hurricane and the increase in effective fetch length because the hurricane is moving. It is suggested for slowly moving hurricanes to use a = 1,0 Once Hq is determined from equation 17c, it is possible to obtain approxi raately the deep water significant waves Hq for other sections of the hurricane by use of Figure 38B, The corresponding wave period will be approximately obtained from T s ( in seconds) (17e) where H is in feet, o Consider now a standard project hurricane, for which R = 34 nautical miles, AP = 29,92 - 27,55 = 2,37 inches of mercury, Vp = 23 mph = 20 knots, Umax “ 1-04 mph = 90,5 knots, whence = 90.5 - ^ (20) * 80,5 knots. Prom equation 17c. H = 16.5 e°’®^ \l + 0-208 x 20 L (16.5) (2.25) (1.462) « 54.4 feet ° ^ y80.5 ^ and the corresponding significant wave period from equation 17d, T = 8.6 2 0 j 8 (8.6) (1.5) (1.226) * 15.8 seconds ^- ^80.5 or T = 2.13 V54.4 = 15.8 sec. Referring to Figure 38B, H© = 54,4 feet corresponds to the relative significant wave height of 1,0 at r/H » 1,0 to right of the hurricane center; that is at 34 nautical miles to the right of the hurricane center H© = 54,4 feet and T = 15,8 seconds; at r/R * 1,0 to the left of the hurricane center, the ratio is about 0,65, whence H© = 0,65 x 54,4 = 35,3 feet and is moving in a direction opposite to that of the 54,4-foot wave. The significant wave period for the 35,3-foot wave is T s= 2,13 35,3 s 12.6 seconds. The most probable maximum wave depends on the number of waves considered applicable to the significant wave. This depends on the length of the section May 1961 50 c Part I Chapter 1 of the hurricane for which near steady state exists and the speed of the hurricane. It has been found that optimum conditions apply for H© i 0,5 foot and corresponds to a distance equal to the radius of maximum wind. That is, for Ho = 54,4 feet, this zone includes all waves from Ho = 53,4 to 54,4 feet with an average of 53,9 feet. The time that it takes the radius of maximum wind to pass a particular point is equal to ts=~ =““=1.7 hours = 6,120 seconds (17f) Vp 20 the number of waves will be N = JL s ^ 287 Ts 15.8 (17g) The most probable maximum waves can be obtained from Hn = 0.707 Ho v^Io^T^ (17h) n The first most probable maximum wave is obtained by setting n = 1, whence H^ = 0.707 (53.9) Vloge 38?= 93 feet Assuming that the 93-foot wave occurred, then the second most probable maximum wave is obtained by setting n * 2, the third from n = 3, etc. The limit of the above equation should not be taken beyond about n = N as the expression is an asymptotic solution only for larger values of N/n, Con¬ tinuing with the above example th e second and third most probable maximum wave will be Ha = 0,707 (53,9) y^loge 193.5 = 87 feet, H3 = 0.707 (53,9) ^loge 129 = 84 feet The problem now is to propagate the deep water waves across the conti¬ nental shelf taking into account the combined effects of bottom friction, refraction and the continued action of the wind. This requires numerical integration, using Figures 38C and 38 d and refraction diagrams. It is also necessary to obtain an effective fetch length, by use of r H -1 2 Fe = I -2-1 (17i) L 0.0555UJ Where Fg is the effective fetch in nautical miles, Hq is the deep water significant wave height, in feet, and U is the maximum sustained wind speed in knots, whence for this example Fg = r 54,4 _ 12 - iij nautical miles L 0.0555 X 90.5 J It might be noted that the effective fetch here is not equivalent to the minimum fetch required to obtain H = 54,4 feet with a 90,5-knot steady May 1961 50 d Fig. 38-B Isolines of relative significant wave height for slowly moving hurricane. May 1961 50 e 50 f May 1961 X V FIGURE 38-D. RELATIONSHIP FOR FRICTION LOSS OVER A BOTTOM OF CONSTANT DEPTH 50 g May 1961 Part I Chapter 1 wind as obtained from the wave forecasting curves; from those curves one would obtain Fmin ® nautical miles. The value of Pg » 117 nautical miles does not take into account the increase in fetch length due to the forward displacement of the hurricane. For this example the increase in fetch length would be approximately A? * 175 - 117 = 58 nautical miles Either the forecasting curves for deep water beginning with = 175 miles can be used for the remaining part of this problem or Fg = 117 nauti¬ cal miles can be used with the following formulas Hq = 0.0555 y Pg AF < 54.4 feet and To = 2.13 yST The latter being a numerical method is easier to use than the forecast¬ ing curves which require a not-so-accurate graphical method. The procedure for computing wind waves over the continental shelf will be that for the bottom profile off the mouth of Chesapeake Bay, using the standard project hurricane developed forthe Norfolk area. The storm surge computed elsewhere and 2,5 feet of astronomical tide are added to the mean low water depths to obtain the total water depth for wave generation. Re¬ fraction is neglected but should normally be included if more accurate results are required. The results of these computations are given in Table 2A followed by examples and explanations, Colvmm 1 of Table 2A gives the distance in nautical miles measured sea¬ ward of the entrance to Chesapeake Bay; the increments are at 5 nautical miles for each section. Column 2, dj^ gives the depth in feet (mean low water) at the end of each section, corresponding to x of column 1, and the computed waves of columns 15, 18 and 20, Column 3 is the depth dj at the beginning of each section and column 3, d 2 at the end of each section, d^ and d 2 ^ correspond to dj and d 2 given in the Table for storm surge except that the corresponding surge plus 2,5 feet astronomical tide is added, and are rounded off to the closest foot. Column 5 is the average of columns 3 and 4 to the closest foot. Column 6 is the effective fetch Pq and is obtained from Fe = Fe* + aF < 117 nautical miles where Fg* is given in Column 14 of one line above in each case. Column 7 is deep water significant wave height and is obtained from May 1961 50 h COMPUTATIONS FOR WIND WAVES OVER CONTINENTAL SHELF X o »o o CO CO 00 CO O’ O' CM o • • • • • • • • • • • • • • (M g 0 nO «o CO O’ CO CO CM CM CM O' 00 o o nO o o oo CM CM nO X X o 00 O' o O’ o r- o CO' oo CM r- X CM r> CO O' O' O' O' O' »o CO CO nO yO X 00 «o CO O’ O' o •H 00 CM NO CO X 0“ CO • • • • • • • • • • • • • • X O' r- O' O' «o CO O' NO CM O' nO X CO O' O' CO CO CO CM CM CM o rr CO 00 o 'O CO CO o X O' O' nO X fO CM CM CM CO m O' O' O' o O' O' o O' O' O' O' O' O' o u • o CJ o 00 O' NO o CM CM CO o CM a 'O CNJ »o rsj O' 00 'O CO CO CO yo nO nO nO CO 'O « o o H 00 CO O' CO 00 CM O' CO X X CO o « • • • • • • • • • • • • • (h CO CO CO O' CO CO CM CM o O' O' X •H »o »o CO o o CM 00 o O' • • • • • • • • • • • • 0) NO nC o CM o CM CM CM CO X X 'O CL. o o NO CO O' CO CM TT 00 r' O' CM nO CO CO CM o O’ • • • • • • • • • • • • • • t- CM 00 '€> CM 00 O' X CO CM X CO CO CO O' O' (O CO CO CM CM CM CO CO 'O o O' CO o 00 O' nO CM o X yO o CvJ o o O' O' 00 00 00 00 00 X r- r- NO O' o CM o o CO CM OJ CM O' 00 O' CO o CO 'O CO f- 0 X r- X o CM CM X 00 O' CO 00 o o o CM o o CM CM CM CM CO CO X X O’ O’ CO o o o O' O' O' O' o O' O' O' O' O' O' u: • • • • • • • • • • • • • • o o o o o o o o o o o o o fO CM CO CO O' CO CO NO CO X nO O’ . f-i fO O' 00 t- yo c- O' O' O' O' o O' J.’O fvNs o o o •H o »o O’ O' CO r> (O O' nO O’ O' yo o X 'O o o CO o 00 'O 'O CO O’ X •o CO CO CO CO O' NO 0“ O' yo r- o X o o o »o o 00 NO nO CO O' fO o CO T3 o o O' nO CM o O' CM o o CM CM CM O’ (M K 2 o o CO o 00 r- yO CO O’ X CM ■o ae CO X CO o CO o CO o CO o CO o »o o CO o >0 NO CO CO O’ O’ CO CO CM CM o U 50 i May 1961 Part I Chapter 1 Column 8 is deep water significant wave period and is obtained from To' 2 Column 9 is Column 10 is the shoaling coefficient corresponding to column 9, and is obtained from Figure 38C, Kg versus t 2/ d-j* Column 11 is the friction loss parameter and is equal to f Ho*Ks AX 0.01 Ho* Kg (5)(6.080) 304 Hq* Kg ( dj.)^ ( ( where f » 0.01, AX « 5(6,080) = 30,400 feet and d^ is average total water depth of the increment aX, where Column 12 is the friction factor Kf and is obtained from Figure 38D 2 _ p f Ho* Kg flX K£ is a function of T /(dy) and -Z—5- ( df) Column 13 is the equivalent deep water wave height and is obtained from Ho* ■ Ho Kf, Column 14 is the equivalent deep water fetch length for Ho* and is obtained from .2 Pe* r H * -]2 p H • LJ ' L<5 ?o2>J Column 15 is To* * 2.137 h^ 2 Column 16 is (T^* ) /d 2 . vdiere d 2 is total water depth at end of section AX, Column 17 is Ks 2 corresponding to column 16, Column 18 is Hg s H^^* x Kg 2 Column 19 is N s -J_ = To* To* Column 20 is H^ax ® 0,707 iTs “n/ loggN, where Hs = (Hs - 0.5) feet. After one line of computations is completed, the next line is begun using Fg » Fg* ♦ AF < 117 where Fg* is from column 14 of the completed line, For exaiiq)le consider the line corresponding to X » 40 being completed. Then 50 j May 1961 Part I Chapter 1 the computation for the next line X = 35 is as follows: Pg* *5 60 from line X = 40 Pg «B 60 ♦ 5 = 65 nautical miles for line 45. Compute Hq s 5.02 40.6 ft. To = 2.13 y40.6 = 13.6 ToV dT « Kg » 0.918 (Pig. 38C) using d-p = 1,65 (304)(40.6)(0.918) 304 H K o s (do.)^ (112) (112) Kf e 0 .89 (Pig. 38D) 2 — using To / dx = 1,65 Ho* « Ho Kf * 40,6 (0.89) = 36.2 feet F/.* s [Ho ] ^ .P6.2 ■|2 » 52 nautical miles L 5 .O 2 J [ 5.02 To* = 2.13 ^36.2 = 12,8 seconds 0.902 CV> /d2 = 1.5 s2 0.915 Hg * 0,915 (36,2) » 33,1 feet, which is the shallow water wave height for depth d 2 = 109 feet, corresponding to MLW of 104 feet, 6120 N s ,To* = 478_or the total number of waves applicable to steady state significant wave of Hg » (33,1 - 0,5) = 32.6 feet, H = 32.6 (0.707) Vloge 478 = 57,2 feet max When refraction becomes important then refraction coefficients must be determined all along the traverse, for the beginning and end of each section AX, and incorporated into the numerical computations. 1.3 CHANGES IN WATER LEVEL 1,31 TIDES - The tide is the alternate rising and falling of the level of the sea caused by the attractive forces of the sun and moon on the rotating earth. There are usually two high and two low waters in a tidal or lunar day. Tides follow the moon more closely than they do the sun. As the lunar day is about 50 minutes longer than the solar day, the tides occur on the average 50 minutes later each day. Because of the varying effects May 1961 51 Part I Chapter 1 of the sun and moon, a diurnal inequality in tides occurs in which, at certain places, there may be little, if any difference between one high water and the succeeding low water of a day but a marked difference in height between the other high water and its succeeding low water. Along the Atlantic coast the two tides each day are of nearly the same height. On the Gulf coast the tides are low but in some instances have a pro¬ nounced diurnal inequality. Pacific coast tides compare in height with those on the Atlantic coast but have a decided diurnal inequality (see Appendix A, Figure A-10). Pertinent data concerning tidal ranges to the nearest foot along the sea coasts of the United States are given in the following tables. Spring ranges are shown for areas having approximately equal daily tides and di¬ urnal ranges are shown for areas having either diurnal tide or a pronounced diurnal inequality. Detailed data concerning tidal ranges are given in Tide Tables, U. S. Department of Commerce, Coast and Geodetic Survey (21) ( 22 ). TABLE 3 TIDAL RANGES Localities Approximates Ranges (feet) From To Mean Diurna1 Spring ATLANTIC COAST Calais, Maine W. Quoddy Head 20-16 23-18 W. Quoddy Head Englishman Bay 16-12 18-14 Englishman Bay Belfast 12-10 14-11 Belfast, Maiae Provincetown, Mass. 10-9 11-11 Provincetovm Chatham 9-7 11-8 Chatham Cuttyhunk 7-3 8-4 Cuttyhunk Saybrook, Conn. 3-4 4-4 Saybrook. Conn. East River, N.Y. 4-7 4-8 Montauk Pt., N.Y. Sandy Hook, N.J. 2-5 2-6 Sandy Hook Cape May, N.J. 5-4 6-5 Cape May, N. J. Cape Henry, Va. 4-3 5-3 Cape Henry, Va, Charleston, S. C. 3-5 3-6 Charleston, S. C. Savannah, Ga. 5-7 6-8 Savannah, Ga. Mayport, Fla. 7-5 8-5 Mayport, Fla. Key West, Fla. 5-1 5-2 GULF COAST Key West, Fla. Apalachicola, Fla. 1-1 2-1 Apalachicola, Fla. Atchafalaya Bay, La. 1-2 * Atchafalaya Bay Port Isabel, Texas 1-1 2-1 PACIFIC COAST Point Loma, Calif Cape Mendocino, Calif. 4-4 5-6 Cape Mendocino Siuslaw River, Ore. 4-5 6-7 Siuslaw River Columbia River 5-6 7-8 Columbia River Port Townsend, Wash. 6-5 8-8 Puget Sound, Wash. 7-11 10-15 * Diurnal tide May 1961 52 Part I Chapter 1 1.32 WIND SET-UP AND STORM SURGE 1,321 General - In most coastal locations the wind may induce a surface current in the general direction of the wind movement, thus causing an increase or decrease in water level above or below that dxie to tidal action. This current results from tangential stresses at the water surface between wind and water and, to a lesser degree, from differences in atmospheric pressure over the water surface. The wind-induced surface current produces a piling up of water at the leeward side and a lowering of water level at the windward side with a return flow along the bottom. The determination of design water elevations, under storm and hurr' ane conditions, represents the most complex problem involving the interaction between wind and water, particularly in shallow depths. The term storm-tide has been used to define that rise above normal water level due to the action of wind stress on the water surface. To differentiate the rise in water level accompanying storms from that due to the astronomical tides, herein the term wind set-up has been used when this action takes place on enclosed lakes and reservoirs and storm surge when applied to the open coast. When the storm surge is due to a hurricane, the term hurricane surge is used and includes that rise in level due to atmospheric pressure reduction as well as that due to wind stress. An example of wind set-up is discussed in a recent study by Hunt^^^^\ He pointed out that the prediction of storm water levels on Lake Erie caused by high winds is becoming increasingly important. Engineers must know ex¬ pected water levels to design harbor improvements and shore protective works. Lake Erie, the shallowest of all of the Great Lakes, with an average depth of some 58 feet, has appreciable wind set-up. Hunt developed a forecasting procedure for wind set-up at Buffalo, N, Y, whereby if the average hourly recorded wind speed and the air-water temperature difference at Buffalo are known, the wind set-up at Buffalo can be obtained. Actual recorded observa¬ tions in Buffalo Harbor show an extreme rise of 9,9 feet above low water datum on 7 December 1909 and extreme drop of 4,2 feet below datum on 30 January 1939, with an extreme range of 14,1 feet. The greatest range for any one year was 12,2 feet in 1909, The foregoing observutions are based on U, S, Lake Survey records. Figure 38E is a typical example of wind effect on Lake Erie. Another recorded example of the inclination of a water surface caused by wind stress is possible as the result of observa¬ tions on Lake Okeechobee, Florida, (155) (158) (112) (159) (160)^ a notable example of this phenomenon occurred during the passage of the hurricane of 26-27 August 1949 over the northern part of Lake Okeechobee, Not only was the lake level caused to incline but particular attention was given to this case when the wind turned 180® during a period of about 3 hours. This turn¬ ing of the wind was accompanied by a turning of height contours of the lake surface, but the latter rotated more slowly than the wind direction so that for some time the wind blew parallel rather than perpendicular to the water level contours (isohypses). Figure 38F gives the contour lines of the lake surface from 1800 hours on 26 August to 0600 hours on 27 August 1949, Special May 1961 52a >jjox MaN ‘aP!l uD3i^ aAoqv 133J ui i3A3"i p3tnduj03 U a: ijj ijj < _i o I— o UJ Li- Ll LlI O I UJ 00 to LU OC Z) o May 1961 52 b «n o May 1961 52 c FIGURE 38F-LAKE OKEECHOBEE, FLORIDA: HURRICANE OF AUG. 26-27,1949 LAKE SURFACE CONTOURS. TABLE 3A-MBAN RANGE AND HIGHEST AND LOWEST TIDES ATLANTIC AND GULP COASTS Place Period of Observation Mean Range ft. Bastpoct, Maine 1930-1953 18.2 Portland, Maine 1912-1953 8.9 Portsaowth, N, H. 1927-1953 wH • 00 Boston, Mass. 1922-1953 9.5 Woods Hole, Mass. 1933-1953 1.8 Newport, R. I, 1931-1953 3.5 Providence, R, I. 1938-1947 4.6 New London, Conn. 1938-1953 2.6 WilletS Point, N. Y. 1932-1953 7.2 Port Haailton, N. Y. 1893-1932 4.7 New York (Battery), N. Y, 1920-1953 4.4 Sandy Hook, N. J, 1933-1953 4.6 Atlantic City, N, J, 1912-1920,1923-1953 4.1 Lewes, Del. (Pt, Miles) 1936-1939,1947-1950,1952- 1957 4.2 Philadelphia, Pa. 1900-1920,1922-1953 5.8 Baltlnore, Md. 1902-1953 1.1 Annapolis, Md. 1928-1957 0.9 Solomons, Maryland 1907-1908, 1937-1957 1.2 Washington, D. C. 1931-1953 2.9 Merehcad City, N. C. 1953-1957 2.8 Norfolk (Sewall Pt.), Va, 1928-1953 2.5 Wilmington, N, C. 1935-1957 3.7 Southport, N. C. 1933-1953 4.1 Charleston, S. C. 1922-1953 5.1 Fort Pulaski, Ga. 1936-1953 6.9 Fernandina, Pla. 1897-1924,1939-1953 6.1 Miami Beach, Pla. 1931-1951 2.5 Key West, Fla. 1926-1953 1.3 Cedar Key, Pla. 1914-1925,1939-1953 2.5 Pensacola, Pla. 1923-1953 1.3 Bayou Rlgaud, La. (Barataria Pass) 1947-1958 1.0* 1.9* Eugene Island, La. (Atchnfalaya Bay) 1939-1958 1.1 Galvestoa, Texas 1908-1953 1.0 Port Isabel, Texas 1944-1957 1.2* 0.0 (* Dlnraal Range) Highest Tides Lowest Tides Alwre Mean High Water_Below Mean Low Water ft. ft. Date ft. ft. Date 4.0 5.0 Nov. 20,1945 3.7 4.2 Jan, 7, 1943 3.0 4.3 Nov. 30,1944, 2.6 3.2 Oct. 8,1915, Nov. 20,1945 Apr. 3,1923, Jan. 7,1943. 2.9 3.9 Nov. 30,1944 2.4 2.8 Mar, 14,1930 3.1 4.3 Apr. 21,1940 2.8 3.5 Jan. 25,1928, Mar, 24,1940 2.8 7.8 Sep. 21,1938 1.8 2.5 Jan. 24, 1936 2.9 10.3 Sep. 21,1938 1.9 2.6 Jan. 25, 1936 3.3 15.6 Sep. 21,1938 2.4 3.0 May 11, 1945 3.5 8.5 Sep, 21,1938 1.9 3.0 Dec, 11, 1943 4.0 9.9 Sep. 21,1938 2.9 3.8 Mar, 24, 1940 2.8 4.0 Feb. 5, 1920 Nov. 10,1932 3.0 4.1 Feb. 2, 1908 3.1 5.5 Nov. 7,1953 2.8 3.8 Mar, 8, 1932 3.1 5.6 Nov. 7,1953 2.7 3.7 Jan, 24, 1936 2.9 5.4 Sep. 14,1944 2.6 3,5 Mar. 8, 1932 3.6 Nov, 23,1953 2.8 Mar, 28, 1955 2.4 4.8 Nov. 25, 1950 3.0 5.1 Jan. 25, 1945 2.7 7.2 Aug. 23,1933 2.9 4.5 Jan, 24, 1908 5.9 Aug. 23,1933 2.9 Sep. 18,1936 Jan. 8, 1929 2.9 Oct. 15,1954 2.2 Mar. 23,26,1928 3.1 8.6 Oct. 17,1942 2.7 2.8 Peb, 15, 1940 Mar. 1, 1949 4.2 Oct. 15,1954 Sep. 19,1955 1.7 Dec, 11, 1954 2.8 6.3 Aug. 23,1933 1.8 2.7 Jan, 23,26, 1928 4.6 Oct. 15,1954 1.6 Peb. 3, 1940 2.4 3.4 Nov. 2,1947 1.2 1.9 Jun. 28, 1934 2.6 5.6 Aug. 11,1940 2.1 2.8 Peb. 15, 1953 2.8 4.5 Oct. 15,1947 2.7 4.1 Mar. 20, 1936 2.7 7.8 Oct. 2,1898 2.4 3.7 Jan. 24, 1940 1.9 3.9 Oct. 18,1950 1.1 1.4 Mar. 24, 1936 1.5 2.6 Oct. 18,1944 1.0 1.4 Feb. 19, 1928 2.5 3.5 Feb. 15,1953 2.6 4.6 Sep. 18, 1947 1.8 7.8 Sep. 20,1926 1.3 2.0 Jan. 6, 1924 3.5 Sep, 19,1947 1.1 Feb. 3, 1951 5.2 June 27,1957 2.3 Jan. 25, 1940 2.8 10.1 Aug, 16,17, 2.3 4.9 Jan. 11,1908 1915 2.4 Aug. 26,1945 1.8 Jan. 17, 1946 Date from! ( Reference 147 and 157) May 1961 52 d TABLE 3B - DIURNAL RANGE AND HIGHEST AND LOWEST TIDES 8 & pn u > w o w 3 M H g « 5 §c 5 5 o; e V u +* X M >- 4> > O Xi < w V T3 ^4= ^.2! SJ ® bO c •H M X V z bO Wi (« cb W 4> o >1 > < (A V J3 tx m O' O' O' O' • • • o «o O' fO fO fM r> m W to to o vO kH »—1 •H o o o tn fO CO to CO (O o (O to 'O O' «o * •o «o to CM W) wo «k O' O' O' O' O' O' O' O' •o O' O' O' O' O' O' O' O' O' fO •H *-l •-t •H •H t-l tH O' •H •-i O' O' CM rH fH iH iH * •k •> «k •k «k •k * •H •> •k •H •k •k •k «k •k •k * (b t" 'C t" 'O C'- (M •> 'O O' •» «k •k O' 00 O' O' to t-l CM •o . c > c e u u o c u c c > > > ft) V ft) ft) 4) ft) ft) (b cb cd o (d (b ft) ft) ft) s 4) X a: •M <0 •n <0 m •o «o 00 'O vO O' CJ O' •H o to o c 0) L. a> H— a> q: c bfi H 10 44 a O' I'- O' to O' o O' fa wo fH 'tr to ■b' o to o 00 6 (b (b (b ft) ft) • • • • • • • • • • • • • • • • • • • • ft) u 4) ft) •H fa to CO to 'b' WO wo fO to to b' fa CM iH o Z ft) > < >■ H— o 00 00 ^-p '<(• to o O' O' O' O (M IH 'O o •H to *H ta wo C" CO pH lO O' • wo •k to WO CO wo wo 00 00 'b' 00 oo fa fa » O' ^ k4 •CJ 00 O' 'O O' O' O' O' O' o O' •k «k •k fH » •H pH pk fH pH Pk Pk >o * (b X (b r- wo f- O' r- o •k o «k m fa Pk • 00 fa b- •o tH ra cva tva oa fH to fO to ta fa ta fa ca iH fH m CM 4) • • • • • • • • • • • • • • • • • • • • • JO B U U c o o u u > p u > > 4* > >• 44 e c c b( ft) ft) ft) «b ft) 3 ft) ft) ft) 0 ft) ft) o o U o o O U (b (b (b •W Wi P P •-) P •-) p P p X (If p 2 X o X x: o o •“> *-0 X +4 X 44 WO to f'a to wo •H O' o to fa oo wo wo 00 O' 'O O' W ft) • • • • • • • • • • • • • • • • p • • • ft) ft) fva fsa ca fsa oa to to O' to to wo wo wo 'b' b- wo fa fa fH o c- wo CO to b" b- wo to wo b' o fa v> • • • p • • • • • • • • • • • p • fa ta ta ta fa fa b" b- to to to fa fa pH ft) tb tx 44 00 ta b- b- t" O' fa ta to f- b" fO e^ O' 0 WO fO c~ c* to C e ft) p p p p p p p p p p p p p P p p p p p p 14 tb ft) wo wo wo wo wo fO 00 00 p4 wo 'O 'O O' 0 0 O' to to to 3 ,,j oc u-> pH pH pH p4 p4 pH ca P c 0 •H to to to to to to to CO to to to pH ca to fO to 00 WO O' to to to •o 44 wo wo wo wo WO wo wo wo WO wo WO b- wo WO wo WO to fa to WO wo wo o tb O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' O' •H > p4 pH fH fH p4 fH pH pH p4 pH pH p4 pH pH pH p4 pH pH p4 pH pH 14 O l4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ft) ft) fO wo b" to 00 to WO WO O' b" 00 'O b' WO 00 0 WO 00 fa b" 'O to to Ql. V) 0 ca ca to O' to fa to O' to pH to b* b- to b- fa pH ca CO b- b- b" p O' O' O' O' 00 O' O' O' 00 O' O' e^ O' O' O' O' O' O' O' O' O' O' O' O pH fH pH fH p4 pH p4 pH pH pH p4 p4 pH p4 p4 p4 pH pH pH pH pH p M (b p p p X M p p <44 <44 p Vi M p <44 ♦H •H JS tb P <44 •H pH pH V) pH X 3 P •H pH tb tb tb tb tb lb 44 p pH tb 0 0 2 fX tb C •0 44 •pH <44 tb 0 c p tA tb tb X X < < pH •H U p p 0 X p p tb tb M (b Vi (b pH p 0 >k bO (0 Si U pH (A cb (A X lb p p p U tb p tb u 44 ft) tb w 0 < tA (b X tb V) pH ki ft) ft) 0 V) u Vi •H (H tb Xi tb pH tn pH tb < 0 > lb U Pk ft) •H *H u 0 pH < tb < pH p .D 0 ffi tb 0 p pH c u p 9 c < pH < ft) (H CJ pH tX) tb ft) 0 c 44 p p X tb •k < - bC lb ft) Of ft) pH bo T. (b c tb tb ft) p >> 44 p tb X IH IH •H pH c l4 ft) •H m pH fH 3 tb • 3 •a k4 K u P 0 < tb Of u U 44 tb xi tb i tb 44 IH 0 X lb *-o 44 (A 0 JC 44 •0 u ft) bb X tb tb X u ft) (A a (A e c ft) 44 tb tb •H 44 C cb 44 X u 44 ft) (A tb tb 0 tb tb l 4 (A ft) ft) iH ft) 3 M •pH tb 4 ) c 3 lb to I-) p-a to to 0 < X to (Ip Ui •-) «0 to >4 (0 < 0 to 2; May 1961 52e Part I Chapter 1 attention is called to the map of contours for 2300 hours on 26 August when the wind at two points was blowing parallel to the highest contours. The effect of storm surge may be significantly higher than normal tidal action in coastal areas; for example, in Galveston, Texas, where the normal tidal range is about 2 feet, hurricanes caused water levels up to 15 feet above mean low water in 1900 and to 12,5 feet in 1915, The amount of surge depends on the wind velocity, the distance over which it blows, and the depth, being greater for lesser depths. This dependence on depth is the reason for the generally greater values of storm-surge observed along the Gulf coast than on the Atlantic, and along the Atlantic as com¬ pared to the Pacific, Surge may also be increased, particularly in coastal areas, by a funneling effect in converging open mouth bays. One method of predicting storm surge in a coastal area is by use of frequency charts obtained from a statistical analysis of water level data. Tables 3A and 3B (147)(157) indicate the extreme conditions of storm surge (including wave set-up)for the period of record of the various locations along the coast of the United States, The frequency of the highest and lowest tide is indicated by the length of the period of observation. The average yearly highest tide is the mean of the highest tides for each year covered during the period of observation. Accumulation of much data over many years in some areas (as the North Sea) has led to relatively accurate empirical methods of prediction for specified points - but these formulas are not applicable to places other than the specific points for which they were derived. However, relatively good estimates may be made for winds approaching more or less perpendicularly to the shore, as discussed by Reid^^O^ and Bretschneider^^l^ These me¬ thods require calibration by at least one storm in the area considered, however, Harris(192) has also given an empirical method relating surge elevation to lowest central pressure, based on a large number of occurrences. This method gives values which are probably correct on the average but, since many other considerations are neglected, may result in values different by several feet from those observed, 1,322 Determination of Wind Set-up and Storm Surge , - The methods presently in use are a recent development, and it is expected that future modifications and revisions will be available after the various theories have been more fully evaluated and substantiated by additional data. The following table (Table 3C) shows the classifications of various wind set-up and storm surge problems. TABLE 3C CLASSIFICATIONS OF WIND SET-UP A, Enclosed Lakes and Reservoirs 1, Rectangular Channel, Constant Depth 2, Regular in Shape 3, Somewhat Irregular in Shape 4, Very Irregular in Shape AND STORM SURGE PROBLEM^ B, Offcoast or on Continental Shelf 1, Bottom of Constant Depth 2, Bottom of Constant Slope 3, Slightly irregular Bottom Prof ile 4, Irregular Bottom Profile May 1961 52f Part I Chapter 1 Table 3C (cont.) C. Coastline 1. Smooth Coastline 2. Coastline S(»iewhat Irregular 3. Jagged Coastline D, Behind Coastline 1, Low Natural Barriers 2, Medium High Natural Barriers 3, High Natural Barriers E. Open Bays and Estuaries 1. Entrance backed by long estuaries 2, and with tidal flow moving freely past entrance 2, Entrance backed by short estuary and with tidal flow moving freely past entrance 3, Entrance constricted sufficiently to prevent free movement of tidal flow past entrance This table may not necessarily be complete and some problems will fall into more than one category, but it is impossible to discuss all of the special cases since there is no one general solution. In the case of an open coast the design still water level must include the astronomical tide as well as the storm surge. Where the astronomical tide is small compared with the surge elevation, it can be added directly to the surge elevation to obtain the design still water level. But where the astronomical tide is large, it should be incorporated in the surge computa¬ tions. The step-by-step procedures for determining the design still water level for either wiixd set-up or storm surge problems are as follows: (1> Select the area to be investigated (2) Obtain all wind and water level data available from past storms and hurricanes (3) Investigate the physical factors which might affect wind set-up or surge elevations and surge computations, (4) With the knowledge of available wind data and the physical features, determine the most suitable approach to the investigation. Perhaps several methods would be equally satisfactory, (5) Outline formulas and procedures for computation, and perform computations and compare with observations. May 1961 52g Part I Chapter 1 (6) Study discrepancies between computations and observations, and attempt to reconcile these discrepancies, either in the data or methods of computation, then make necessary logical and justified changes in the procedures, (7) Repeat the above procedures until satisfactory agreement is reached, (8) Apply the standard project or design storm or hurricane to the area, using the calibrated method, (See following paragraph on standard project hurricane,) (9) Study the results obtained from the design storm or standard project hurricane, and determine whether they are reasonable since it is possible that adverse situations for past condi¬ tions might have resulted in a deceiving type of calibration. Based on detailed analysis of all past hurricanes, statistical evalua¬ tions have been compiled of the hurricane parameters for various zones or areas along the Atlantic and the Gulf coasts of the United States, TTie parameters considered were R, radius of maximum wind, Po» minimum central pressure of the hurricane, Vp, the mean forward speed of the hurricane upon approaching and/or crossing the coast, and U^axt. the maximum sustained wind speed applicable at an elevation of 30 feet above the mean water level. From the statistical analysis the average high and low central pressure for the standard project hurricane, the average high and low radii of maximum wind, and the average, high and low forward speeds are presented for each zone. Theoretical considerations, calibrated by use of actual data, are used to construct the standard project hurricane from the various selected parameters. In the construction of the standard project hurricane, modi¬ fications of the wind and pressure fields are taken into account as the hurricane crosses the coast. Various hurricane reports have been compiled in the Hydro-Meteorological Section of the U, S, Weather Bureau and can be used to compute hurricane surge and waves for certain locations. Wind Set-up in Enclosed Lakes and Reservoirs, - For a wind of constant speed and direction along a rectangular channel and of constant depth, the equations for wind set-up are given by Hellstrom^^®\ Langhaar^^^\ Keulegan among others. ds dx pq (d + s) (slope of water surface) ( (d + s) dx = Ld ^ o /•L J (d + s) dx = Ld (conservation of volume of water for non-exposed bottom) (conservation of volume of water for exposed bottom) (17j) (17k) (17 X) May 1961 52 h Part I Chapter 1 where S is wind set-up X is horizontal distance Tg is wind stress p s w/g mass density, where w = unit weight of water and g is acceleration of gravity, L * length of channel « length of exposed bottom (when applicable), and / s: 1 ♦ where is bottom stress From studies of Lake Okeechobee, Florida a stress parameter k can be determined as follows: ^ I’s/ Pg = k U^/ g, where k = 3,3 x 10~^for enclosed bodies of constant depth, and for the open coast, where the bottom slopes 1‘b/T’s is assumed negligible, k s' 3,0 x 10“^, The solution of equations 17j, k and Ji have been presented in dimen¬ sional form in Tables 3D and 3B from reference (191), These tables can be used to obtain the water elevation S, measured from the undisturbed still water elevation for various distances from X = 0 to X = L, Negative values of S represent a drawdown in water elevation and positive values a rise in water elevation. Near the center of the channel where S goes from negative to positive (S =0) there is a nodal point given by Xj^ for both the non-exposed and the exposed bottom conditions. For the exposed bottom conditions there is a point Xq, denoting that distance be¬ tween X = 0, and X = Xq where the bottom is exposed. Consider the following example: the length L « 30 statute miles, d = 12 feet and U = 50 mph. Compute L e 30 (5280) « 158,400 feet U » 50 (1,47) = 73,5 feet/second use k =3,3 X 10“^ Compute kU^L ^ 3,3 X 10“^ (73,5)2 (158,400) _ q 61 gd^ 32,2 (12)^ 2 From Table 3D for ^ = 0,61 52 i May 1961 TABLE 3D PARAMET® RELATIONS FOR WIND SET-UP IN RECTANGULAR CHANNEL OF CONSTANT DEPTH FOR NON-EXPOSED BOTTOM X/L 0 0.1 0.2 0.3_0^4_0^5_0.6 0.7 0.8_0^^9_1^ VALUES OF S/d CORRESPONDING TO X/L AND 0.201 0.492 -0.104 -0.082 -0.060 -0.039 -0.018 0.002 0.021 0.041 0.060 0.079 0.097 0.209 0.492 -0.109 -0.086 -0.063 -0.041 -0.019 0.002 0.022 0.043 0.063 0.082 0.101 0.218 0.491 -0.114 -0.089 -0.066 -0.043 -0.020 0.002 0.024 0.045 0.065 0.086 0.106 0.228 0.491 -0.119 -0.094 -0.069 -0.045 -0.021 0.002 0.025 0.047 0.068 0.089 0,110 0.239 0.490 -0.125 -0.098 -0.072 -0.047 -0.022 0.002 0.026 0.049 0.072 0.094 0.115 0.251 0.489 -0.132 -0.103 -0.076 -0.049 -0.023 0.003 0.027 0.052 0.075 0.098 0.121 0.265 0.488 -0.139 -0.109 -0.080 -0.051 -0.024 0.003 0.029 0,054 0.079 0.104 0,127 0.280 0.488 -0.147 -0.115 -0.084 -0.054 -0.025 0.003 0.031 0.058 0.084 0.109 0,134 0.296 0.488 -0.157 -0.122 -0.089 -0.057 -0.026 0.004 0.033 0.061 0.089 0.116 0.142 0.315 0.487 -0.167 -0.130 -0.095 -0.061 -0.028 0.004 0.035 0.065 0.094 0.123 0.150 0.337 0.486 -0.180 -0.140 -0.101 -0.065 -0,029 0.005 0.038 0.070 0.101 0.131 0.160 0.361 0.485 -0.194 -0.150 -0.109 -0.069 -0.031 0,006 0.041 0,075 0.108 0.140 0.171 0.390 0.484 -0.211 -0.163 -0.117 -0.074 -0.033 0.006 0.044 0.081 0.117 0.151 0.184 0.423 0.482 -0.230 -0.177 -0.127 -0.080 -0.035 0.008 0.049 0.088 0.126 0,163 0.199 0.463 0.480 -0.255 -0.195 -0.140 -0.087 -0.038 0.009 0.054 0.097 0.138 0,178 0.217 0.511 0.478 -0.286 -0.217 -0.154 -0.096 -0.041 0.011 0.060 0.108 0.153 0.196 0.238 0.571 0.476 -0.324 -0.244 -0.172 -0.106 -0.044 0,014 0.069 0.121 0.171 0.219 0.265 0.648 0.472 -0.377 -0.280 -0.195 -0.118 -0.048 0.018 0,080 0.138 0.194 0.247 0.298 0.750 0.467 -0.452 -0.329 -0.226 -0.134 -0.052 0.024 0.095 0.162 0.224 0.284 0.341 0.894 0.464 -0.587 -0.409 -0.274 -0.160 -0.059 0.032 0.115 0.192 0.265 0.334 0.399 0.930 0.458 -0.614 -0.421 -0.278 -0.159 -0.055 0.039 0.125 0.205 0.280 0,350 0.A18 0.971 0.455 -0.659 -0.443 -0.290 -0.164 -0.055 0,043 0.132 0.215 0.292 0.365 0.434 1.015 0.452 -0.715 -0.467 -0.302 -0.169 -0.055 0.047 0.140 0.226 0.306 0,382 0.453 1.066 0.440 -0.794 -0.494 -0.315 -0.174 -0.054 0.053 0.150 0.239 0.322 0.401 0.475 1.125 0.444 -1.000 -0.526 -0.329 -0.178 -0.051 0.061 0.162 0.255 0.342 0.423 0.500 TABLE 3E PARAMETER RELATIONS FOR WIND SET-UP IN RECTANGULAR CHANNEL OF CONSTANT DEPTH FOR EXPOSED BOTTOM kU^L gd^ X 0 X n XA 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 L L VALUES OF S/d CORRESPONDING TO XA AND XU^L gd^ 1.125 0 0.444 -1,0 -0.526 -0.329 -0.178 -0.051 0.061 0.162 0.255 0.342 0.423 0.500 1.20 0.021 0.438 -1.0 -0.565 -0.345 -0,182 -0.047 0.072 0.179 0.276 0.367 0.452 0.533 1.40 0.070 0.428 -1.0 -0.712 -0.397 -0.198 -0.039 0.097 0.218 0.328 0.429 0.524 0.613 1.60 0.112 0,423 -1.0 -1.000 -0.466 -0.222 -0.038 0.116 0.251 0.373 0.485 0.589 0.687 1.80 0.145 0.423 -1.0 -1.0 -0.556 -0.253 -0.042 0.130 0.280 0.413 0.536 0.649 0.754 2.00 0.175 0.425 -1.0 -1.0 -0.681 -0.292 -0.050 0.141 0,305 0.450 0.582 0.704 0,817 2,20 0.200 0.428 -1.0 -1.0 -1.00 -0.338 -0,063 0.148 0.326 0.483 0.624 0.755 0.876 2.40 0.223 0.432 -1.0 -1.0 -1,0 -0.393 -0.079 0.153 0.345 0,513 0.664 0.802 0.931 2,60 0.244 0.436 -1.0 -1.0 -1.0 -0,459 -0,099 0.155 0.361 0.540 0.701 0.847 0.983 2.80 0.262 0.441 -1.0 -1.0 -1.0 -0.539 -0.121 0,155 0.376 0.566 0.736 0.890 1.033 3.00 0.279 0.446 -1.0 -1.0 -1.0 -0,643 -0.147 0.152 0.388 0.590 0.768 0.931 1.080 3.20 0.294 0.450 -1.0 -1.0 -1.0 -0.806 -0.177 0.148 0.399 0.612 0.799 0.969 1.126 3.40 0,308 0,455 -1.0 -1.0 -1.0 -1.0 -0.210 0.142 0.408 0.632 0.829 1.006 1.169 3.60 0.321 0.460 -1.0 -1.0 -1,0 -1.0 -0.248 0,134 0.416 0.651 0.856 1.041 1.210 3.80 0.333 0.465 -1.0 -1.0 -1,0 -1.0 -0.289 0.125 0.423 0.669 0.883 1.075 1.251 4.0 0.345 0.470 -1.0 -1.0 -1.0 -1.0 -0.336 0.114 0.429 0.686 0.908 1,108 1.290 4.20 0,355 0,475 -1.0 -1.0 -1,0 -1.0 -0.388 0.102 0.433 0.701 0.933 1.139 1.327 4.60 0.375 0.483 -1.0 -1.0 -1.0 -1.0 -0.517 0.074 0,440 0.730 0.978 1.299 1.399 5.00 0.392 0.492 -1.0 -1.0 -1.0 -1.0 -0,712 0.041 0.443 0.756 1.021 1.255 1.466 5.40 0.407 0.505 -1.0 -1.0 -1.0 -1.0 -1.0 0.0011 0.443 0.778 1,060 1.307 1.520 6.00 0. 428 0.511 -1,0 -1.0 -1.0 -1.0 -1.0 -0.068 0.438 0.808 1.114 1.381 1.621 7.0 0.456 0.528 -1.0 -1.0 -1.0 -1.0 -1.0 -0.218 0.418 0.847 1.194 1.492 1.759 8.0 0,480 0.542 -1.0 -1.0 -1.0 -1.0 -1.0 -0.433 0.386 0,877 1,263 1.593 1.885 9.0 0.500 0.555 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 0.342 0.897 1.323 1.683 2.000 (Prom Ref. 191) May 1961 52j Part I Chapter 1 Interpolate and obtain the following values: X/L S/d X (Statute Miles) S (feet) 0 -0.351 0 -4.22 0.1 -0.262 3 -3.14 0,2 -0.183 6 -2.20 0.3 -0.112 9 -1.35 0.4 -0.046 12 -0.55 0.5 0.016 15 0.19 0.6 0.075 18 0.90 0.7 0.130 21 1.44 0,8 0.183 24 2.20 0.9 0.233 27 2.80 1.0 0.282 30 3.44 The nodal point is obtained as 82 miles from the lee end of the X„/L lake* s 0.474, whence X^ s 0.474 (30) a If the wind speed had been 76 raph instead of 50 mph, one would obtain 1.40 and the bottom would be exposed and Table 3E would be use^, the far end of the lake would be at X/L » 1,0, and for (0.613) = 7.35 miles. The bottom would be exposed for Xq » 0.07 (30) « 2,1 miles. The set-up at = 1.40, S = 12 The above examples illustrate the ideal situation, which seldom occurs in practice. The depth, d, may vary from one end of the lake to the next and wind speed may vary in magnitude and direction. It then becomes neces¬ sary to resort to numerical computations. When numerical computations are made, it is necessary to have a starting point. It appears that the easiest approach is to begin at the nodal point. An estimate of the nodal point can be made by computing an equivalent value of kU^L gd2 i * m Y BikUUjjAX i = 1 ism i Z Bi fiX L i = 1 where the lake is now divided into M sections, aX is the length of each section (L » MAX if aX is the same for each section), U is the absolute value of the wind speed and Ux is the component of wind speed along the axis of the lake for which computations are made, di is the mean depth May 1961 52 k Part I Chapter 1 corresponding to each section, and is mean width of the corresponding kU^L section. Once — 5 - has been determined, use Tables 3D or 3E to determine gd^ the nodal point. The equations for computing the wind set-up are given as follows: AS. 1 / 2NkUU„ AX X ■*•1 -1 (17m) g “Ir = d. ♦ 1 i = M-1 1 ‘Si i = 1 (17n) S B i s I i s M 1 AS. 1 (17o) In the above equations: SS. is incremental rise (or fall) in water elevation over the corres¬ ponding increment &X dj^ * mean depth of section, exclusive of any set-up. d^ = total water depth including the rise from all preceding sections but not including that for the section under consideration. S represents the total set up. When computations are performed along the axis in the direction of Ux positive, beginning at the nodal point will be positive. In the opposite direction from the nodal point ASf will be negative. In equation 17m there is incorporated a planform factor, given the symbol N, This is in analogy to a refraction coefficient and takes into account the convergence or diverg¬ ence of the energy prism due to changes in bottom depth and width of section. The planform factor is not usually required, provided the sections are taken at small enough intervals, A check on the computations can be made by com¬ puting the volume balance of water using the following equation. i = m X * 0 i = 1 where Bi is the mean width of the section corresponding to S^, the average value over the increment ASi . The above method for making determinations can be used easily for ob¬ taining reasonable estimates of wind set-up. Refinements in the procedures and computations can be made to obtain better estimates if required by project considerations. May 1961 52 I Part I Chapter 1 X (Naut. Mi.) TABIJB 3P WIND STRESS (Based on Figure 38G) k = 3.0 X 10“^ (k/g) UU^ dx (k/g) UU^ fix X X (feet^) (feet^) k = 3.3 X 10"^ (k/g)UU^ AX (feet^) 0 0 0 0 5 5 5 5.5 10 15 10 11.0 15 30 15 16.5 20 50 20 22.0 25 80 30 33.0 30 120 40 44.0 35 170 50 55.0 40 230 60 66.0 45 295 65 71.5 50 365 70 77.0 55 435 70 77.0 60 500 65 71.5 65 500 60 66.0 70 615 55 60.5 75 665 50 55.0 80 715 45 49.5 85 760 45 49.5 90 800 40 44.0 95 840 40 44.0 100 875 35 38.5 105 915 35 38.5 110 945 30 33.0 115 975 30 33.0 120 1000 25 27.5 125 The third column in the above table represents the stress to be used for the hurricane over a sloping bottom and the last column over an enclosed body of constant depth. May 1961 52 m 002‘l<-^^-1- ^^^^ -'000*21 May 1961 52 n Part I Chapter 1 The following example illustrates the procedure for a lake of variable depth and width using a variable wind field (see Figure 38G and Table 3F foi wind stress relations). In this particular example it will be assumed that the overall length of the lake is 50 nautical miles, and the mean width Bland mean depth di of each section is as given in Table 3F, It will also be assumed that 10 sections of the lake are sufficient in number to neglect any planform factor, (See Figure 38H and Table 3G,) Perhaps 20 sections would be more satisfactory. The top of the table will correspond to the lee side of the lake where drawdown occurs and the bottom of the table will correspond to the end of the lake for which set-up occurs. Several positions of the stresses given in Table 3P should be used to obtain the optimum con¬ ditions, For the purpose of this sample, it will be assumed that X = 40 kUlL , 2 nautical miles in Table 3P where ^ » 66 ft, corresponds to the end of the lake for set-up. The last column of Table 3P is used for the stresses, since the bottom is more-or-less constant in depth. It might be noted that if the hurricane passed over land to the lake in question, the wind speeds would be reduced and there would be a corresponding reduction in wind stresses. The following table is prepared. TABLE 3G -WIND SBT-UP kUU^ AX kUU AX B. A 1 X Bi ^i g gdi2 Section XA. (Naut. Mi.) (Naut. Mi.) (feet) (feet^) (Naut. Mi.) 0 0 1 0.1 5 10 12 49.5 3.44 2 0,2 10 15 16 49.5 2.91 3 0.3 15 20 24 55.0 1.91 4 0.4 20 25 20 60.5 3.78 5 0.5 25 25 20 66.0 4.12 6 0.6 30 20 24 71.5 2.13 7 0.7 35 25 36 77.0 1.48 8 0.8 40 20 24 77.0 2.67 9 0.9 45 15 20 71.5 2.68 10 1.0 50 10 16 66.0 2.58 Av. = 18. 5 Av. = 21,2 Tot. 27.70 May 1961 52 0 Part I Chapter 1 B£ ® “lo X ®i “ 18,5 nautical miles kUU^ fiX B. - - - - = 27,70 nautical miles gdi The equivalent value of “Sr = 2TJ0 , 1 50 18.5 gd Using Table 3E for = 1,50, the nodal point is estimated to be at Xjj/L = 0,426 or = 8,426 (50) » 21,3 nautical miles. The bottom is exposed to X^/L = 0,087 or to Xq = 0,087 (50) * 4,35 nautical miles. At a point 21,3 nautical miles from the beginning of the lake is the nodal point where computations begin. The first increment will be AX » 25 - 21,3 * 3,7 nautical miles in length followed by 5-mile increments for the others. For this increment, one must use = 66 (Id) . 49 g The total depth to use for this section will be df * 20 feet (section 5) and the incremental rise over this short sectiwi will be AS. = 20 [ -a -1 1 * 2,31 feet, total rise at end of ^ 20^ section 5, The total water depth for the ext Section, Section 6, will be dj = 24 ♦ 2,31 = 26,31; the incremental rise over Section 6 will be 6 S. = 26.31 r /8.(71.5) ~ .i] , 2.60 feet ‘ (26.31)'’ J The total rise at the end of Section 6 will be S = 2,31 + 2,60 = 4,91 feet. The total water depth to use 40,91 feet. The incremental rise for Section 7 will be dj = 36 + 4,91 « over Section 7 will be: AS. s 40,9ir /2(77) ^ -ll= feet " IV (40,91) J The total rise at the end of Section 7 will be S = 4,91 ♦ 1,85 s 6.76 feet. 52 p May 1961 |33j u; uo||DAai3 52 q May 1961 Distance in Nautical Miles FIGURE 38-H. EXAMPLE OF WIND SET-UP COMPUTED SURFACE PROFILE FOR AN ENCLOSED LAKE (FIRST APPROXIMATION) Part I Chapter 1 The above computations are repeated for the other sections and the re¬ sults are given in Table 3H. Computations are also given for the drawdovm end of the lake where AS£ are negative. For example, the remaining incre¬ ment Section 5,is 5-3,7 = 1,3 nautical miles from the nodal point, and one must use kUUx &X g 66 (lii) 5,0 17,2 feet^ and AS -20 [.yi^ +1 20 ‘ = -0,85 feet For Section 4, dj » 20 - 0,85 « 19,15 feet and A S^ 19 .15 r /SiSlIII ^ -ll* -2,95 feet LV (19.15)2 J andS * -0.85 ^(-2.95) * 80 feet Similar computations as above for both +S and -S have been made for all sections and finally the check for volume balance has been made. All these computations are summarized in Table 3i. TABLE 3H - WIND SBT-UP COMPUTATIONS X “i S ‘^T B i s B.S (Naut, Mi.) (feet) (feet) (feet) (feet) (Naut, Mi.) (feet) (ft.-Naut.M 0 12 (limit) (-12) 1.33 10 -11.33 -113.3 5 16 -4.20 -10.67 9.53 15 -8.57 -128.5 10 24 -2.67 -6.47 20.20 20 -5,64 -113.0 15 20 -2.95 -3.80 19.15 25 -2.34 -58.5 20 20 f -0.85 ^ +2.31 -0.85 ) 20 25 +0.73 + 18.1 25 24 ♦2.60 2.31 26.31 20 3.61 72.2 30 36 1.85 4.91 30.91 25 5.84 146.0 35 24 2.40 6.76 30.76 20 7.96 159.5 40 20 2.35 9.16 29.16 15 10.33 155.0 45 16 2.30 11.51 27.51 10 12.66 126.6 50 13.81 L B^S = 264.1 ft.- -Naut .Miles 52 r May 1961 Part I Chapter 1 The value of ^ BiS = 264.1 ft.-Naut, miles represents an error re¬ sulting from the approximation used to obtain the nodal point. The error in equivalent feet of elevation is obtained by dividing by ^ Bi = 185, whence 264,1/185 = 1,43 feet. The set-up at the far end of the lake will be between 13,81 feet and (13.81 - 1.43) s 12.39 feet. The error of 1,43 feet also means that the nodal point is actually closer to the center of the lake than was computed by use of theory. An estimation of the next approximation of the nodal point can be made by plotting S versus X of Table 3F and shifting the origin upward by 1,43 feet. When this is done the next approximation for the nodal point will be at X s 22.7 nautical miles approximately instead of at 21,3 nautical miles as originally used. Finally a repeat of the above computations will result in a more accurate answer. Figure 38H shows the results of the first approxi¬ mation. It might now be mentioned that several additional traverses might have been selected parallel to the main axis of the lake, and there would be different values of the stress parameters for these other sections. Con¬ sequently additional refinements can be made by use of additional traverses through the lake. It must be determined, however, whether such refinements are justified, because of the additional work required and the incorporated errors present in any numerical method» However, it is necessary to repeat the above computations by moving the stress diagram at various positions along the lake until critical or maximized conditions are reached. For example, it may be necessary to repeat the above example by shifting the nodal point until it has been determined that optimum conditions have been reached by selecting various positiwis of the hurricane stress diagram. Once the optimum position has been found, then refinements can be made. It might be of interest at present to make an estimate of set-up by use of one simple computation. In the above example: vti2t —* 1.48, and from Table 3F gd at XA = 1.0, S/d = 0.635 Using the average depth of the lake from Table 3G, di = 21,2 feet, S = 21,2 X 0,635 = 13,4 feet which in this case is as accurate as 13,8 feet given in Table 3H and is certainly sufficiently accurate as a reasonable estimate of the hurricane wind set-up. For final design, however, a more accurate value might be necessary, in which case the problem should be investigated in greater detail. Hurricane Surge Determinations on the Open Coast - The determination of hurricane surge for the open coast is a more complex problem than that for an enclosed body of water. The problem becomes even more complex when May 1961 52 s Part I Chapter 1 the surge from the open coast is to be routed through the entrance of an open bay or estuary, where it is required to take into account the con¬ figuration of the body of water, the propagational speed of the surge, and possible effects of cross winds. Typical examples of these complexities can be surmised from the details and efforts made for various particular locations given in references (213) (214) and (215), wherein it was re¬ quired to obtain very refined answers for hurricane surge elevations. The methods presented here are limited to those which might be used to obtain reasonable estimates for hurricane surge elevations on the open coast. On the open ocean for hurricane conditions, in addition to the wind stress, there will be an increase in water elevation due to atmospheric pressure reduction from normal, and in some cases, particularly for the steeper continental shelf areas where large waves are propagated shoreward there will be an increase in water elevation due to the transport of water by wave action, i.e., breaking waves. The speed of the hurricane in re¬ lation to the speed of the free gravity long wave for the particular depth may result in an amplification of the hurricane surge.(See reference (190).) Storm surge estimations can be made by use of reference (190), or reference (191). Reference (191) requires only numerical integration of the following formulas: d = d (MLW) ♦ As ♦ Si * S2 (17p) where d is the design still water depth d (MLW) is mean low water depth Ag is astronomical tide is rise in water level due to atmospheric pressure reduction from normal, S- is rise due to wind stress r -- 1 Sj = 1.14 1^1 - e J (17q) where APq = PN “ Po» atmospheric pressure reduction at the center of the hurricane from normal, P© is central pressure, Pn is normal or some¬ times called the asymptotic pressure. The factor 1,14 converts the pressure from inches of mercury to feet of water, R is the radius of maximum wind, and is the radial distance measured from the center of the hurricane to the zone of maximum winds, r is the radial distance to some point of interest measured from the center of the hurricane. For example at r = R, position of maximum windjS^ * 1.14 APq ^ 0.63, and at r = 0, center of hurricane, = 1.14 aPq. The rise in water level due to wind stress is given by: N ^2 = Z (17r) 52 t May 1961 Part I Chapter 1 where 2kUU ax X g(d^)^ ♦ 1 -1 (17s) In the above $2 is equal to the sum of ASi over all increments of AX at the beginning of the first increment N = 1 to the beginning of the Nth increment. That is, dT ® d minus aS^ of the last increment for each step of integration, or a_ a d (MIW) ♦A + S, + Z AS, (17t) sill 2 where g = acceleration of gravity (32,16 ft/sec ) U a absolute value of the wind speed U a Component of wind speed in direction of computation (along the selected traverse). AX is the increment of horizontal distance over which UU acts, UU^ changes with distance through any section of a hurricane, k a stress parameter and is 3,3 x 10 ^ for enclosed bodies of water of constant depth. On the continental shelf where the bottom is not of constant depth, it appears that k a 3,0 x 10“^ is most suitable. Equation 17s can be used for the continental shelf in a manner similar to that discussed previously for an enclosed lake, except that that compo¬ nent due to atmospheric pressure reduction and the astronomical tide must be taken into account. In case of the enclosed lake a nodal point deter¬ mination was required, and the amount of water above the normal still pool level forward of the nodal point must be equal to the amount of water below the normal still pool level behind the nodal point. For the continental shelf, however, there will be a continued influx of water from an unlimited supply source, and one can assume the nodal point at the edge of the con¬ tinental shelf and no drawdown seaward but only a continued rise shoreward. The procedure then is to begin at the continental shelf and make computations shoreward, taking into account the atmospheric pressure reduction from normal. The following example is given to illustrate these procedures, using wind stresses from Table 3F and the continental slope off the mouth of Chesapeake Bay, Figure 381 shows the bottom profile off the mouth of Chesapeake Bay, based on an average of three east-west traverses. Summary of sturge computations are given in Table 31 for the condition May 1961 52 U still Water Level o 00 iaej uj io q;daa May 1961 52 V Distance from Coast in Nautical Miles FIGURE 38-I.MEAN BOTTOM PROFILE OFF MOUTH OF CHESAPEAKE BAY Part I Chapter 1 kUU^ AX 2 of the stress diagram located over the traverse with --- * 50 ft, at the last section, corresponding to X » 35 nautical miles of Table 3P, Colmn 3, TABLE 31 Summary of Surge Computations X “i (feet) -*2 (feet) d (feet) kUUx AX g ‘^T S 65 1000 500 750 35 750 0,05 0,05 60 500 190 385 40 385,05 .10 .15 55 190 154 172 40 173,15 ,23 ,38 50 154 116 135 45 135,38 .33 .71 45 116 112 114 45 114,71 .39 1,10 40 112 110 111 50 112,10 .45 1,55 35 110 104 102 55 103,55 ,53 2,08 30 104 82 93 60 95,08 ,63 2,71 25 82 70 76 65 78,71 ,82 3,53 20 70 60 65 70 68,53 1,02 4,55 15 60 52 56 70 60,55 1.15 5,70 10 52 42 47 65 52,70 1,22 6,92 5 42 32 37 60 43,92 1,35 8,27 0 32 24 28 50 36,27 1,35 9,62 The value of S « 9,62 feet is close to the optimum value. but other positions of the stress diagram should be selected and computations made for a check. Actually if the stress diagram is moved a sufficient number of times over all increments. one would obtain a steady state hydrograph of S versus the distance that the hurricane is from the coast. Knowing the speed of the hurricane the time hydrograph of S for steady state conditions can be determined. To the above value of 9,62 feet must be added that component due to atmospheric pressure reduction from normal, as well as the astronomical tide range. In fact, it is more accurate to include the atmospheric pressure rise and the astronomical tide in the numerical integration pro- dure. However, in this particular example the atmospheric pressure rise May 1961 52 w Part I Chapter 1 and the astronomical tide are small compared with the depth of water, and can therefore be added linearly. In this example the atmospheric pressure reduction at the center of the hurricane is about 2.4 inches of mercury corresponding to 1,14 x 2,4 » 2,73 feet of water rise at the center of the hurricane. The 9,62 feet of surge due to wind stress is not at the center of the hurricane but is near the radius of maximum wind. In general this component of surge can be found from equation 17q: Si . 1.14 [ 1 - e * ] where R is the radius of maximum wind and r is the radial distance to the point of interest, and in this example r s R, whence s 1.14 (2.4) (0.63)*: 1,73 feet Thus the total hurricane surge is equal to 9,62 -f 1,73 » 11,35 feet. May 1961 52 X Part 1 Chapter 1 1.33 SEICHES^ ^ Seiches are standing waves of relatively long period which occur in lakes, canals, bays and along open sea coasts. Originally the word seiche (pronounced sash) designated long-period, free oscillations in lakes. liowever, common usage now applies this descriptive term to free oscillations of all relatively small bodies of water. The mechanics of seiche generation are not completely understood, although all available evidence proves rather conclusively that lake seiches are the result of a sudden change, or a series of intermittent-periodic changes, in atmospheric pressure and similar changes in wind velocity. Standing waves in canals can be initiated by suddenly adding or substracting appreciable quantities of water. Seiches in bays can be generated by local changes in atmospheric pressure and wind, the same as lake seiches, and by oscillations transmitted through the mouth of the bay from the open sea. Open-sea seiches can be caused by changes in atmospheric pressure and wind, earthquakes and sub¬ marine landslides. Standing waves of large amplitude are likely to be generated if the causative force which sets the water basin in motion is periodic in character, especially if the period of this force is the same as, or is in resonance with, the natural or free oscillating period of the basin. Free oscillations have periods which are dependent upon the horizontal and vertical dimensions of the basin, the number of nodes of the standing wave, and friction. Friction can usually be neglected unless the basin is very long and shallow. The period of a true forced-wave oscilla¬ tion is the same as the period of the causative force. Forced oscillations, however, are usually generated by Intermittent external forces and, in this case, the period of the oscillation is determined partly by the period of the external force and partly by the dimensions of the water basin and the mode of oscillation. Oscillations of this type have been called forced seiches(18) to distinguish them from pure seiches in which the oscillations are free. In a closed basin, wave loops must be situated either at the ends (longitudinal seiche) or the sides (transverse seiche). A closed rectangular basin with length L, average depth d, and n the munber of nodes, has a natural free oscillating period; 2L Tn = - (18) nxfid The fundamental and maximum period, when n = 1, becomes: T = - (19) Equation 19 is called Merian’s formula(123). In a rectangular bay the simplest form of standing wave is one with a node at the opening and the loop at the closed end of the bay. The period of the free oscillation in this case is: T = \l^ (the total length of the bay is occupied by only one-fourth of a wave ( 20 ) length.) May 1961 53 Part I Chs5)ter 1 Considerable modification is necessary in applying the above single theory to lake or bay oscillations because of the variations in width and depth along the axes of actual basins. The theory of free oscillations in basins of various particular shapes, has been developed by many workers (18) (19)(U8)(52)(5U)« Defant (37) developed a convenient but rather laborious ■athod of determining the possible periods of free oscillation in lakes of any shape, Defant*s method is considered the most useful in engineering work because it permits the conputation of periods of oscillation, relative magni¬ tudes of the vertical displacements along chosen axes, and the positions of nodal lines and loops. This method, which is applicable only to free oscilla¬ tions, can be used to determine the modes of oscillation of binodal as well as uninoded. seiches. The the^^ for a particular case of forced oscillations was also dertved by Defimt (37) and is discussed by Sverdrup (123); Recent investi¬ gations by Hunt (156) pointed out some of the coirplexities involved in the hjrdraulic problems associated with Lake Erie and he has offered an interim solu¬ tion to the problem of the vertical displacement of water at the eastern end of the lake which is so important in engineering design, 1,3U LAKE LEVELS - The Great Lakes have only insignificant tidal variations, but are subject to seasonal and annual changes in water level and to changes in water level caused by wind set-up, barometric pressure variations, and by seiches. The average or normal elevations of the lakes* surfaces vary irregu¬ larly from year to year. During the course of each year the surfaces are sub¬ ject to consistent seasonal rises and falls, reaching their lowest stages during the winter months and attaining their maximum stages during the summer months, Ifydrographs of monthly lake levels from the year I860 to the present are shown in Figure 39 (13U)# Table U summarizes certain lake level data, TABIE U - FLUCTUATIONS IN WATER LEVELS - THE GREAT LAKES _ PERIOD 1860 - I960 INCLUSIVE _ tow* Wean Change in Stage Water Surface Between Monthly Alltime Monthly Means Datura Elev, Means Max, Max, fegh Lew differ- rise fall ence take ^erior 601,60 602,20 6oh.O^ ^99.90 h,07 Lake Michigan- Huron 578.50 580,54 +0,85 -0,69 583.68 577.35 6,33 Lake Erie 570,50 572,34 +1.57 -0.73 57U.70 569.)i3 5.27 Lake Ontario 2Uli,00 2U6,03 +1.90 -0.95 2U9.29 2U2,68 6.61 Elevations are in feet above mean tide at New York, 1935 datum, ♦Low water datum is the plane to which depths in Federal navigation improve¬ ments and on Lake Survey charts are referred. In addition to the seasonal and annual fluctuations, the lakes are subject to occasional seiches of irregular amount and duration. Sometimes these result from variations in barometric pressure, which may produce changes in water surface elevation ranging frtMn a few inches to several feet (75), At other times the lakes are affected by wind set-up which raises the level at one end and lowers it at the other end of a lake (5l)o In general, the maximum amounts of these irregular changes in lake level must be determined for each location under consideration. May 1961 54 UNITED STATES LAKE SURVEY CORPS OF CNCINEERS. U S ARUT UuA>|ui FIGURE 39a. HYDROGRAPH OF MONTHLY MEAN LEVELS OF THE GREAT LAKES 55 MOF 1961 Part I Chapter 1 1.4 CHARACTERISTICS OF BEACH MATERIALS The beach, the nearshore and in some cases the offshore area to con¬ siderable depth is included in the littoral zone. The composition of the material therein may be defined by analyzing samples of surface material. Customary procedure is to obtain samples of the top 2 inches of material at the mean water line and along the profile at 5-foot or 1 fathom depth intervals seaward as far as regular sorting pattern along the profile is indicated. Any sampling device that will secure a representative san^le without loss of fines is suitable. For sandy material, l/2 to 1-pint samples are usually adequate. Spacing of sampling profiles depends upon regularity of the shore and slopes, and on long straight beaches profiles quite widely spaced will usually provide adequate definition of the distribution of materials in the littoral zone. The physical properties of the littoral material include size (usually in millimeters) of the individual grains, the shape and roundness of the grains, mineral composition, porosity, and permeability. Of these, size and mineral composition are the most important and generally will be the only properties examined in detail. Since the size distribution is of primary concern, all samples should be subjected to mechanical analysis. Sieving or settling velocity methods are equally acceptable for this purpose. Uniform size classifications (based on Casagrande classification) approved by the Corps of Engineers, March, 1953 are given in the following table. TABLE 5 - STANDARD SIZE CLASSIFICATIONS * Name Grain size limits (diameters) Cobbles Coarse gravel Fine gravel Coarse sand Medium sand Fine sand Silt or clay Above 3 inches (above 76mm) 3/4 inch to 3 inches (19 mm to 76 mm) No. 4 sieve to 3/4 inch (4*76 mm to 19mm) No. 10 sieve to No. 4 sieve (2.0 mm to 4*7 mm) No. 40 sieve to No. 10 sieve (0.42mm to 2.0 mm) No. 200 sieve to No. 40 sieve (0.074 inm to 0.42 Below No. 200 sieve (below 0.074 nim) * Corps of Engineers Uniform Soil Classification U. S. Standard Sieve Size Preparatory to mechanical analysis, the samples are washed free of salt, dried and quartered down to about 50 grams for sieving, or to 5-7 grams if a settling velocity tube is used. After determining the relative percentages by weight of the grade sizes contained in the material, cumulative size distribution curves are constructed (76) from which the grain diameter (in millimeters) at the first-quartile (Q^),third quartile (Q^), and the median are read. Twenty-five percent of the sample by weight has a grain diameter larger than the diameter of the first quartile and seventy-five percent larger than the diameter of the third quartile. The 57 Part I Chapter 1 median is the point at which 50 percent of the material has a larger grain diameter and 50 percent smaller. The three statistical parameters employed to express the characteristics of the size distribution are; the median diameter, the coefficient of sort¬ ing, and the skewness. All of these parameters are derived from the cumulative size distribution curves. The median diameter is the mid-point of the grain size diameters in millimeters contained in the sample, and is read directly from the 50 percent figure on the cumulative size distribution curve. Fifty percent of the total weight of the sample is composed of particles with a diameter greater than, and fifty percent is smaller than the median diameter. The coefficient of sorting (Sq) Is a measiire of the spread in grade sizes represented in the sa mple of the littornl material. It is determined by the formula Sq = '\/Qp/Q 3 . If there is perfect sorting the value of would be unity. A value of 1.25 is Indicative of good sorting in the beach material, and 1.45 for material from the nearshore bottom. The skewness (S^) is a measure of the degree of symmetry of the size distribution with respect to the median. It is derived from the formula Sk = Qi Q 3 /M ^2 . If the value of the skewness is unity, the point of maximiam sorting coincides with the median diameter, if the value is greater than unity the maximum sorting lies on the fine side of the median diameter, and if it is less than unity the maximum sorting lies on the coarse side of the median diameter. The greater the value of the median diameter the coarser is the material, the larger the value for the coefficient of sorting the more poorly sorted is the material, and the more the value for skewness diverges from lanity the more unsymmetrlcal is the size distribution curve. Small values for both Sq and indicate the material is in adjustment with its environment. A large value for Sq and a small value for indicate the material is spread through many grade sizes. A small value for Sq and a large value for indicate that the material ranges through many grade sizes and that one set of environmental factors is dominant, though traces of others are still retained. Large values for both Sq and Sj^ indicate that the sediment is completely out of adjustment with its environment. If the source of material is uncertain, petrographic analysis may provide evidence by comparison of mineral content on the beach and at possible sources. Samples should be about 250 cc. or about l/2 pint in size. This size sample will permit both a sieve analysis and microscopic examination for mineral content. The first step in the microscopic examination for mineral content is to separate the "heavy minerals", (minerals with a specific gravity greater than 2 . 85 ), from the quartz and feldspar. This is accomplished by panning, by use of heavy liquids, by use of electromagnets, or by some special method or device. The minor accessories, or so-called "heavy minerals", even 58 Part I Chapter 1 though present in very small amounts (tenth of 1 percent or less) are the important elements in determining source sands. The next step involves the Identification of minerals and determination of the frequency of their occurrence. In this the percentage of total heavy minerals in the sample is computed. The heavy minerals are then subdivided to a quantity sufficient for mounting on a microscopic slide. After identification, the frequencies are determined by actual count or estimation and recorded for comparative piarposes. The abundance of a mineral species can be expressed in percent by number or in percent by weight. The frequency in percent by number is obtained by counting the mineral grains in carefully spaced fields under the microscope. To determine the frequency by weight, the relative per¬ centage of the mineral species in closely sized fractions and the weight of the total separate is first determined, and then the weight percentage is computed from the relative percentage. Corrections for size and specific gravity must be made to make the values obtained comparable with the percent by number. Mineral frequencies in the different grade sizes of a sample are not alike, hence, in the comparative study of many samples it is necessary to study the same grade size in all samples or to study the same relative size within the distribution curves. The variations in the mineral composition of a suite of samples within an area are related to the physical conditions of deposition. The reasons for the variation in the frequency of a given mineral species may be due to contamination by the addition of locally derived species to the littoral drift; by selective breakage and abrasion, since all minerals are not equally resistant to these two processes tending to eliminate the mineral dioring transportation; and by selective sorting by littoral processes. An increase in the percent of the frequency of occurrence of a mineral does not mean an absolute increase but may mean only a relative increase which is the complement of the decrease of another mineral, since the total must always add up to 100 percent. The probable error in the value for the frequency of occurrences of a mineral species is greatest for the rare constituents and lowest for the abundant species. Complete descriptions of laboratory procedures in making mechanical and petrographic analyses of sands are given by Krumbein and Pettljohn(76). Table 6 lists the fifty most common detrital minerals of sands. The most common minerals in beach sands are capitalized. Light minerals are underlined; all others are the heavy minerals. 59 Part I Chapter 1 TABLE 6 DETRITAL MIMRALS IN SANDS (Pettijohn, lOO) 1. Actinolite - tremolite 26. HORNBLENDE 2. Anatase 27. HYPERSTHENE - ENSTATITE 3. ANDALQSITE 28. ILMENITE 4. APATITE 29. KYANITE 5. AUGITE 30. LEUCOXENE 6. Barite 31. Limonite 7. Biotite 32. MAGNETITE 8. Brooklte 33. Monazite 9. GALCITE 34. Muscovite 10. Cassiterite 35. Olivine 11. Chalcedony 36. RUTILE 12. Chloritoid 37. Serpentine 13. CHLORITE 38. Slderite u. Clinozoisite 39. Sillimanite 15. Cellophane 40. Spinel 16. Cordierite 41. SPHENE 17. Corundum 42. STAUROLITE 18. DIOPSIDE 43. Topaz 19. dolomite 44. TOURMALINE 20. Dumortierite 45. Vesuvianite 21. EPIDOTE 46 . Xenotime 22. Fluorite 47. ZIRCON ■ 23. GARNET 48. ZOISITE 24. V Glauconite 49. FELDSPAR 25. Hematite 50 . QUARTZ 60 Part 1 Chapter 2 CHAPTER 2 LITTORAL PROCESSES 2.1 GENERAL - The existence of an erosion problem is prima facie evidence that one of two conditions exist. The first is that the water level and the land have not become adjusted in terms of shore slope, as in the case of exceptionally high lake levels or storm tides, and upland material is being eroded to establish equilibrium slopes. The second and more common con¬ dition, which may exist concurrently with that cited above, is that material being removed from the area exceeds that being supplied. This material, designated "littoral drift" may be defined as the material moved in the littoral zone under the Influence of waves and currents. The term is some¬ times considered as the movement of the material as well as the material itself, however, in a strict sense the term littoral drift should be re¬ stricted in meaning to the material, and the term littoral transport used where movement is meant. A stable shore line is one in which the supply of material to the area under consideration is approximately equal to losses of material from the area. On an accreting shore line the supply of material exceeds the losses, and the reverse is true of an eroding shore line. Accordingly, the need for protective works and the choice of type of protective works to be provided are dependent on the net balance between supply and loss of material. Analysis of an erosion problem preparatory to functional planning of remedial measures requires that conclusions be reached from the most re¬ liable data available concerning the nature of the littoral drift and littoral transport within the problem area. The information required can be placed in three basic categories: a. Sources and characteristics of littoral materials b. Modes and direction of littoral transport c. Rates of supply and loss of material It will not always be possible to reach well supported conclusions for these categories; nevertheless careful investigation v^ill provide knowledge in place of conjecture in many cases. 2.2 SOURCES AND CHARACTERISTICS OF MATERIALS 2.21 GENERAL - The characteristics of materials found in littoral zones have been discussed in Chapter 1, Section 1.4. The three main natural sources of material to any beacn segment are: (a) material moving into the area by natural littoral transport from adjacent beach areas; (b) con¬ tributions by streams; and (c) contributions through erosion of coastal formations, other than beaches, exposed to wave attack. In addition. 61 Part I Chapter 2 there may occasionally be some long range net movement of material onshore apart from normal seasonal or other periodic fluctuationso The latter might occur, for example, with permanent or semipermanent changes in water level. Considering coasts as a whole, maintenance of beaches must be attained at the expense of erosion of the land mass. For any individual segment of beach, the largest source of material moving into the area is generally littoral drift eroded from the adjoining updrift segment 'unless some major sediment bearing stream enters the segment in question or cliff or dune erosion is sufficiently rapid to provide appreciable supply. Caution should be exercised when determining the source, as the material on any one beach may be the product of several source areas or it may be obtained from only one of several possible so'urces. A study of the beach environment, the relative availability of material in the possible source area, the agents of erosion that are active, and the conditions favorable to transport material from the source areas to the beach site will generally indicate the source or sources of supply. Petrographic analysis of samples of the littoral material and of samples of possible source materials may establish a correlation between the mineral content of the littoral material and that of a source area. The correlation might be established through a similarity in the frequency of occurrence of a particular mineral or mineral suite, or by identifying a specific mineral or variety of a mineral unique to the littoral material and to the material from one of the source areas. 2.22 CONTRIBUTIONS BY STREAMS - The amounts of the various contributions to the littoral supply by sand carrying streams can be determined approx¬ imately by these general methods; (a) direct measurements, (b) studies of terrestrial sedimentation, and (c) computation of the sediment carrying capacity of the streams. To date, the only method upon which any great degree of reliance can be placed is that of direct measurements. Direct measurements may be made with considerable accuracy 'under certain conditions. Delta measurements by successive hydrographic surveys are adequate to determine the amount contributed by those streams which carry sediment to the ocean or lake only during flash floods lasting a re¬ latively short period of time. Similar comparative surveys are adequate to determine the amoixnt contributed by streams which bring material to the shore continuously, or over an extended period of time, where those streams termiriate in navigable channels or other natural settling basins. Some correction may be necessary to account for sediment deposited outside of the channels or basin area, for material moving out of the area between siarveys by natural littoral transport, and for material removed from the area artificially during maintenance of navigation channels and basins. Should investigation show that the principal sources of beach-building materials are the drainage basins tributary to the shore under considera¬ tion, a detailed study of the geology of these basins may be required if direct measurements are impracticable. Such a study should include data on hydrology, physiography, petrology, and sedimentology, and the sediment supply deduced from measured or estimated rates of terrestrial sedimentation. 62 Part I Chapter 2 For mountainous watersheads, the Forest Service has developed empirical methods for estimating the sedimentation rate(l). The determination was for a specific area of known geologic characteristics, but the results are applicable for other areas if corrections are made for several variables, i.e., vegetation, hydrology, rock type, etc. Even if there are excellent data on terrestrial sedimentation rates, it may be very difficult to estimate how much material reaches the shore. The measurement of losses must be indirect. If the streams are degrading or appear to be at grade, it can be assumed that all soiorce material ultimately reaches the shore. But if the streams are aggrading, deposition rates along the channel must be estimated and losses subtracted from the total sedimentation to determine the net sediment supply to the beaches. The method generally employed in the computation of the sediment carrying capacity of a stream is the direct measurement of the wash-load of a river by suspended-load sampling(35). This method is expensive and time consuming, taking from 1 to 10 years of continuous observation to predict the wash-load of a river, depending on the regularity of its flows. A different method(40) (41) has been developed which is based on grain size of the bed materials. Its basic concept is that bed material always moves according to the capacity of the stream. The capacity rates for bed material may be computed by formulas which were developed to permit the prediction of the individual bed-load rates of the different bed components in terms of stream discharge. The solutions are laborious, but are not difficult to follow. However, the method would probably be used only if a determination of the stream carrying capacity was of great importance and could not be determined by direct measurements or historic records. 2.23 CONTRIBUTIONS BY EROSION OF COASTAL FORMATIONS - Eroding coastal formations are the last major source of beach material. Along the Great Lakes this is a major source, whereas, along much of the sea coast it is of comparative unimportance. As long as a beach berm is maintained between the formation and the action of the waves, the formation contributes negligibly to the littoral supply. At some locations, littoral transport has been interrupted by artificial barriers and the ocean has turned to the upland for its supply, causing serious recessions of the coast line. The amount of such contribution can only be estimated through comparative surveys, sub-division plot maps, property surveys, and statements of long-time residents of an area. The formations frequently contain much material too fine to remain on the beach. The proportion of beach material supplied out of total material eroded may be determined by mechanical analysis of a composite sample. Each stratum should be represented in proportion to its thickness. In the Great Lakes area, rises in lake level may allow waves to attack bluffs, which are generally of a very friable material. This causes recession of the shore line and contributes to the supply of beach material. Where erosion of coastal formations is important, a geological study may be required. The extent of field work and investiga¬ tion will depend on the importance of erosion of coastal formations as a source of littoral material. 63 Part I Chapter 2 2o3 MODES OF LITTORAL TRANSPORT - Waves and currents supply the necessary forces to move the littoral materials. The mechanics of littoral transport are not precisely known, but it may be generally stated that littoral material is moved by one of three basic modes of transport: (a) Material, known as ’’beach drift”, moved along the foreshore in a more or less scalloped path due to uprush and backwash of obliquely approaching waves; (b) Material moved principally in suspension in the surf zone by littoral currents and the turbulence of breaking waves; (c) Material moved close to the bottom by sliding, rolling, and saltation, collectively known as ’’bed load", within and seaward of the surf zone by the oscillating currents of passing waves. Significant bottom movement has been observed in depths exceeding 100 feet on exposed sea coasts. Figure 40 illustrates the three basic modes of transport, and Figure 41 (ill) depicts trends of proportionate bed load transport related to wave energy and steepness from laboratory experiments. Regardless of the mode of transport, the direction and rate of littoral drift depend primarily upon the direction and energy of waves approaching the shore. Exceptions exist on short reaches of shore adjoining tidal inlets where the tidal currents pattern may be dominant. 2.31 DEPTHS AT WHICH MATERIAL MOVES - Wherever sandy beaches exist or where the surf zone and nearshore bottom are composed of sand, the grain size of material along a shore profile decreases generally as the water depth increases until depths are reached where normal wave currents are incapable of moving bed material. The coarsest material is usually found in the surf zone in the vicinity of the plxinge point of waves, though in protracted periods of mild wave action the foreshore and surf zone material may be nearly equal in size. This gradation of grain sizes along the beach profile is due to the process of the slope sorting of beach materials which is Illustrated in Figures 42 and 43. Slope sorting is ascribed generally to the differential velocity in the oscillating wave currents in shallow water. Whereas, the velocity of water particles is theoretically uniform about the orbital path for particles in deep water, moving in the direction of wave propagation while the crest is passing and in the opposite direction with passage of the trough, the uniformity ceases when the wave begins to feel the bottom. Deformation takes the form of steepening the crests, shortening them in relation to lengths of the troughs, so that the particles move forward with the wave crest in less time than they return with the trough. Figure 44 illustrates laboratory measurements of particle movement in water of various depths. The effect of this phenomenon is to transport coarser particles of bed material shoreward. Where the slope becomes sufficiently steep, gravity counterbalances the current effect and an equilibrium con¬ dition results. Currents introduced by reflected waves likewise affect the balance of forces in this region near the shore. Because of slope sorting, the material in littoral transport moves generally within a depth range compatible with its size or resistance to transport. The actual path or rate of transport of irldividual particles or groups of particles cannot be stated from present knowledge. It is known that foreshore and nearshore slopes are related to the grain size 64 BACK SHORE CREST OF BERM-^, ,1 ^ STILL WATER LINE t 1 PATH OF BEACH DRIFT V 1 1 ' DIRECTION OF WAVE INDUCED CURRENT IN THE SURF ZONE MATERIAL PLACED IN SUSPENSION BY BREAKERS IS MOVED LATERALLY BY THE LONGSHORE CURRENT, AS WELL AS LATERALLY AND IN A ZIGZAG PATTERN ALONG THE BED. PATH OF SAND GRAINS OUTSIDE SURF ZONE BED LOAD MOVES UP OR DOWN COAST IN A ZIGZAG PATTERN. MOVEMENT IN ALL THREE ZONES ILLUSTRATED IS IN A DIRECTION AND AT A RATE DEPENDENT ON THE LONGSHORE COMPONENT OF WAVE ENERGY. LATERAL MOVEMENT OF LITTORAL DRIFT FIGURE 40 Wave Steepness Ho/Lo LABORATORY RESULTS INDICATING TRENDS OF PROPORTIONATE BED LOAD AND SUSPENDED LOAD TRANSPORT RELATED TO WAVE ENERGY AND WAVE STEEPNESS. (Saville, 1950) FIGURE 41 65 Percentogc Variation Of fl^edion Diometers Compored With Reference Somple SEASONAL VARIATION OF SIZE DISTRIBUTION IN SEDIMENTS ALONG "D" RANGE OFF LA JOLLA, CALIFORNIA (Inman, 1953) FIGURE 43 66 Part I Chapter 2 material (see Figure 45) of which they are formed, however, this relation¬ ship is not the same at all localities since it is also influenced by water level variability, wave exposure, and ground water level. Although particle density and shape are factors in transportability, median grain size is a satisfactory parameter for evaluating generally the transport¬ ability of littoral material„ 2.32 DETERMINATION OF DIRECTION AhD DIRFCTION VARIABILITY - It is not only necessary to know the direction of littoral transport at any one time — which can generally be determined by observation of shore con¬ figuration in the vicinity of existing structures — but the predominant direction of littoral transport over a normal climatic cycle must be establishedo This may also involve locating the position of natural and lunnatural littoral barriers and those areas called nodal zones in which the net littoral transport changes direction. In these zones the net littoral drift is zero or, in other words, the downdrift components of littoral drift are equal to the updrift components. Although the methods used in determining the direction of littoral transport may differ from place to place, determination of the Instantaneous and predominant directions of littoral transport and the location of littoral barriers and nodal zones may ordinarily be accomplished by analysis of such of the following factors as may be required to reach conclusions; Accretion or erosion effects of existing structures; Shore patterns in the vicinity of headlands; The configuration of the banks and beds of inlets and streams; Statistical analysis of wave energy; Characteristics of beach and bed materials; Current measurements (particularly in the vicinity of inlets)o 2o321 Effects of Existing Structures - This provides the most reliable means of determining littoral transport characteristics, and will ordinarily outweigh all other evidence. The use of existing structures to determine the direction of littoral transport is illustrated in the series of aerial photographs. Figures 46 to 51, inclusive. Considering th4 evidence pre¬ sented by groins. Figures 46 and 47, the condition of the beach at the time of Inspection probably indicates the direction of littoral transport during the immediately preceding periodo To determine the predominant direction of littoral transport requires ovservations at regular intervals over a period of at least one year to avoid misinterpretation due to seasonal effect. Considering the evidence presented by breakwaters and entrance jetties. Figures 48, 49, and 50, the quantities involved are generally large enough so that the condition observed at any one particular time probably is in¬ dicative of the predominant direction of littoral transport, with incremental changes showing the short term direction variability. 2.322 Evidence at Headlands - The evidence presented by headlands as to the direction of littoral transport is not ordinarily as clear as 67 a: Ui H < a. UJ UJ o 1 It IM — Trova _ s 4 0 4 0 4 1 0 L 2 1 4 14 t/T H/L>0 0S4. tf/L> 0.449 H T C d g (ft) (MC) (f»/Mc) (fO _ 0146 0.74 y69 1.218 IIM SURFACE TIME HISTORY x«0l M«on perlicle position ot y:0f e = Oond S ’1037ft • Meosured position at equol time miervols H *0 146 ft. • Meosured position,second cycle T’0.74sec • Meosured position ,third cycle C’ 369ft/5ec d ’I 218ft 0 «l 50 PARTICLE ORBIT ABOUT MEAN PARTICLE POSITION WITH NO MASS TRANSPORT ac UJ H- < $ O < X tft , \ ff 1.x ^■%i 0 at 04 M 04 10 f/T _Approiimote eiperiinenlol surfoce . Wove22-23,M/L’002l.d/L=0057 wove 26-2T H/L •• 002l.d/L = 0-057 H L T C d (ft) tftj Isec) {ft/soc3 Ift) Wove 23 0 106 5 10 162 2 83 0 292 wove 2 7 0 105 5 10 1 62 3 64 0 292 — Approiimote experimental distribution . e ’ O.Wove 23. H/L «0 02l.d/L *0 057 e ’ 0,Wove2 7, H/L* *0 021,d/L* 0057 . e -180,Wove 23 • 0 :i80,Wave27 H L (ft.) (ft) Wove 23 0106 5 lO Wove 27 0 105 5 10 T C d (sec) (ft/sec) (ft) 162 2 83 0 292 162 3 64 0 292 SURFACE TIME HISTORY HORIZONTAL MAXIMUM PARTICLE VELOCITY DISTRIBUTION . Meosured position ot equol time mtervols • Meosured position, second cycle Wove 22-23 H ’0 i 06 ft L ’ 5.10ft C»2 03 ft /sec T =162 sec d’0 292fl particle ORBIT ABOUT MEAN PARTICLE POSITION WITH NO MASS TRANSPORT UJ > < $ o z < UJ IT (D • I >0. d’ 0330lt . t * 0. d= 0 330ft ot i/(CgT)’000 ond l.OOIvoned for eochi/fC^T)) Hp T ® (ft) (sec) (ft/sec) (ft) _ ■oTre Tei 400 0 330 I 50 0 0262 SURFACE TIME HISTORY - crest _ --SWL -/] -Ifouqh-- A. 1 • L - Theory i Equotion 2) _Approximote experimentol = 0 = 180 y(ft) 0 Wd ve irov r. • ♦ /' .* N 'lEnd N. Beqin ling I* ■ 04 .0 3 .0.2 .0.1 (ft) (sec) (ft/sec) (ft) 1276 141 400 0 300 0 344 1 50 0 0262 HORIZONTAL MAXIMUM PARTICLE VELOCITY DISTRIBUTION Mean porticle position ot 8=0ond S=0 267 Measured position ot equol lirhe mtervols Measured position,second cycle Q276ft d = 0 330fi I 41 sec 0 = I 50 4 00fl /sec H/L =00262 0 330 ft PARTICLE ORBIT ABOUT MEAN PARTICLE POSITION WITH MASS TRANSPORT Legend S s elevofion of fhe porficle obove the bottom(ft), in this cose the surfoce porticle d : still woter depth (ft) H : wove height (ft.) L = wove length (ft ) t s time (sec) T : wove period (sec) u : honzontol porticle velocity,(ft per sec ) e = ongulor position of porticle in its orbit meosured counterclockwise ond where there is no moss tronsport, (degrees) * s horizontol position of porticle in its meon honzontol position (ft) C = wove velocity of propogotion (ft./sec) y = verticle position of porticle in its meon honzontol position « : beoch slope e ’ subscript - refers to breaking conditions Equotions i,4,5,0ond lO ore from Stokes' second opproximotion Equotion 2 is from Stokes' first opproximotion Equotions 9 ond ii ore reduced forms of Stokes' second opproximotion TIME HISTORIES, PARTICLE VELOCITY DISTRIBUTIONS AND ORBITAL PATHS. OF WAVES IN DEEP WATER,IN SHALLOW WATER, AND AT BREAKING FIGURE 44 ( Monson ond Crooke.i953) 68 69 iTzo iT30 iT4(5 Tso Heo TTfO iTeO iigo Foreshore Slope RELATIONSHIP BETWEEN GRAIN SIZE AND FORESHORE SLOPE FIGURE 45 Part I Chapter 2 that presented by structures because of the frequency of rocky shores on both sides of headlands. In some instances the headland is so oriented as to cause a reversal of direction of littoral transport under all wave con¬ ditions, thus compartmenting the coast line. Figure 52 show two types of headlands. The headlands depicted in Figure 52a permits passage of littoral drift even though no beach exists on the headland itself. Figure 52b is illustrative of a headland which may act as a littoral barrier. Wave cut cliffs with no sand beach usually mark the downdrlft shore, whereas relatively wide stable beaches are found on the updrift shore. 2.323 Evidence at Tidal Inlets and Streams - The location and formation of tidal inlets may also be Indicative of the direction of movement of littoral drift. In general, over a long period, such Inlets tend to migrate in the direction in which littoral drift is moving. Brief re¬ versals, associated with the shifting of the bar channel, are often ob¬ served. Natural closure and break-through at an npdrift location may confuse the evidence. Figures 53 and 54 show typical inlet formations and the manner in which they indicate generally the direction of littoral transport. Offset of the channel and delta at stream mouths, where they are not artificially controlle(4 is often indicative of the direction of littoral transport. The channel offset is usually toward the downdrift side. 2.324 Wave Analysis - A complete knowledge of the directional components of wave energy acting upon the littoral zone will permit deduction of the direction and energy value of the longshore component of the primary force responsible for littoral transport. No satisfactory instrument for measuring wave direction has yet been devised, thus there are no con¬ tinuing instrumental records which can be used for tnis purpose. Two methods have been employed for developing statistical wave data, the first being reports over a long period by ships at sea, compiled by the U. S. Navy Hydrographic Office and published as "Sea and Swell Charts." The second method, applied first by Scripps Institution of Oceanography on the California coast (117) and later by the Beach Erosion Board for the Great Lakes(ll3) and a portion of the North Atlantic coast,(102) involves ap¬ plying wave forecast techniques to produce wave statistics from historical synoptic weather maps, (see section 1.23) Both methods provide statistical data on wave heights and directions in deep water. The latter method also provides wave periods associated with height and direction, enabling energy evaluation. When other reliable evidence as to predominant littoral transport direction is lacking, the longshore wave energy component has been employed for that purpose. There are two general methods for making this computa¬ tion, the first by simple vector force diagrams to determine the resultant force in deep water parallel to the general shore alignment. The second method is a refinement of the first, involving projecting the deep water wave energy to a position near the shore by refraction analysis, and com¬ puting the longshore component at that point. For comparatively regular shore alignment and bottom topography (see Figure 55), there will be little difference in results from the two methods. For more complex 70 Direction Of Littoral Transport Santa Monica, Calif. Jan. 1946 FIGURE 46. EFFECT OF SINGLE GROIN ^ I rtf Groins/^ Direction of littoral transport Alomitos Bay Beach Long Beoch, Colif. Dec. 1948 FIGURE 47. EFFECT OF A SERIES OF GROINS 71 FIGURE 48 EFFECT OF OFFSHORE BREAKWATER Accretion Erosion Entrance jetties Direction of littoral transport FIGURE 49 EFFECT OF ENTRANCE JETTIES 72 Bollona Creek ,Calif. Jan. 1946 Erosion Accretion Breakwater Accretion Direction of littoral transport Sonto Barbara FIGURE 50 EFFECT OF SHORE CONNECTED BREAKWATER Direction of littoral transport Slight erosion ’ile type pier Santa Monico Beach,Calif. 9’3 Photo by Fairchild FIGURE 51 EFFECT OF PILE CLUSTERS Aerial Surveys, Inc. 73 7 , J*' Sand passing around headland Headland Direction of littoral transport Pt. Mugu, Colif. Jon. 1946 'FIGURE 52a DIRECTION OF LITTORAL TRANSPORT INDICATED BY HEADLANDS J^lv^^i^nstoble Zone Socket Beoch. Wider At Down- drift End.-^ Direction Of Littoral Tronsport- ‘Narrowing, Fairly Stable Beoch, X / 1 ' • § P ^ ' ^ V ut Cliffs^ )r No BeachL. Wave Cut Cliffs ^ Little Or No Beach'^ San Juan Capistrano Point FIGURE 52 b. DIRECTION OF LITTORAL TRANSPORT INDICATED BY HEADLANDS 74 erosion Direction of littorol transport Mugu Lagoon, Calif. Jon. 1946 Barrier beach FIGURE 53 TIDAL INLET TO BAY OR LAGOON FIGURE 54 TIDAL INLET THROUGH A BARRIER BEACH 75 Part I Chapter 2 topography, local inconsistencies are usually found. There is at present no proven technique for employing refraction analysis to determine transport direction, and when results of this method are in conflict with other more reliable evidence, its use for general practice should be tempered with the available evidence from other sources. Beach FIG. 55— REFRACTION DIAGRAM-WAVE CONDITIONS BY ORTHOGONALS 2.325 Variations in Material Characteristics - A comparison of the median diameters of a series of samples taken in the same manner along a coast frequently indicates a progressive trend in the variation of the grain size of the material. In ordinary cases of sediment transport (upland) the median grain size will decrease with distance from the source, thus indicating the predominant transport direction. Because of the wide variation in grain size along beach and nearshore slopes; the effect of exposure and underwater topography upon slope sorting; and the disruptive effects of varied sea conditions which may occur just preceding or during a sampling program, such evidence is not a reliable indication of predominant littoral transport. May 1961 76 Part I Chapter 2 A progressive variation in the total heavy mineral content in a series of samples taken along a coast may indicate the direction of littoral trans¬ port since these are less subject to frequent dislocation. Heavy minerals have a tendency to remain on the beaches, while the light minerals often are carried out beyond the plunge zone. When this occiirs, the relative percentage in total heavy mineral content contained in the samples will Increase with distance from the source of the light minerals. A progressive decrease in the frequency of a particular mineral may also indicate the direction of littoral transport; the frequency decreasing with distance from the source area. 2.326 Current Measurements - Measurements of littoral current may some¬ times give an indication of the direction of littoral transport, but they require much time and are frequently unreliable. They must be made at frequent intervals over a full year to be of value, and if reversals in direction and wind velocity variations are observed, they cannot be evaluated in terms of littoral drift. The most common methods used to obtain the direction and velocity of the currents are by use of floats outside the breaker zone and fluorescein inside the breaker zone. Fluorescein is a yellowish-red crystalline compound which receives its name from the brilliant yellowish-green fluorescence of its alkaline solutions. Fluorescein can be purchased at moderate cost from most chemical firm.s in quantities of one pound or more. A common method of employing fluorescein is to place a handful of dry sand in a paper towel, or other substance, which would disintegrate readily, with a heaping teaspoon of fluorescein crystals. The entire mass is twisted within the towel and tossed seaward into the breaker zone. As the towel disintegrates and the crystals dissolve, a small patch of water is dyed an easily distinguish¬ able brilliant yellowish-green. The movement of this patch of colored water in a longshore direction can be traced from the beach, noting the distance traveled and the time required for the travel. All measurements should be to the center of the colored area, which will gradually disperse until no longer distinguishable. From the distance moved and the time elapsed during the movement, the velocity of the current can be computed. Time permitting, 3 readings should be taken at each station and averaged. For the best effect, these readings should be taken twice a day at regular intervals along the beach under study, with concurrent measurements of wave height, period and direction. As the wind affects the shallow water currents, its velocity and direction should also be recorded. Current measurements seaward of the breaker zone are generally made with floats. In general, these react still more erratically than does the fluorescein. In the low current velocities common to this area, wind conditions have considerable bearing on the floats unless great care is taken. Although many types of floats have been used, most of them follow the same general pattern of movement. Floats should be low in the water and designed to offer the least wind resistance and the maximum water resistance possible. The floats should be released in sequence along the entire length of a profile, spaced at regular Intervals from 200 to 400 feet. Each float should have a distinctive color flag or marker to permit its identification. Locations of the floats at regular time intervals may 77 Part I Chapter 2 be made by transit intersection. These measurements are generally con¬ tinued through one or more tidal cycles. Because of the short time interval, this type of current measurement does not represent the seasonal changes and can be used only in connection with other observations. Two general types of subsurface floats are used, the rod float and the vaned float. The rod float, of uniform dimensions, gives an approx¬ imation of the integrated values over the depth covered by the float. The vaned float gives an approximation of the current velocities at the depth of the vanes. Floats of too great length will drag on the bottom and give improper readings. Floats of comparatively short length record only surface currents. The float lengths should be such as to penult selection of the proper length float for the range of depths along a profile. This selection of proper float lengths is based on experience at the site after a trial run. One design for a vaned float is shown in Figure 56. Pegram current meters are used on extensive studies to measure bottom currents. These are not generally used in the determination of direction of littoral transport because the information gained is seldom commensurate with the cost of the operation. Balls of slightly greater density than the water have been used to roll along the bottom dragging a very light line and float to make their location. 2.33 MTES OF LITTORAL TPtANSPORT - The rate of littoral transport is as important as the direction of movement of littoral drift in the functional and structural design of shore protective structiares. The rate of littoral transport can only be measirred accurately at a substantially complete artificial littoral barrier. At such barriers this rate can be computed by measuring either the accretion at the updrift side of the barrier or the erosion at the downdrift side. Accretions can be measured at partial barriers, but no method has been devised to determine what proportion of the total littoral drift is trapped by each partial barrier. Until some such method is devised, the measurement of material trapped by groins or short jetties is a most Inadequate way of determining the rate of movement of this drift. Natural littoral barriers are of little use in determining the rate of littoral transport because over geologic time the beaches either updrift or downdrift from these barriers tend to reach a condition of stability where the sand supply equals the sand losses. Typical examples of essentially complete, substantially complete, and temporary artificial barriers are shown in Figures 57, 58, and 59. In these examples, and in all similar cases, the rate of littoral transport is determined by measuring the amount of accretion or erosion occurring during a known period of time. To compensate for seasonal changes, surveys should be taken at about the same time each year. To compensate for annual fluctuations, the period of time between surveys should be extended as con¬ ditions permit. The rate of movement should be expressed in amount of drift per unit time, usually a year. Where the rate of littoral transport is to be established at a littoral barrier, the base surveys should be extended a sufficient distance updrift and downdrift from the barrier to include the entire accretion and erosion 78 Varies FLOAT ASSEMBLY White Orange FLAGS 30g. Aluminum Flags in each set should differ in shape and color FIG. 56 TYPICAL VANE FLOAT ASSEMBLY 79 80 FIGURE 57 ESSENTIALLY COMPLETE LITTORAL BARRIER 81 FIGURE 58. SUBSTANTIALLY COMPLETE LITTORAL BARRIER bgrrier unti bor reaches 82 FIGURE 59. TEMPORARY LITTORAL BARRIER Part I Chapter 2 zones at the end of the study period. Where erosion is anticipated, the base line should be referenced to points at a considerable distance from the ocean as recessions of the shore line of 1,000 feet or more are not uncommon. Profiles are run from the base line seaward at least to the 30-foot depth contour, although extension of the profiles to greater depths may be required in areas of severe exposure or in the vicinity of sub¬ marine canyons. Profiles should be spaced in conformity with accuracy desired and the degree of regularity of the area.(114) Profiles may be run by any standard hydrographic survey method. Where the amoiint of such work to be done is large emd continuing, echo sounding equipment and amphibious vehicles are advantageous. Care should be taken to insure accurate vertical control and measurement as small uncompensating vertical errors result in large quantity errors. Measurement of accumulations on the updrift side of jetties or long groins provides a good basis for estimating the rate of littoral transport. Depth of water to which the structures extend, and the character of material trapped, must be considered in evaluating the im¬ poundment rate in comparison with total littoral drift. Short groins provide a poor means for measuring the rate of littoral transport, because the amount of material trapped is usually a small and indetermi¬ nable part of the entire quantity of littoral drift. Shoaling rates in entrance channels may provide an estimate of littoral drift in locations where maintenance dredging is done frequently. However, this method can seldom be used because of the difficulty of separating shoaling caused by reversals in the direction of littoral transport from that caused by the predominant direction. When there are no suitable littoral barriers at which the transport rate may be determined from surveys an approximate "order of magnitude" value can be obtained from the relationship between alongshore wave energy component and rate of transport shown in Figure 59-A. This re¬ lationship was compiled by Savage (200) utilizing data of other in¬ vestigators [Caldwell (201), Watts (136), Savage (204), Shay and Johnson (202), Saville (ill), Krumbein (77)J €Uid later supplemented by data of Fairchild (205). When using this relationship, the along¬ shore wave energy component should be obtained by using the computed deep water wave energy and the refraction analysis described in Section 2.324* If the wave characteristics are measured in relatively shallow water depths of 30 feet or less, these characteristics may be used to compute the energy directly, ignoring refraction. Energy values em¬ ployed in compiling the relationship shown in Figure 59-A were derived as follows: (a) Energy data at Anaheim, Calif. (201) and South Lake Worth Inlet, Fla. (136) were computed using wave ch€U*acteristics measured in about 20 feet of water; (b) Energy data from model studies (77), (202), (204) were obtained using deep water wave characteristics ignoring refraction; May 1961 83 Part I Chapter 2 (c) Beach Erosion Board model energy data were computed using deep water wave characteristics and considering refraction to the point where the waves breedc; and (d) Fairchild's energy data (205) were obtained from wave hindcasts to obtain deep water wave characteristics, and wave refraction to the 30-foot depth contotu* was considered. (It is suggested that computations of this type using deep water wave characteristics should include consideration of refraction to the breaking point of the waves,) Use of the above relationship for energy versus rate of transport requires that ample consideration be given to sources and availability of littoral materials to determine whether quantities of the order given by the relationship are available to be transported by the available alongshore energy component. 2,34 LOSSES OF LITTORAL MATERIAL - Principal avenues of loss of littoral material from a specific beach area include (a) movement of material laterally out of the eureaj (b) movement of material offshore into water of sufficient depth that it is lost to the littoral supply; (c) loss of material into submarine ciuiyons; and (d) loss of material inland. Loss of material by abrasion of sand has been found of slight importance.(83). 2.341 Losses by Longshore Transport - The movement of materials out of the area is measured by the net rate of transport at the downdrift end of the beach segment under study. It may be that this loss can be measured directly as outlined previously. If it cannot be measured directly at a particular location, it may be possible to estimate it by considering the rates of transport at the two closest known points above and below where the rate has been established or can be measured directly. At best this is a rough estimate as the unknown factors of added supply and losses throughout the area must also be taken into consideration. 2.342 Movement Offshore - It has been observed that changes in the shore profile occur with changes in water depth or in wave characteris¬ tics. Profile adjustment due to change in water depth is relatively slow, and therefore, minor with respect to single tidal cycles. Meas¬ urable change has been detected when compeurisons are made with water level expressed as daily mean sea level. The profile adjustment due to change in wave characteristics is rapid, though usually temporaiTr. A single storm of a few hotirs duration may cause a major change in pro¬ file. Profile changes may be ascribed primarily to the onshore or off¬ shore shifting of beach and bottom material. Generally, the shift is offshore as the water level rises or the waves steepen, and onshore as the water level lowers or the waves are flatter. Laboratory studies indicate that the critical wave steepness defining the boundary between onshore auid offshore movement is in the order of Ho/Lq = 0,025. This has not yet been confirmed in nature. The continual onshore and offshore shifting of material results in a longshore movement of material at a sluggish rate and may be classed as a type of transport, the relative importance of which is unknown at this time. May 1961 83a FIGURE 59-A RELATION BETWEEN ALONGSHORE COMPONENT OF WAVE ENERGY AND LITTORAL TRANSPORT RATE May 1961 83 b Part I Chapter 2 The quantity of material lost to the offshore depths cannot in itself be determined in the light of present knowledge. It is possible that as information on material sorting with respect to slope and wave characteristics is developed, equations may be evolved by which this possibly important avenue of material loss may be evaluated. At present, it can only be assumed as the amount of loss remaining after all known losses have been subtracted. As the rate of material supply into an area is increased to exceed the transport capacity out of the area, or as the transport capacity along a beach decreases, either sediments acciimulate along the coast line or losses occur to the offshore depths. As these deposits reduce the depth, the beach slope assumes a profile governed by the littoral forces and the beach material. Assuming material character¬ istics to remain constant in gradation, the profile of equilibrivim would be reached when all of the beach material has been sorted roughly. Each size gradation assumes its characteristic slope, depending on the wave competence, between minimum and meiximum depth limits governed by the material size gradation. Continued excess supply would advance the berm seaward without appreciable change in profile, causing deposit of sediments in greater depths. 2.343 Losses in Submarine Canyons - The existence of a submarine canyon in the littoral zone provides a repository for important losses of material into the offshore depths. When combined with a jetty or break¬ water, the sxjbmarine canyon may constitute an essentially complete littoral barrier by drawing off all material passing around the jetty or break¬ water. Comparative surveys have been in insufficient detail to enable determination of the extent of the losses of littoral material into a submarine canyon. 2.344 Losses by Deflation - As a beach widens and the expanse of per¬ manently dry sand increases, the losses by deflation (removal by wind action) increase, generally resulting in the development of a dune belt immediately behind the beach. Rates of loss by deflation are generally difficult to determine. In some instances loss can be determined by measurement of the changes in dune size between successive surveys. Such a measurement would generally be more costly than the information would bo worth unless the problem of dune control must be considered as well as the losses of material from the beach. In general, losses by de¬ flation are not an important factor in the design of shore stinictures. However, because of the aspects of dune control, some attempts have been made to devise means of measuring the amount of deflation and to relate the quantities of beach sand moved to the wind velocities. Experiments at the mouth of the Columbia River(96) give an indi¬ cation of the order of sand losses by wind deflation. According to typical sieve analyses, these beach sands had a median diameter of 0.19 millimeter. Three types of sand traps were tried, two of which were in good agreement as to the measured rate of sand movement. Wind veloc¬ ities were measured during each mm at points ranging between 0.25 and 12 feet above groimd. Sieve analyses of the sand caught in the traps and of the material from the simface of the beach showed variations in median diameter between 0.165 8uid 0.216 millimeter. The specific May 1961 84 Part I Chapter 2 gravity of the sand was 2.65 and the grains were well rounded. The rate of sand movement was related to the wind velocity 5 feet above the beach. The measurements showed that when the velocity at this elevation was less than 13.4 feet per second (9 miles per hour) no movement of sand occurred, but movement was general at this velocity and above. Figure 60 shows wind velocity gradients taken diiring typical runs. Figure 61 shows the relation between wind velocity and rate of sand movement. The rate of movement is in terms of the nmber of pounds per linear foot of beach passing a given line in one day. May 1961 85 velocity in ft./sec. .sje, WIND VELOCITY GRADIENTS FIGURE-60 Velocity of Wind in ft./sec, 5 ft. obove Beoch RELATION BETWEEN WIND VELOCITY OF SAND MOVEMENT FIGURE - 61 AND RATE (O'Brien, 1936] 86 Part I Chapter 3 CHAPTER 3 PLANNING ANALYSIS 3.1 GENERAL - In selecting the shape, size, and location of works the ob¬ jective should be to design an engineering work which will accomplish the desired results most economically and with full consideration of its effects on adjacent shore lines. The cost of maintenance, as well as interest on and amortization of the first cost, must always be evaluated. If any plan considered would result in elongating or preventing the elongation of the existing problem area, the economic effect of each such consequence should likewise be evaluated. A convenient yardstick for comparing various plans is the total cost per year per foot of shore pro¬ tected. The following sections describe the most common engineering solutions now used to meet functional requirements, and give guides for their application. 3.2 SEAWALLS. BULKHEADS AND REVETMENTS 3.21 FUNCTIONS - Seawalls, bulkheads and revetments are structures placed parallel, or nearly parallel, to the shore line, separating a land area from a water area. The primary purpose of a bulkhead is to retain or prevent sliding of the land, with the secondary purpose of affording pro¬ tection to the fill against damage by wave action. The primary purpose of a seawall or revetment is to protect the land and upland structures from damage by wave forces, with incidental functions as a retaining wall or bulkhead. There are no really sharp distinctions between the three structures, and many cases exist where the same type, of structure in different localities bear different names. Thus, it is difficult to say whether a stone or concrete facing designed to protect a vertical scarp is a seawall or a revetment, and often just as difficult to determine whether a retaining wall subject to wave action should be termed a seawall or bulkhead. All these structures, however, have one featiore in common, in that they separate land and water areas, and are generally used where it is necessary to main- ♦ tain the shore in an advanced position relative to that of adjacent shores, where there is a scant supply of littoral material,to the area and little or no protective beach, as along an eroding bluff, or where it is desired to maintain a depth of water along the shore line as for a wharf. 3o22 LIMITATIONS - These structures afford protection only to the land immediately behind them, and none to adjacent areas up or down coast. When built on a receding shore line, the recession will continue downdrift. In addition, any tendency for loss of beach material in front of such a structure may well be intensified; where it is desired to maintain a beach in the immediate vicinity, companion works may be necessary. 3.23 FUNCTIONAL PLANNING OF THE STRUCTURE - The planning of these structures is a relatively simple process, since their functions are restricted to the maintenance of fixed boundaries. The features which must be analyzed 312838 O -T 54 - 7 87 Part I Chapter 3 in adequately planning such a structure are: its use and its overall shape, its location with respect to the shore line, its length, its height, and often the ground level in front of the wall. 3.24 USE OF THE STRUCTURE - The shape to be chosen must be determined by consideration of desired collateral uses. Face profile shapes may roughly be classed as: vertical or nearly vertical face, sloping face, convex curved face, concave curved and re-entrant face, or stepped face. Each silhouette has certain functional applications and so may be used in comination with any other if diverse functional criteria are to be met. A vertical or nearly vertical face structure lends Itself to use as a quay wall or landing place, where other shapes need to be provided with additional work to be so adapted. In addition, especially where a relatively light structure is required, a vertical face (of sheet pile, for example) may often be constructed more quickly and more cheaply than any other type. This may be an important consideration where emergency pro¬ tection is needed. Against wave attack, and specifically in regard to reduction of overtopping, a vertical face is more effective than any but the concave c\irved and re-entrant face, A backward sloping or convex curved face is the least effective of all types against wave attack for a given height of structure. It is, however, more adaptable to use as emergency protection, (sand bag, or dumped stone mounds, for example) than the other types. Actually the use of such a face type should be restricted to those areas in which wave over¬ topping is not a problem, or where esthetic, emergency, or structural con¬ siderations prohibit the use of other shapes. Concave curved or re-entrant faced structures are the most effective in reducing wave overtopping to a minimum. Where the structure’s crest is to be used (for a roadway or promenade for example), a wall so designed will be of the most desirable shape for protecting the crest. This is especially true if the beach in the vicinity is narrow or entirely absent, or if the water level is over the structures base. A stepped face provides the most ready access to beach areas from protected areas and in addition acts to disrupt the scouring action of the wave backwash. 3.25 LOCATION OF STRUCTURE WITH RESPECT TO SHORE LINE - In general, a seawall or bulkhead would be constructed along that line landward of which further recession of the shore line is not to be permitted. Where an areas is to be reclaimed, a wall may be constructed along the seaward edge of the reclaimed area. (A seawall constructed in the water, isolated from shore, becomes an offshore breakwater). 3.26 LENGTH OF STRUCTURE - A seawall, bulkhead or revetment protects no more than the land and improvements immediately behind it. No protection is afforded either to upcoast or downcoast areas as in the case with beach fills. It must be emphasized that in the usual case where erosion 88 Part I Chapter 3 may be expected to occur at either end of a structure, wing walls or tie- ins to adjacent land features must be provided to prevent flanking and possible progressive failure of the structure from the ends. Short term beach changes due to storms, as well as seasonal and annual changes must be considered. It must be remembered that changes updrift from such a struc¬ ture will continue unabated after the wall is built, and that downdrift., these changes will, if anything, be intensified. 3.27 HEIGHT OF STRUCTURE - Seawalls, bulkheads, revetments, and beach fills can be built to such a height that no water would overtop the crest of the structure, regardless of severity of wave attack, though it is frequently not economically feasible to do so. Wave run-up and overtopping criteria on which the height of structure should be based are not completely defini¬ tive at the present time (1960); however, tests to determine the relation of wave run-up and quantity of overtopping water to various wave parameters have been carried out and others are presently under way at several labora¬ tories. The results of tests conducted to date are presented in the follpw- ing two sections. 3.271 Wave Run-up (150,186,187) _ vertical height to which water from a breaking wave will run up on a given structure determines the top eleva¬ tion to which the structure must be built to prevent wave overtopping and resultant flooding on the landward side,,and to prevent possible structural damage by erosion. The actual run-up value depends on the characteristics of the structure (i.e., shape and roughness), the water depth of the struc*- ture.toe, and the incident wave characteristics, but sufficient data have not yet been obtained to completely define these relationships. However, a large number of small scale model tests have been made in an attempt to relate wave run-up to structure and wave characteristics, particularly on straight smooth slopes. The results of these tests for smooth slopes are summarized in figures 61A through 61E. These five figures show the model-determined relation between relative run-up (R/HA) and structure slope as a function of wave steepness (Hq/T^). These relationships are shown for five different ranges of structure depth-deep water wave height ratios (d/H6). For these five sets of curves, R is the wave run-up (the vertical height above still water level to which water will rise on the face of the structure), H6 is the equivalent deep water wave height, T is the wave period, and d is the depth at the toe of the structure. The parameter Hq/T^ is directly proportional to the dimen¬ sionless value of deep watfer wave steepness (H6/Lo) and is equal to 5.12 H6/Lo. As these curves are all referenced to a deep water relative depth (i.e. d/Lo >0.5 or deep water parameters) the designer in using them must compute the deep water wave steepness and height from whatever design wave informatio he has, and use these for computing the run-up. The deep water wave height, if not known initially, may b€ obtained from the non-breaking wave height at any depth of water by using .the table of functions of relative depth (d/Lo) May 1961 89 Part I Chapter 3 Table D-l,App,D), or from the relations for breaking depth or breaking height obtained from solitary wave theory (see Section 1,26U), These latter rela¬ tionships, as rearranged to utilize the generally more easily available value of wave period, T, rather than the deep water wave length, are: d^ - 1.28 (16) Hi ’ (^7a) - 1.92 ^ where db and are the breaking depth and height respectively, Althou^ (as shown in Figure 37) use of these equations will result in a value of HbAo less than 1,0 for waves of steepness H^Ap greater than 0,0278 (or H^2 > o,llA), it is felt that in general values of greater than need not be used for design purposes. Where such values are indicated by the for¬ mula, a value of may be used. Although these five figures ( 61 A-E) are referred to a deep water relative depth, similar curves referred to any other relative depth could be drawn, and would be equally usable. However, in using them one must know the relative depth of reference and be careful to use wave characteristics for that par¬ ticular referenced relative depth. This is true even though the referenced depth may not exist in the actual field location (as ’’deep water" does not for many lakes, for example). Except for the case of structure depth being greater than 3 times the wave height (dA^^ 3) (Figure 6lA), the other four figures refer to struc¬ tures fronted by a smooth 1 on 10 beach slope. Slightly different values of run-up might be observed, particularly for the lesser’ structure depths, if different beach slopes fronting the structure are assumed. Calculations for such cases (and even for the cases presented in Figures 61C-E) may be made by the composite slope method discussed in a following paragraph. The ranges of structure depth - wave height ratio (dA©) used for these four sets of curves were arbitrarily selected for convenience in analyzing some of the laboratory data, and have no other physical significance. These data indicate that the run-up increases with depth at the toe of the structure until a depth-height ratio between about 1 and 3 is reached, and then run-up apparently decreases somewhat. This apparent decrease with greater depth appears generally to be quite small (particularly in the range of greatest interest of value H^A^ greater than about 0,0^) and may in fact be a biased indication caused by difficulties in accurately measuring waves of such low an?3litudo and steepness, and the large scatter inherent in these low amplitude measurements coupled with the scarcity of measurements in this range. The initial increase in run-up with depth may be e^qjlained by the location of the breaking wave. When the wave breaks offshore from a structure. May 1961 89q Part I Chapter 3 it acts over a relatively large distance of gently beach slope before reaching the actual structiire. It may be thought of then as essentially acting over a gentler (total combined or composite) slope than when it breaks more nearly on the structure itself, and consequently has a somewhat lower run-up. Small scale tests have also been made for certain other essentially iapermeable walls (l50, l5l); the actual model data and the resultant curves for these are shown in figures 6lF (vertical wall), 6lG (curved wall - Galveston typo), 6lH (recurved wall-Galveston type, with top recurvature added), 6ll (stop-faced wall, 1 on 1-1/2 slope), and 6lJ (1 on 1-1/2 slope wall covered with single layer of riprap). These ciirves show the relation¬ ship between relative run-up (R/H*) and wave steepness (H^/T^) as a function of structure depth-wave height ratio (d/R*) for the individual walls. These walls weire also fronted by a 1 on 10 beach slope and for reac^ comparison with the smooth slope data, the curves for the 1 on 10 and 1 on 30 slopes are also shown. The curves for the straight smooth slopes are based on a large number of small scale observations, and are believed to be fairly accurate repressn- tatiOBS of the model data; however, the curves for the other structures (vertical, curved, recurved, stepped, and riprap) are based on a relatively small number of observations and consequently may be regarded as indications only. The data for single straight smooth slopes may also be used to predict relatively accurate values of wave run-up for any complex shaped slope, if the actual composite slope is replaced by a hypothetical single uniform slopej this hypothetical slope is obtained by using the breaking depth and an esti¬ mated value of wave run-up (186). For example, a composite slope consisting of a beach slope, a very gently sloping berm, and a steeply sloping face of a structure, may be replaced by a single hypothetical slope extending from the breaking point to an arbitrarily estimated point of maximum run-up on the structure. Using the hypothetical slope, a value of run-up may be determined by interpolation from figure ,^1A or 6lB« In general the value of run-up will be somewhat different from that initially chosen to obtain the hypothetical single slope; the process is then repeated using the new value of run-t^ to obtain a new single slope value, which in turn determines a new value of run¬ up. The process is repeated until identical values are obtained for two successive trials. VThile the curves presented in Figures 61A-E are believed to be rela¬ tively accurate representations of the small scale model data, recent tests with much larger scale waves (1,5 to U.5 feet in height) for 1 on 3 and 1 on 6 smooth slope structures have shown the existence, at least for these slopes, of a significant scale effect (195), This scale effect results in predictions of wave run-up from small scale tests which are lower than those actually May 1961 89 b Part I Chapter 3 observed. Although data were obtained for only two slopes, correction curves for a much greater range of slopes may be inferred from this data, and from the fact that apparently little or no such effect is observed on a beach slope of 1 on 15. Estimated correction curves based on these data are given in Figure 61K. These curves give estimated correction percentage values to be added to the small scale model-determined run-up (given in Figures 61A-E) to obtain a prototype run-up value. The degree of correction must probably also depend on the actual scale increase involved. Therefore, although only one test scale was used, in addition to the curve shown for this data (that marked for waves 1.5 to 4.5 feet in height) another correc¬ tion curve for higher waves (4 to 12 feet in height) has been inferred and is also shown. These correction values are estimates based on rather scanty data, but it is felt that their use with Figures 61A-E will increase the accuracy of run-up predictions. In interpreting and using the data presented in figures 61A through 6ir, the designer must also bear in mind that these data were obtained for impermeable slopes only, with exception of the riprap test (Figure 61J) involving'only a single layer of riprap on an impermeable 1 on 1-1/2 slope. In this test the riprap layer should be regarded almost purely as increas¬ ing the roughness factor rather than the permeability. Observations indi¬ cate that structure faces of relatively high degree of permeability, as riprap or rubble mound structures, generally serve to decrease the amount of run-up considerably. A reasonable estimation of design values for riprap covered structures may be made by determining the run-up value for a smooth impermeable structure of the same slope, and assuming a decrease in run-up of 20% for a single layer of riprap and a decrease of roughly 507o for a structure covered with a large number of layers of riprap. As noted below, values derived from Figure 61L should be used for a completely rubble slope. Some tests (187) have indicated that permeability does have to be of a relatively high degree to affect run-up, and that sand beaches, for example, behave (as far as run-up considerations are concerned) essentially the same as solid impermeijble structures. The run-up curves presented for smooth slopes on the order of 1 on 10 or less are probably very close to actual criteria for which sand beaches of those slopes should be designed. A certain amount of small scale laboratory data has also been obtained for wave run-up on rubble mound slopes. The data available for rubble slopes all pertain to structure depth - wave height ratio (d/H^) values greater than 3. Most of these tests have been conducted by Hudson (162) and Savage (187)^ Hudson's data are for (model) stone sizes about 4.5 times larger than Savage's and show somewhat lesser run-up. Model stone size apparently represents a noticeable, but as yet undetermined scale factor - although Savage has made test comparisons (206)^ Since Hudson's data are for the larger stone sizes (i.e. larger scale), curves derived from his data are felt to probably be more nearly applicable to prototype conditions. May 1961 89 c . ol w X 1 I- fVi \n m o to x| I- q: t'x° May (961 89 d Slope (Cot. a ) FIGURE 61-A. WAVE RUN-UP FOR SPECIFIC VALUES OF 2.5 May 1961 89 e Slope ICot a) Soville 1958 (186) FI(^RE 6l-i| WAVE RUN-UP FOR SPECIFIC VALUES OF ^ 10.0 Slop« (cotGC) FIGURE 61-C. WAVE RUN-UP ON SMOOTH IMPERMEABLE FRONTED BY I ON 10 BEACH SLOPE SLOPE May 1961 89 f 0 Slope (cot OC) FIGURE 61-D. WAVE RUN-UP ON SMOOTH IMPERMEABLE SLOPE FRONTED BY I ON 10 BEACH SLOPE I 89g Moy 1961 IS .2 .25 .3 .4 .5 .6 .7 .6 .9 r.O 5 2.0 2.5 3.0 4.0 S:0 6.0 7.0 6.0 9.0 10.0 Slope ( cot (X ) FIGURE 61-E. WAVE RUN-UP ON SMOOTH IMPERMEABLE SLOPE FRONTED BY I ON 10 BEACH SLOPE 89 h May 1961 0) w 3 U 3 ♦- (/) a> o e o > 89 i May 1961 RUN-UP AS A FUNCTION OF STRUCTURE DEPTH AND WAVE STEEPNESS. Relative Run-Up (^/uO V ' ■ y ■ a -T.5 0.750” 0.38 V V >w, _ L E G E N D SYMBOL d/Hi 1 on 10 slope O 0.3 □ 0.7 X 1.5 3 5 D >3 Curves are for va lues ol 1 \\ or • L slopes ind coted. 0.001 0.002 0.004 o.ooe 0.01 0.02 , 0.04 0.06 0.08 .1 0,2 0.3 04 06 0.8 1.0 Curved Wall-Galveston Type, With I On 10 Beach Slope From Toe Of Structure, FIGURE 61-G FIGURE 61-H RUN-UP AS A FUNCTION OF STRUCTURE DEPTH AND WAVE STEEPNESS. 89 j May 1961 Relative Run-Up (R/, FIGURE 6I-I 0.001 0.002 0004 0.006 0.01 0.02 0.04 0.06 0.08 .1 0.2 0.4 0.6 0.8 1.0 I From Saville, 1956 a Riprop Covered Wall, Ion I 5 Slope With Ion 10 Beach Slope From Toe of Structure. FIGURE 61-J RUN-UP AS A FUNCTION OF STRUCTURE DEPTH AND WAVE STEEPNESS. May 1961 89 k Slope Percent increase in run-up FIGURE 61-K RUN-UP CORRECTIONS FOR MODEL SCALE EFFECT > 0.09 May 1961 89 JL May 1961 90 ( FIGURE 61-L. WAVE RUN-UP ON RUBBLE MOUND AND SMOOTH SLOPES FOR VALUES OF >3. Part I Chapter 3 Curves derived from his tests are shown in Figure 61L. These curves show the relation of relative run-up (R/H^) to wave steepness for slopes ranging from 1 on 1,25 to 1 on 5, For convenience in comparing values for permeable rubble structures with smooth impermeable slope structures, similar curves for the smooth slopes are also shown on Figure 61L, As noted previously, tests on smooth slopes have indicated the existence of a scale effect whereby predicted prototype run-up values should be in¬ creased over those indicated by small scale tests. Tests now (1960) underway with a 1 on 1-1/2 rubble slope indicate that such a scale effect may exist for rubble slopes also. Accordingly, it is believed that although the curves presented in Figure 61L represent a reasonable estimate (and the best now available) of run-up values on rubble slopes, these should be used with some caution. It is particularly felt that values of R/H© less than 0,9 should be used with caution for slopes steeper than 1 on 3, at least until large scale data are available. 3,272 Wave Overtopping - The problem on wave overtopping has been important in the design of protective structures along the shores of rivers, lakes, reservoirs, and the oceans for as long as such structures have been built. However, since very little quantitative work has been done on this subject, predictions of the amount of water overtopping structures by wave action have usually been made with very meager information. This amount of over¬ topping becomes important, not only from the standpoint of the design of a safe structure, but also from the standpoint of flooding, and resultant damage, in low-lying areas behind such structures, and the design of an adequate drainage and pumping system to remove the overtopping water. Model study determination of quantity of overtopping water for various structure types under various wave conditions (115,151) conducted at the U, S. Waterways Experiment Station for the Beach Erosion Board gives dimen¬ sional qualitative results as shown on Figures 62A - 62Q, The curves es¬ tablished by the test data fall into a general family, and values for a particular field condition may be obtained by interpolation. As with wave run-up, large scale model tests (195) have demonstrated the existence of a large scale effect in wave overtopping quantities. This scale effect in¬ creases with a decrease in overtopping rate—probably because of the greater relative importance of the boundary effect. These tests have indicated that, at least for 1 on 3 arri 1 on 6 slopes, overtopping rates on the order of 2 cubic feet per second or greater should be increased by about 25 percent, and rates on the order of 0,5 to 1 cubic foot per second should be increased by about 50 percent. It should be noted that the overtopping values were derived from tests utilizing uniform wave conditions generated in a laboratory flume, so that the same height value is common to all waves in the test. Waves in nature, however, are not of constant height, but vary substantially from wave to wave, having a relatively wide height spectrum. In order to determine over¬ topping quantities for an actual wave train in nature, it is first necessary May 1961 90 a Part I Chapter 3 to determine the partial value of overtopping associated with each height segment in the wave spectrum. These must then be weighted according to the relative frequency of occurrence of the particular height (as given by sta¬ tistical analysis of wave frequency in Figure 13, for example) and then combined in order to get the final value of overtopping associated with a wave train of given significant height. Where waves or structural dimensions are such that these curves may not be used,,other criteria must be adopted. If for example, a vertical or curved re-entrant faced structure is to be located in or landward of the breaker zone, overtopping should be reduced to minimum if the structure’s crest height is located as indicated in section 3.271 (i.e., to an elevation equal to or greater than the run-up) or at least one and one-half times the wave height, 7above the highest anticipated water level at the structure’s location under storm conditions. (The ordinary still water depth at a structure's location may be increased, under storm conditions by wind set¬ up, seiches, scour, etc.) Where some overtopping is allowable, but it is still desired to minimize the effects of overtopping water which has damag¬ ing horizontal wave-induced momentum, structure crest heights may be set as low as seven-tenths the breaking wave height above the anticipated storm water level. In such a case under storm conditions, the overtopping water may still cause significant damage by being blown inland. The shape of stepped or sloping faced structures is cbnducive to per¬ mitting water to overtop them. For these, it is advisable that the'wall crest height be at least one and one-half times the breaking wave height above storm water level. For structures located seaward of the breaker zone, the Sainflou method (see Wave Forces, section 4.2) may be used to determine the proper structure crest height. Some model studies have been conducted to find the relative effective¬ ness of walls placed with their crests at, and at various depths below, the still water level. A summary of the results of these is shown on Figures 76 and 77. Bulkheads so located as to have a permanent beach berm to protect them from the direct impact of the waves may have their crest height reduced to a minimum of 2 feet above the height of maximum wave uprush or to the height of fill the bulkhead is designed to retain. Note however, that some seem¬ ingly permanent protective beaches have been scoured, under certain storm conditions, to the point where little or no protection was afforded. May 1961 90 b Rote Of Overtopping (cfs per foot of crest) Elevation Of Wall Crest Above Still Woter Level (feet) FIGURES 62 A.62B RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CREST February 1957 90-c Rote Of Overtopping (cfs per foot of crest) Elevotion Of Woll Crest Above Still Woter Level (feet) FIGURES 62C,62 D RELATION OF RATE OF WALL OF OVERTOPPING TO CREST ELEVATION Februory 1957 90 d Rote Of Overtopping (cfs per foot of crest) Figures62E.62F relation of rate of overtopping to elevation OF WALL CREST February 1957 90-e Rote Of Overtopping (cfs per foot of crest) 0 ---------- 0 2 4 6 8 10 12 14 16 18 20 22 26 28 Elevotion Of WoH Crest Above Still Water Level (feet) FIGURES 62 0,62 H RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CREST 90 f February 1957 Rote Of Overtopping (cfs per foot of crest) 0 2 4 6 8 10 12 14 16 18 20 22 24 26 26 Elevotion Of Wall Crest Above Still Woter Level (feet) FIGURE 621 RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CREST February 1957 90-g Rate Of Overtopping (cfs per foot of crest) FIGURE 62 J RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CREST February 1957 90 h Rote Of Overtopping (cfs per foot of crest) Elevation Of Wall Crest Above Still Water Level (feet) FIGURE 62 K RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CREST February 1957 90-i Rote Of Overtopping (cfs pet foot of creet) Elevotion Of Woll Crest Above Still Woter Level (feet) FIGURE 62 L RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CREST 90 j Februory 1957 Rate Of Overtopping (cfs per foot of crest) Elevation Of Wall Crest Above Still Water Level (feet) FIGURE 62 M RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CREST February 1957 90-k Rote Of Overtopping (cfs per foot of crest) Elevation Of Woll Crest Above Still Wafer Level (feet) FIGURE 62 N RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CREST February 1957 90-1 Rote Of Overtopping (cfs per foot of crest) FIGURE 62 0 RELATION OF RATE OF WALL OF OVERTOPPING CREST TO ELEVATION February 1957 90-m Rate Of Overtopping (cfs per foot of crest) FIGURE 62 P RELATION OF RATE OF OVERTOPPING TO ELEVATION OF WALL CREST February 1957 90-n Rate Ot Overtopping (cfs per foot of crest) Elevation Of Wall Crest Above Still Water Level (feet) FIGURE 62 Q RELATION OF RATE OF OVERTOPPING TO .ELEVATION OF WALL CREST February 1957 91 Part I Chapter 3 3.28 DETERMINATION OF GROUND ELEVATION IN FRONT OF A STRUCTURE - Seawalls and revetments are usually built to protect the upland area from the effects of continuing erosion as well as to protect shore improvements from damage by wave attack. Although the exact nature of the effect of such a wall on the processes of erosion cannot be determined (in certain instances these processes seem to have been halted or reversed), for safety in design they must be considered to continue. An exact determination of the beach profile that will exist after the construction of the wall is impossible; therefore, approximate methods must be relied upon. As an initial short-term effect, scour may be anticipated at the toe of the structure in the form of a trough, dimensions of which are governed by the type, of structure face, the nature of wave attack* and resistance of the bed material. In the case of a rubble mound seawall, the effect of scour is to undermine the toe stone, causing it to sink to an ultimately stable position. This will result in settlement of stone on the seaward face, which may be provided for by overbuilding the cross-section to provide for the settlement. Another method is to provide excess stone at the toe in order to fill the anticipated scour trough. The face of a vertical structure may similarly be protected against scour by use of stone. A gravity wall must be protected from undermining as a result of scour by providing impermeable cut-off walls at the base.. As a rule of thumb method, the maximum depth of scour trough below the natural bed, may be approximated as being equal to the height of the maximum unbroken wave that can be supported by the original depth of water at the toe of the structure. For example; if the depth of water at the face of the structure is 10 feet, the maximum wave height is 7.8 feet,.therefore the maximum depth of scour at the toe of the structure would be 7.8 feet below the original bottom or a total of 17.8 feet below the design water level. For long-term effects, it is preferable to assume that the structure would have no effect on reducing the erosion regime fronting it. In other words, the beach seaward of the wall would erode in the same manner as if the wall were not there. Since the determination can only be very approxi¬ mate, rules of thumb may be adopted. Consider a beach as shown in Figure 63 where the line DACEB represents an average profile. It is desired to place a seawall at point A as shown. From prior records, either the loss of beach width per year, or the annual loss of material over an area which includes the profile, is known. In the latter case, the annual quantitative loss may be converted to an annual loss of beach width by the rule of thumb "loss of 1 cubic yard of beach material is equivalent to loss of 1 square foot of beach area at the berm". In general, beach slopes are fairly steep shoreward of a depth of from 0 to 10 feet, and fairly flat seaward of that depth. Analysis of beach profiles on eroding beaches would indicate that it may reasonably be assumed that the beach seaward of a depth of 30 feet will remain May 1961 92 Part I Chapter 3 essentially unchanged, that the point of slope break will remain at about the same elevation, and that the profile of the beach shoreward of the point of break in slope will remain essentially unchanged} accordingly the ultimate depth at the wall may be estimated as follows: (a) - In Figure 63 let B represent the beach at a water depth of 30 feet, E the beach at the point of slope break (say) at a depth of 5 feet, and C the present position of the berm crest. If at A it is desired to build a structure whose economic life is (say) $0 years, and it is found that n is the annual loss of beach width at the berm, then in $0 years without the wall this berm will retreat a distance $0 n to point Dj (b) - From D to the elevation of point E draw a profile DF parallel to CE, and connect points B and F, This line DFB will re¬ present the approximate profile of beach after ^0 years, without the presence of the wall. The new beach elevation at the wall’s location will be approximated by Point A*. Similar calculations may be made for anticipated shc^ time beach depredations caused, for example, by storms, 3,3 PROTECTIVE BEACHES 3,31 FUNCTIONS - Beaches are very effective in dissipating wave energy, and, when they can be maintained to adequate dimensions, afford protection for the adjoining upland. Therefore, a protective beach can be classed as a shore protection structure. When studying an erosion problem it is nearly always advisable to investigate the feasibility of mechanicaU.y placing beach May 1961 93 Part I Chapter 3 material on the shore in such a manner that an adequate beach will be maintained in addition to any other remedial measures considered. The method of placing the beach fill to insure sand supply at the required replenishment rate is an inportant factor. Stockpiling of suitable beach material at the updrift sector of the problem area may be utilized where stabilization of an eroding beach is the problem. The establishment and periodic replenishment of such a stockpile is termed artificial beach nourishment. If the solution of a problem involves restoration of the eroded beach and stabilization at the restored position, direct placement of fill along the eroded sector is first carried out and thereafter artificial nourishment is accomplished by the stockpiling method. When conditions are suitable for artificial nourishment, long reaches of shore may be protected by this method at relatively low cost as corrpared with costs of other adequate defensive structures. An equally important advantage is that this treatment remedies directly the basic cause of most erosion problems, that is, a deficiency in natural sand supply, and thereby benefits rather than damages the shore beyond the immediate problem area. Also, the fact that the widened beach may have considerable value as a recreation feature is to be considered, A protective beach may also be provided, under certain conditions, by properly designed groins. This method must be used with caution for if a beach is restored or widened by impounding the natural supply of littoral material, a corresponding decrease in supply will occur in adjoining areas with resulting expansion or transfer of the problem area. Detrimental effects of groins may be prevented in most cases by artificial fill in suitable quantity placed concurrently with groin constructionj this is referred to as ''filling the groins". When groins are considered for use in conjunction with artificial fill it is desirable to evaluate benefits attributable to them carefully in order to determine justification for their inclusion, 3*32 LIMITATIONS - An obvious consideration in determining whether to provide a protective beach with or without groins, is the availability of suitable sand for the piirpose. Also, the beach fill method of protection is usually quite costly on a unit length basis when applied to short segments of shore. The latter is not necessarily a limitation if, by artificial nourishment, the enlargement of a problem area can be prevented. However, difficulties could be encountered in financing a shore protection method designed to provide protection beyond the immediate problem area, 3,33 PLANNING CRITERIA - The planning of a protective beach involves the following: a. Determination of the predominant direction of littoral transport and deficiency of material supply to the problem'area, b. Evaluation and selection of borrow material for initial beach fill and/or periodic nourishment, c. Determination of beach berm elevation and width. May 1961 94 Part I Chapter 3 d. Determination of wave-adjusted foreshore slopes. Determination of feeder beach (stockpile) location, 3.331 Direction of Material Movement and Deficiency of Supply - The various methods by which the predominant direction of littoral movement may be de¬ termined are outlined in section 2,32 (Littoral Drift), The deficiency of material supply is actually the rate of loss of beach material and is the rate at which the material supply must be increased to balance the transport capacity of littoral forces so that no net loss will occur. If there is no natural supply available, as may be the case on shores downdrift from a major littoral barrier, the deficiency in supply will be equal to the full rate of littoral transport. Comparison of surveys over a long period of time is the only accurate means of determining the rate of nourishment required to maintain stability of the shore. Since surveys in suitable detail for volumetric measurement are rarely available at problem areas, approximations computed from changes in the shore position as determined from aerial photographs or any other suitable records are often necessary. For such computations a relationship wherein one square foot of change in beach surface area equals one cubic yard of beach material appears to pro¬ vide acceptable values on exposed seacoasts. For less exposed shores this ratio would probably result in volumetric estimates somewhat in excess of the true figure and would thus produce conservative values, 3.332 Selection of Borrow Material - Having established the characteristics of the littoral transport processes in the area, the next problem is that of selecting borrow material for the beach fill and/or for periodic nourishment. When sand is deposited on a shore the waves operating in the area immediately start a sorting action on the surface layer of the fill moving the finer part¬ icles seaward, and leaving the coarser material shoreward of the plunge point. This sorting action continues until a layer, composed of a distribution of the coarser particles compatible with the wave spectrum of the area, armors the beach and renders it relatively stable. However, if the armor is broken due to a storm, the underlying material is again subjected to the sorting process. In view of this sorting process, beach fill materials containing some or¬ ganic material or materials finer than sand may be used with the assurance that natural processes will clean the fill material and make it acceptable material for nourishment. This has been confirmed by experience with fills containing foreign matter at Anaheim Bay, California, and Palm Beach, Florida, Material finer than that occupying the natural beach will, when exposed on the surface, move seaward to a depth compatible with its size thus tending to form inshore and foreshore slopes that are flatter than those considered normal prior to placement of fill. Fill.material of coarser size distribution characteristics than that on the natural beach will remain in the foreshore zone and may be expected to produce a steeper than normal beach. Limited data on the relationship between grain size and slope are shown in Figure 45, Criteria have not been established for accurately evaluating the quality of the borrow material relative to the material occupying the natural beach. May 1961 95 Part I Chapter 3 However some studies have been made by Krumbein^ ^ on this relationship and certain techniques may be utilized which will provide an indication of the probable behavior of the borrow material on the beach. This latter approach involves the procurement of a sufficient number of core borings in the borrow zone and samples from the beach and nearshore zones, mechanical size analysis of the borings and samples, and comparison of the resulting particle size distribution curves of the two sets of data. Almost any source of borrow near the shore will produce some material of suitable size. Since the source of artificial nourishment will control its cost to a major degree, 4iValtiati6n of the p6rtional volume of material of optimum characteristics in the borrow areas under consideration is an important factor in economic design, 3,333 Berm Elevation and Width - Beach berms are formed by the deposit of material by wave action. The height of a berm is related to the cyclic change, in water level, foreshore and nearshore slopes, and the wave pattern. Some beaches have no berms, others have one or several. Figure A-1 of Appendix A illustrates a beach zone with two berms. In this case the lower berm is the natural or normal berm and is formed ty the uprush of normal wave action for the ordinary range of water level fluctuations. The higher berm is referred to as the storm berm and is formed by wave action during storm conditions. During storm conditions the water level may or may not be higher than normal on the beach| however, most stonh conditions will create higher water levels. Wave overtopping reduces the normal beach berm width and may conpletely oblit¬ erate it if the overtopping prevails for a sufficient length of time. The fluctuations of the normal beach berm are reflected in the storm berm, and the degree of protection to the backshore is very much dependent on the effective¬ ness of the storm berm. Therefore in the planning of a beach fill the beach berms must be given careful consideration. If a beach fill is placed to an elevation lower than the natural berm crest height, a ridge will subsequently develop along the crest. Concurrent high water stage and high waves will over¬ top the crest or ridge and cause ponding and temporary flooding of the back- shore, Such flooding, if undesirable, may be avoided by constructing the berm at the height of or slightly above the natural berm crest. Although several alternative techniques may be employed to estimate the height of the berm, there is at present no proven theoretical basis for accurately predicting beach crest height. If there is an existing beach at the site, the natural crest height dan be determined therefrom. An estimate may be made on a basis of comparison with other sites possessing similar exposure characteristics and beach material. If sufficient wave data (wave data developed frcan synoptic surface weather charts or actual recorded wave data) are available and applicable to the project site, these data may be applied to the relationships of wave run-up, given in section 3,271, to establish a design berm crest height. Criteria for specifying berm width depend upon a number of factors. If the purpose of the fill is to restore an eroded beach to protect backshore improvements from damage by major storms the width may be determined by the protective width which experience had demonstrated to be required, i,e,, the beach width lost during storms of record plus minimum required to prevent wave action from reaching improvements. Where use of the beach for recreational May 1961 96 Part I Chapter 3 purposes is also a significant factor justifying its impro'\ement the width of the beach may be governed by area required for recreational use. The optimum area for recreational use is presently considered to be about 75 square feet of dry beach area per bather. Where the beach fill is to serve as a stockpile, to be periodically replenished, the berm width should be sufficient to provide for expected recession during the intervals between replenishment operations, 3.334 Slopes - It is generally considered that the toe of fill of a stock¬ pile of beach material should not extend to such depth that transport of any material forming the surface of the fill would be retarded. There is no firm specification for this limiting depth at present but available data indicate that depths of 20 feet below low water datum on seacoasts and 12 feet on the Great Lakes may be used safely. The initial overall slope of any beach fill will necessarily be steeper than that of the natural profile over which it is placed. Subsequent behavior of the slope depends principally upon the characteristics of the fill material. In ordinary practice the initial fill slope is designed parallel to the local or comparable natural beach slope above low water datum. Assumed design slopes usually in the order of 1*20 to 1:30 from low water datum to the intersection with the existing bottom are used for computation of quantities only. It is not only impracticable but unnecessary to grade beach slopes artificially below the berm crest since they will be naturally shaped by wave action. 3.335 Feeder Beach Location - The location of a stockpile or feeder beach along a problem area is dependent on local conditions. In general the dimensions of a stockpile will be governed primarily by economic considera¬ tion involving comparisons of cost for different replenishment intervals; therefore planning of a stockpile location must generally be considered in conjunction with stockpile dimensions. If the problem area is part of a continuous and unobstructed beach, the stockpile is located at the updrift end of the problem area. Until the stockpile material is transported by littoral forces to the beach zone downdrift of the stockpile location, that beach zone may be expected to recede at the same rate as determined from historical survey data. If economically justified, appropriately dimensioned stockpiles may be placed at points along the problem area thereby decreasing the interval of time between stockpile placement and complete nourishment of the problem area. Since the updrift end of a stockpile will be depleted first, long stockpiles are usually most suitable where a bulkhead or seawall exists to protect the backshore as erosion progresses along the stockpile. Lengths from a few hundred feet to a mile have been employed successfully. If the plan involves a feeder beach of significant quantity along the downdrift area from a coastal inlet, wave refraction and inlet currents must be considered in order to establish the optimum location of the feeder beach so maximum nourishment will be provided to downdrift shores with minimum loss of the feeder beach material into the inlet. The possibility of loss into the inlet by littoral forces during those periods when material is moved in an upcoast direction must also be considered. The latter requirement may necessitate supplementary structures to minimize material movement into the inlet during periods of drift reversal. May 1961 96 a Part I Chapter 3 The more or less continuous interception of littoral materials on the updrift side of an inlet and mechanical transportation of the materials to a point on the downdrift shores constitutes a form of stockpiling for arti¬ ficial nourishment to the downdrift shores. In this type of operation, the stockpile or feeder beach will generally be small in size as the stockpile material will be transported on downdrift by natural forces at a rate about equal to or greater than the rate of deposition. For the location of the stockpile or feeder beach for this type of operation see section 5,33, The need for a jetty or groin between the stockpile or feeder beach and the inlet to prevent return of the material to the inlet must be evaluated in any case where such structures do not already exist, 3,34 SAND DUNES _ Sand dunes are an important form of protective struc¬ ture,.The dune ridges along the coast act as barriers against the action of normal and storm tides and waves. They prevent storm waters from flooding the low interior areas; and they also reduce the action of onshore winds, which would erode more of the interior of the coast if no obstruction were present. Other dune ridges farther inland also are protective to a lesser degree than the frontal barrier dunes. If they are well stabilized the ridges serve as a second line of defense against the water and wind erosion should the foredunes be destroyed by storm wave action. Suitable stabiliza¬ tion measures should be undertaken to hold this sand that might otherwise migrate over adjacent areas and damage improved property as seen in Figure 64, Stabilizing the sand dunes also prevents the loss of protection. At those locations which have an adequate natural supply of sand, and which are subject to inundation by storm tides and high seas, a belt of sand dunes may provide more effective protection at a lower cost than either a bulkhead or seawall (see section 5,4), The sand dunes near the beach are not only protective barriers against high water and waves. Their role as stockpiles to feed the littoral stream is also important. Deposition often accumulates on the seaward slope rapidly enough to extend or build the dune toward the shore line. This material may be returned to the foreshore by wave action on a high tide and thus serve to nourish the beach, 3,4 GROINS 3,41 DEFINITION - A groin is a shore protective structure devised to provide, build or widen a protective beach by trapping littoral drift or to retard loss of an existing beach. It is usually perpendicular to the shore, extending from a point landward of possible shore line recession into the water a sufficient distance to stabilize the shore line at a desired location. Groins are rela¬ tively narrow in width (measured parallel to the shore line) and may vary in length from less than 100 feet to several hundred feet. Groins may be classified as permeable or impermeable, high or low. May 1961 96 b High, well- stabilized barrier dune. Migration of unstabililized dune across a road. Courtesy of Notional Pork Service FIGURE 64. STABILIZED NATIONAL SEASHORE AND MIGRATING DUNES RECREATIONAL AREA, AT CAPE HATTERAS north CAROLINA. May 1961 96 c h »/! id ' f r * .v^ ,< -■V ■ tj(i .:^;,4i’' '*.U,I-7 « 1 *■ f■.«» 'l^•' Jp p »< '*’ *a*' ^ ♦'. •>'’ it'f lif/ ' •'♦ fi*i -1X ^ ‘>f ’ • *■ '"•'* ' ■'■" ’ >•' ‘f.', '•■■»*- .« dUif* ' • tl fife, f - -■ 4 • .^U *fWft(f(L . . M*. »ir»f# " ’ •4" - 'I'UA'ii’H t; *4**« ii . t»*r #*rC */5if m .'»§ i *•» ^ ^ "''■'i ». t * ' V ^ * '‘t"- 'i.i t t'B -rf:. ii. i •l4|i"'4. . -.i' ;-» •M'.I v'i *'•4* ■'* ;r * i*%i 4j|y. * '••v.■^r' ■ ■ ti’-jortJ h i.=ff , ,*';H .if •’*4 - .'41 .t-J - •i- * ^ 4 . ii' If'r* ' V ,*' ' ' . *1* / ' I'r ..' <*■' ■• . . . * ^ .. ^.<0»>•■' ■ '’iUtfeianw-lO'i.-Hijicw^K^f*- -...:— ' • ■ -- ir> ,.- >•,. , ' • ' U [•n^Mnog^H - ^ ^ t \dkii‘ tc»ii ’.vl .4 r!‘: Part I Chapter 3 and fixed or adjustableo They may be constructed of timber, steel, stone, concrete, or other materials, or combinations thereof. Impermeable groins have a solid or nearly solid structure which prevents littoral drift passing through the structure. Permeable groins have openings through the structure of sufficient size to permit passage of appreciable quantities of littoral drift. Some permeable stone groins may become impermeable with heavy marine growth. A series of groins acting together to protect a long section of shore line is commonly called a groin field, "Groin system" is a preferable term. Groins differ from jetties in that jetties generally are larger with more massive component parts, and are used primarily to direct and confine the stream or tidal flow at the mouth of a river or entrance to a bay and prevent littoral drift from shoaling the channel. In some sections of the country groins are commonly referred to as jetties or piers, 3.42 GROIN OPERATION - A groin interposes a total or partial barrier to littoral drift moving in that part of the littoral zone between the sea¬ ward end of the groin and the limit of wave uprush. The extent to which the littoral drift is so modified depends on the height, length, and permeability of the groin. The manner in which a groin operates to modify the rate of littoral drift is approximately the same whether it operates singly or as one of a system, provided spacing between adjacent groins is adequate. However, under some conditions a single groin or the updrift groin of a system, may have a somewhat smaller capacity than the other individual groins of the system. A typical groin is illustrated in Figure 65. In this figure, the groin extends from some distance landward of the top of berm to the 6-foot depth contour. The net direction of wave attack, as typified by the orthogonals shown, is such as to cause a net movement of littoral drift. The shore line and 6-foot depth contour are represented by e a i and £ £ h respectively, occurring in a state of nature prior to the construction of the groin _j_ c. Prior to the construction of the groin the offshore beach slope had stabilized between a and _c in a manner dependent on the character of the beach material and the type of wave attack, as shown by the profile d a k c. The groin acts as a partial dam to intercept a portion of the normal drift. As material accumulates on the updrift side, supply to the down- drift shore is correspondingly reduced and the latter shore recedes. This results in a progressively steepening slope on the updrift side and a flattening slope on the downdrift side, since both slopes must reach a common elevation at or near the end of the groin. Since the grain size of the beach material normally increases to establish a steeper than normal slope, the residual accreted material is probably by selective process the coarser fraction of the material that was in transport. When the accreted slope reaches ultimate steepness for the coarser fraction of the material available, impoundment ceases and all littoral drift passes the groin. If the groin is sufficiently high that no material may pass over it, all transport must be in depths beyond the end of the 97 LAND Direction of Littoral Transport WATER FIGURE 66-EFFECT OF GROIN SYSTEM 98 Part I Chapter 3 groin. Because of the nature of transporting currents the material in transit does not move directly shoreward after passing the groin, and transport characteristics do not become normal for some distance on the downdrift side of the groin. Thus a groin system too closely spaced would divert littoral transport offshore rather than create a widened beach. The accretion fillet on the updrift side of the groin creates a departure from normal shore alignment, tending toward a stable alignment normal to the resultant of wave attack. The impounding capacity of the groin is thus dependent upon both the stability slope and stability align¬ ment of the accretion fillet. These in turn depend upon characteristics of the littoral material and the direction of wave attack. Figure 66 shows the general configuration of the shore line to be expected for a system of two or more groins. It assumes a well establish¬ ed net littoral transport in one direction. 3.43 PURPOSE - Under some conditions and subject to definite limitations imposed on their use, groins may be used to: (a) Stabilize a beach,subject to intermittent periods of advance and recession; (b) Provide upland protection through prevention of removal of a protective beach; (c) Reduce the rate of littoral transport out of an area by reorienting a section of the shore line to an alignment more nearly perpendicular to the major direction of wave approach; (d) Build or widen a beach by trapping littoral material; (e) Prevent loss of material out of an area by compartmenting the beach; (f) Prevent advcuice of a downdrift area by acting as a littoral barrier; and (g) Stabilize a specific area by reducing the rate of loss from the area. All of the foregoing ends are attained through modification of the rate of littoral transport. Also, all are forms of shore protection. 3.44 LIMITATIONS ON THE USE OF GROINS - Because of their limitations, the use of the groin as a major protective feature should be decided upon only after careful consideration of the problem and the many factors involved. Principal factors to be considered are: (a) The extent to which the downdrift beach will be damaged if groins are used; 99 Part I Chapter 3 (b) The economic justification for groins in comparison with stabilization by noiirishment alone; (c) The adequacy of natural sand supply to insure that groins will function as desired; (d) The adequacy of shore anchorage of the groins to prevent flanking as a result of downdrift erosion; (e) In providing protection by groins, it must be remembered that the alignment of the shore adjacent to a groin is dependent upon the direction of wave attack and will vary. If minor fluctuations of the shore cannot be allowed a solution other than the use of groins must be found; and (f) Where the supply of littoral material is insufficient to permit the withdrawal from the littoral stream of enough material to fill the groin or groin field without damage to downdrift areas, artificial fill may be required to fill the groin or groin system and thus permit the natural littoral supply to pass without interruption. 3.45 TYPES OF GROINS 3.451 Permeable Groins - Many types of permeable groins have been employed in efforts to avoid the abrupt offset in shore alignment which normally occurs at impermeable groins. The primary effect of permeability is to reduce the impounding capacity, which can ordinarily be accomplished at less cost with a properly designed impermeable groin. An important disadvantage of permeable groins is their relative ineffectiveness in holding a beach under storm conditions. As a means of river bank control where sediment transport results from ordinary current flow their value is well established. (Such structures are normally called permeable dikes.) Where wave action is the principal cause of transportation, permeable groins are unlikely to prove fully satisfactory as a shore pro¬ tection measure. 3.452 High and Low Groins - The amount of sand passing a groin depends to some extent on the height of the groin. Groins based on a headland or reef, or at the entrance to a bay or inlet where it may be either unnecessary or undesirable to maintain a sand supply downdrift of the groin, may be built to such a height as to block completely the passage of material moving in that part of the littoral zone covered by the groins. Where it is necessary to maintain a sand supply downdrift of a groin, it may be built to such a height as to allow overtopping by storm waves, or by waves at high tide. Such low groins serve the same purpose as that intended by designers of permeable groins. 3.453 Adjustable Groins - The great majority of groins are fixed or permanently built structures. In England, the Case and Du-Plat-Taylor adjustable groins have been used with reported success. These groins are essentially adjustable batter boards between piles, with a raising 100 Part I Chapter 3 and lowering device so that the groin can be maintained at a fixed height (usually one to two feet) above the sand level, allowing a considerable part of the sand to pass over the groin and maintain the downdrift beach. Adjustable groins are reported to be particularly useful where an attempt is being made to widen a beach with a minimum of erosion damage to the downdrift area. However, they are effective only where there is an ad¬ equate supply of littoral material, 3.46 DIMENSIONS OF GROINS - The width and side slope of a groin depend on wave forces to be withstood, the type of groin, the materials with which it is constructed, and the construction methods used. These features are considered \mder struct\iral design. The length, profile and spacing are important considerations with respect to functional success. The length of a groin is determined by the depths in the offshore area and the extent to which it is desired to intercept the littoral stream. The length should be such as to interrupt such a part of the littoral drift as will supply enough materials to create the desired stabilization of the shore line or the desired accretion of new beach areas. Care must be exercised that these ends are attained without a corresponding damage to downdrift areas. For functional design piarposes, a groin may be considered in three sections; (a) The horizontal shore section; (b) The intermediate sloped section; and (c) The outer section 3.461 The Horizontal Shore Section - This section would extend from the desired location of the crest of berm as far landward as is required to anchor the groin to prevent flanking. The height of the shore section depends on the degree to which it is desirable for sand to overtop the groin and replenish the downdrift beach. The minimum height of the groin is the height of the desired berm, which is usually the height of maximum high water that occurs frequently plus the height of normal wave uprush. The maximum height of groin to retain all sand reaching the area (a high groin) is the height of maximum wave uprush during all but the least frequent storms. This section is horizontal or sloped slightly seaward, paralleling the existing beach profile or the desired slope, in case a wider beach is desired or a new beach is to be built. 3.462 The Intermediate Sloped Section - The intermediate section would extend between the shore section and the more or less level outer section. This part of the groin should approximately parallel the slope of the foreshore the groin is expected to maintain. The elevation at the lower end of the slope will usually be determined by the construction methods used, the degree to which it is desirable to obstruct the movement of the material, or the requirements of swimmers or navigation interests. 3 . 463 o The Outer Section - The outer section includes all of the groin extending seaward of the sloped section. With most types of groins, this section is horizontal at as low an elevation as is consistent with economy of construction, since it will be higher than the design updrift slope in 101 Part I Chapter 3 any case. The length of the outer section will depend upon the design slope of the updrift beach. 3.464 Spacing of Groins - The spacing of groins in a continuous system is a function of the length of the groin and the expected alignment of the accretion fillet. The length and spacing must be so correlated'that when the groin is filled to capacity the fillet of material on the updrift side of each groin will reach to the base of the adjacent updrift groin with sufficient margin of safety to maintain the minimum beach width desired or prevent flanking of the updrift groin. Figure 6? shows the desirable resulting shore line if groins are properly spaced. The solid line shows the shore line as it may develop when erosion is a maximum at the toe of the updrift groin. The erosion shown occurs while the updrift groin is filling. At the time of maximum recession, the solid line is nearly normal to the direction of the resultant of wave approach and the triangle of recession ”a” is approximately equal to the triangle of accretion ”b". The dashed line m-n shows the stabilized shore line which will obtain after material passes the updrift groin to fill the area between groins and in turn commences to pass the downdrift groin. It will be noted that the fillet of sand between groins tends to become and remain perpendicular to the predominant direction of wave attack. This alignment may be quite stable after equilibrium is reached. However, if there is a marked variation in the direction and intensity of wave attack either seasonally or as a result of prolonged storms, there will be a corresponding variation in the alignment and slope of the shore between groins. In areas where there is a periodic reversal in the direction of littoral transport an area of accretion may form on both sides of a groin as shoivn in Figure 68. Between groins the fillet may actually oscillate from one groin to the other as shown by the dashed lines, or may form a U-shaped beach somewhere in between depending on the rate of supply of littoral material. With regular reversals in littoral drift, the maximum line of recession would probably be somewhat as shown by the solid line, with the triangular area ”a" plus triangular area ”c_” approximately equal to the circular segment "b”. The extent of probable beacU recession must be taken into account in establishing the length of the horizontal shore section of groin and in estimating the minimum width of beach that may be biilt by the groin system. 3.465 Length of Groin - Before the total length of a groin, and the position of the shore line adjacent to a groin can be determined, it is necessary to predict the ultimate stabilized beach profile on each side of the groin. The steps involved for a typical groin are: (a) Determine the original beach profile in the vicinity of the groin; (b) Determine the conditions of littoral transport; (c) Plot a refraction diagram for the mean wave condition, i.e. the wave condition which would produce the predominant direction and net rate of littoral drift; 102 Original LAND FIGURE 68 GROIN FIELD OPERATION WITH REVERSAL _ OF TRANSPORT 103 Part I Chapter 3 (d) Determine the minimum width beach desired updrift of the groin. This may be a width desired to provide adequate recreational area, adequate protection of the upland area, or in the case of a groin system, adequate depth of beach at the next groin updrift to prevent flanking of this groin by wave action. The latter criterion is depicted at point m on Figure 67, if line m-n represents the berm crest of the beach; (e) The position and alignment of the desired beach relative to the groin vinder study is indicated by the line m-n , Figure 67, the line being constructed approximately normal to the orthogonals based on mean wave conditions from m to n; (f) Apply the distance c-n from Figure 67 to Figure 69 and this distance, plus sufficient length landward of c to prevent flanking, will represent the length of the horizontal shore section; (g) The slope of the ground line from the crest of the berm seaward to about the mean low water line will depend upon the gradation of the beach material and the character of the wave action. This section of groin, the intermediate sloped section. Figure 69, is usually designed parallel to the original beach profile. The ground line will assume the slope of the groin section n-p or, if the material trapped is coarser than the original beach material, will assume a steeper slope. The length of the outer section p-r depends upon the amount of littoral drift it is desired to intercept. It should extend to sufficient depth so that the new profile p-s will intercept the old profile c-d-p within the toe of the groin; (h) The final gro\ind line on the updrift side of the typical groin shown in Figure 69 is indicated by the line p-n-p-s. The groiind line on the downdrlft side of a groin will be different for an intermediate groin in a system than it will for a single groin or for the farthest downdrlft groin in a system. If the system is properly planned and constructed, the ground lines would be about the same for the latter two. Considering first an intermediate groin in a groin system, the maximum shore recession on the downdrlft side of the groin would occur before the updrift groins fill. During this time the maximum recession would occur when the shore line between the intermediate groin and the next downdrlft groin has reoriented to a position normal to the net wave orthogonals such that area a = area b in Figure 70. To determine the maximTim recession of the downdrlft shore line, draw the proposed groin on the original beach profile as in Figure 71. From the crest of berm at station 0, lay off distance cd. taken from Figure 70. Draw the foreshore from crest of berm to datum plane (MLW) parallel to the original beach slope, and connect that point of intersection with the original profile at the seaward end of the groin. 104 TYPICAL GROIN PROFILE FIGURE 69 February 1957 105 Part I Chapter 3 After the position of maxiiiiuin recession has been reached, as shown by c-£ on Figure 70, the shore line will begin to advance seaward maintain¬ ing its alignment normal to the net wave orthogonals until siifficient material flows around or over the downdrift groin to produce a stabilized shore line as shown by the line m-n in Figure 70. To determine the stabilized downdrift line, see Figure 71. From the crest of berm at station 0 lay off the distance ^ taken from Figure 70. Draw the foreshore from the crest of berm to datum plane (MLW) parallel to the original beach line and connect that point of intersection with the original profile at the seaward end of the groin. 3.47 ALIGNMENT OF GROINS - Examples may be found of almost every conceiv¬ able groin alignment, and advantages are claimed by proponents of each type. Based on the theory of groin operation, which establishes the depth to which the groin extends as the critical factor affecting its impounding capacity, maximum economy in cost is achieved with a straight groin normal to the shore line. Various modifications such as a '•!” or "L" head are usually designed with the primary purpose of limiting recession on the downdrift side of a groin. While these may achieve the intended purpose in some cases, the zone of maximum recession is often simply shifted to a point some distance away from the groin (on the downdrift side) and benefits are thus limited. Storm waves will normally produce greater scour at the extremities of "T” or ”L" head structures than at the end of a straight groin normal to the shore, delaying the return to normal profile after storm conditions have abated. Curved, hooked or angle groins have been employed for the same purposes as the "T” or "L" head types, and have the same objectionable featiires, that is, they Invite excessive scour and are more costly to build and maintain than the straight groin normal to the shore. In cases where the adjusted shore alignment expected to result from a groin system will differ greatly from the alignment at the time of construction, it may be desirable to align the groins normal to the adjusted shore alignment in order to avoid angular wave attack on the structures after the shore has stabilized. This condition is most likely to be encountered in the vicinity of inlets and along the sides of bays. 3.48 ORDER OF GROIN CONSTRUCTION - This applies only to sites where a groin system is under consideration. Here two conditions arise: (l) where the groin system will be filled artificially and it is desired to stabilize the new beach in its advanced position; and (2) where littoral transport is depended upon to make the fill and it is desired to stabilize the existing beach or build additional beach with a minimum of detrimental effect on downdrift areas. In the first instance the only interruption of littoral transport will be between the time the groin system is constructed and the time the artificial fill is made. In the interests of economy, the fill is normally placed at one time, especially if it is being accomplished by hydraulic dredge. Accordingly to reduce the time Interval between groin construction 106 February 1957 Part I Chapter 3 and deposition of fill, all groins should be constructed concurrently or as rapidly as practicable if constructed consecutively. Deposition of fill should commence as soon as the stage of groin construction will permit. In the second instance no groin can fill until all of the preceding updrift groins have been filled. The time required for the entire system to fill and the material to resume its unrestricted movement downdrift may be such that severe damage will result. Accordingly only the groin or group of groins at the downdrift end should be constructed initially. The second groin, or group should not be started until the first has filled and material passing around or over the groins has again stabilized the downdrift beach. Although this method may increase costs, it will not only aid in holding damage to a minimum but will provide a practicable guide to spacing of groins to verify the design spacing. 3.5 JETTIES 3.51 DEFINITION - A jetty is a structure generally perpendicular to the shore extending into a body of water to direct and confine a stream or tidal flow to a selected channel, and prevent or reduce shoaling of the channel by littoral materials. Jetties at the entrance to a bay or river also serve to protect the entrance channel from storm waves and cross currents, and when located at inlets through barrier beaches they also serve to stabilize the inlet location. 3.52 TYPES - Generally jetties built on the open coast are of rubble-mound construction, but steel sheet pile cells have been used. Jetties have also been built of caissons and cribs using timber, steel or concrete in the Great Lakes Area. Single rows of braced and tied Wakefield timber piling and steel sheet piling have been used in sheltered areas. 3.53 SITING . The proper siting and spacing of jetties for improvement of harbor entrances are very important. Careful study must be given to direction and strength of tidal currents, and the channel section needed for navigation. Jetties should generally be laid out so that the channel will be controlled in the position and direction corresponding with the natural tidal flow. The spacing, however, must be great enough so that the jetties will not be sub¬ ject to undermining and will freely admit the flood tides so that the tidal prism will not be materially reduced. In locating the entrance position, investigation should be made of waves approaching the harbor to avoid locations where waves build up. Construction of wave refraction diagrams is a means of locating such areas. In spacing jetties, care must be taken that the current velocities between them during the strength of the tide are not too great for practicable navigation. Furthermore, jetties should be far enough apart to allow for safe passage of vessels through the entrance even during adverse weather. 3.54 EFFECTS ON THE SHORE LINE . The effects of entrance jetties on the shoreline are illustrated by Figure 49. A jetty interposes a total littoral May 1961 108 Part I Chapter 3 barrier in that part of the littoral zone between the seaward end of the structure, generally terminated in a depth of water equivalent to the project depth of the channel, and the limit of wave uprush. In view of the above, accretion takes place updrift from the structures at a rate equal to that of the littoral transport, and erosion downdrift at the same rate. The quantity of the accumulation depends on the length of the structure and the angle at which the resultant of the natural forces strikes the shore. If the angle that the shoreline of the impounded area makes with the structure is acute, the impounding capacity is less than it would be if the angle was obtuse. In the former case channel maintenance will be required in a lesser time due to littoral material passing around the end of the structure. Planning for a jettied entrance should include some method of bypassing the littoral material to eliminate or reduce channel shoaling and erosion of the downdrift shore. 3.6 BREAKWATERS - SHORE CONNECTED 3.61 DEFINITION. A breakwater is a structure employed to reflect and/or dissipate the energy of water waves and thus prevent or reduce wave action in an area it is desired to protect. Breakwaters for navigation purposes are constructed to create sufficiently calm water in a harbor area, thereby providing protection for safe mooring, operating and handling of ships and protection for shipping facilities. Breakwaters are sometimes constructed in the interior of large established harbors to protect shipping and small craft in an area that would otherwise be exposed to damaging wave action. 3.62 TYPES . Breakwaters may be rubble mound, composite, concrete caisson, sheet piling cells, crib or mobile. Generally breakwaters built on the open coast are of rubble-mound construction. Occasionally they are modified into a composite structure by using a concrete cap for stability. Precast con¬ crete shapes such as tetrapods are also occasionally used for armor stone when rock of sufficient size is not obtainable. In the Great Lakes Area timber, steel or concrete caissons or cribs have been used. Breakwaters are infrequently built of a single row of braced and tied wakefield timber piling or steel sheet piling in relatively sheltered areas. Several types of floating breakwaters have been designed and tested, but as far as is known, none is in use at this time. 3.63 SITING . Shore-connected breakwaters are sited mainly to accomplish the specific mission of providing a protected harbor for vessels. Numerous factors affect the siting of a breakwater such as the dirrection and magni¬ tude of the littoral drift, the harbor area to be attained by the location of the breakwater, the character and depth of the bottom material in the proposed harbor, etc., but the most important is that of determining the optimtam location which will produce a harbor area with minimum wave and surge over the greatest period of time in the year. Almost without exception this determination is made through the use of refraction and diffraction analysis. May 1961 109 Part I Chapter 3 3.64 EFFECT ON THE SHORE LINE . The effect of a shore-connected breakwater on the shore line is illustrated by Figure 50. As in the case of a jetty the shore arm interposes a total littoral barrier in that part of the littoral zone between the seaward end of the shore arm and the limit of wave uprush. The same accretion and erosion patterns result from the installation of this type of structure. The accretion, however, in this case is not limited to the shore arm but generally extends along the seaward face of the sea arm building a berm over which littoral material is transported to form a large accretion area at the end of the structure in the less turbulent waters of the harbor. This type of shoal creates an ideal condition for sand bypassing procedures, A pipeline dredge can lie in the relatively quiet waters behind the shoal, and transfer the accumulated material shoreward to nourish the downdrift shores, 3.7 BREAKWATERS - OFFSHORE 3.71 DEFINITION . An offshore breakwater is a structure employed to reflect and/or dissipate the energy of water waves and thus prevent or reduce wave action in an area it is desired to protect. Offshore breakwaters may serve as aids to navigation or shore protection or as a combined purpose,and differ from other breakwaters in that they are generally parallel to and not connected to the shore, 3.72 TYPE , Almost without exception offshore breakwaters are of rubble- mound construction. If however suitable rock is not available, concrete armor units such as tetrapods and/or a concrete cap can be used. 3.73 SITING . Offshore breakwaters are sited to provide shelter to a harbor entrance, to create a littoral reservoir, and to provide a relatively calm area in which small craft may seek refuge and/or a pipeline dredge can operate to pump sand to downdrift shores. An excellent example of this type of siting or use is illustrated in Figure 139C which shows the recently con¬ structed harbor entrance at Ventura, California, Offshore breakwaters have also been sited off massive seawalls to provide a first line of defense, 3.74 EFFECTS ON THE SHORE LINE, The effects of an offshore breakwater on the shore line are partially illustrated by Figure 73. An offshore breakwater has an effect bn the shore regime similar to that of any other structure, such as a groin, which modifies the rate of littoral transport. It is probably the most effective means of completely intercepting littoral transport of all such modifying structures now in use. Being usually in deeper water than the seaward end of a jetty or groin, it controls a wider portion of the bank of littoral drift than do structures attached to shore. Because littoral trans¬ port is the direct result of wave action, the extent to which the breakwater intercepts the littoral drift is directly proportional to the extent of interception of wave action by the breakwater. 110 May 1961 Part I Chapter 3 3.75 OP ERATION OF AN OFFSHORE BREAKWATER . An offshore breakwater at first tends to cause sand to deposit in its lee by damping the wave action in that area, thus reducing or eliminating the forces responsible for transport. Diffraction causes some wave action within the geometric shadow, but such wave action is much less than that which would exist in the area in the ab¬ sence of the breakwater. The typical diffraction diagram drawn on Figure 73 shows that wave heights within the breakwater’s geometric shadow are less than one-half the wave heights outside the breakwater. As sand is deposited, a shore salient is formed in the still water behind the breakwater. This projecting shore alignment acts as a groin, which causes the updrift shore line to advance. As the projection enlarges and the zone of littoral transport is closer to the breakwater, the salient becomes increasingly efficient as a littoral barrier. If the breakwater is of sufficient length in relation to its distance from the shore to act as a complete littoral barrier, the sand depositing action may continue until a tombolo is formed with the breakwater at its apex. Such a tombolo accretion is shown on Figure 74, an aerial photograph of the offshore breakwater at Venice, California. The precise shape of the deposit is difficult to determine. In general there will be accretion updrift from the breakwater and erosion downdrift. The area immediately behind the breakwater customarily assumes a form convex seaward. It has been found for complete barriers that a large percentage of the total accumulation collects in the breakwater lee during the first year and that the ratio of material in the lee of the structure to total material trapped decreases steadily until such time that the trap is filled and lit¬ toral material begins moving seaward around the structure. 3.76 OFFSHORE BREAKWATERS IN SERIES . It is not necessary to build offshore breakwaters as a single unit. A series of relatively short structures will have the same general effect as a single one, but the efficiency of the series as a sand trap will be decreased; a condition which is sometimes de¬ sirable. The tendency for a tombolo to form will be increased. Figure 75 is an aerial photograph taken in 1949 of the breakwater off Winthrop Beach, Massachusetts constructed in 1931-1933, The characteristic convex accretion in the breakwater lee is evident, as is also the erosion zone downdrift. Note that here shoals formed from the breakwaters, extending landward, indi¬ cating that these breakwaters are in the littoral movement zone. 3.77 HEIGHT OF AN OFFSHORE BREAKWATER . One of the factors determining the effectiveness of an offshore breakwater as a sand trap is its height in relation to the wave action at the site. That structure which can, except for diffraction at its ends, completely eliminate wave action in its lee will, if long enough, function as a complete littoral barrier. In this sense, then, the most efficient type of breakwater is one whose crest ex¬ tends to such a height that no significant overtopping by waves takes place. It has been common practice to assume that essentially no wave overtopping takes place in the case of rubble-mound structures, the most common type in this country, if the structure crest is one and one-half times the design wave height above the design still water level, and this criterion is May 1961 112 OFFSHORE BREAKWATER; VENICE,CALIFORNIA FIGURE 74 113 114 OFFSHORE BREAKWATER} WINTHROP BEACH,MASSACHUSETTS FIGURE 75 > > ^ -- < > < L V (feet) landward ft.) seaward > > < ( > A i / / / _ > / water level dth ot wave , (ft lbs. per dth ot wave per ft.) > < o > «> > d < < > / / / • 'J / / / / V ' below still cture (feet) per unit wl jre. ~ p g H*L per unit wi jre (ft. lbs. 1 > > > < < > < > r < J X N / / / LU 3 >« ^ ^ w a» o o* u O w — fc. M o ° 4. » , ^ o* o* 5? f £ o £ o £ > > > : : > < < - -- > > 3 3 > <>. X ‘ < * ' \ 1 \ X < ^ > f < a> » > ■o £ o H (1 It T) N - liJ o o o M J .«» UJ ■ > 1 1 1 1 V J > > — < 115 AVERAGE EFFECT OF STRUCTURE ON WAVE HEIGHT RATIO OF ENERGY LANDWARD TO SEAWARD FOR VARIED RATIO WITH SUBMERGENCE SUBMERGENCE OF STRUCTURES FIGURE 76 _ _ FIGURE 77 _ Part I Qiapter 3 suggested as a measure of the effectiveness of such a breakwater as a lit¬ toral barrier. That is, if a breakwater’s crest above still water level is one and one-half times the anticipated wave height, it should be considered to be as complete a littoral barrier for a given length of structure as could be built at the site. Recent model studies (162) indicate that the required crest elevation of a rubble type of breakwater depends to considerable extent on wave height, wave length, and face slope of the structure. Data given in Figure 61-L indicate values for wave run-up on rubble-mound structures and until further verification can be made run-up values at least as great as in¬ dicated by curves in Figure 61-L should be considered if no overtopping is a design criterion or to evaluate the extent of permissible overtopping. In the case of offshore breakwaters it may be desirable to build a structure so that it is not completely effective as a littoral barrier. This may be accomplished by construction of the breakwater to a height less than that sufficient to prevent overtopping. Subject to adequate marking or other means of preventing a navigation hazard, such partial barriers need not extend above low water. The maintenance requirements of a submerged breakwater are often so much less than those for a high breakwater that, on an economic basis alone, serious consideration should be given to such a structure where only a partial bar¬ rier is desired. (Some submerged offshore breakwaters are built primarily to retain a beach fill. They function primarily as bulkheads, though sub¬ merged, and may be treated as such). The relative effectiveness of partial and complete barriers is difficult to determine. Model studies of the effects of submerged structures on wave heights and wave energy have been made at the Beach Erosion Board (8) and at the University of California at Berkeley (89), These showed that of the vari¬ ous shaped barriers tested, a vertical-faced structure was most efficient in reducing transmitted wave energy, though the difference in effectiveness for the other shapes tested was small. The efficiency of all structures increased with increase in width in the direction of wave approach. One possible measure of the effectiveness of a submerged breakwater com¬ pared to a complete barrier is the ratio of transmitted wave energy to incident wave energy. The model studies indicated that this ratio depends on both the depth of water at a structure’s site, and the depth of its submergence, A summary of the data taken during one of these tests (8) for both relative transmitted wave height, and wave energy is shown on Figures 76 and 77 plotted against the ratio (d/z) where d indicates the depth and z the height of the structure. The approximate range of effectiveness of structures whose crest heights are greater than those plotted on these figures, yet less than one and one-half times the anticipated wave height above still water level, may be deduced by considering that both the relative transmitted wave height and energy will be reduced to zero when the structure’s crest height is one and one-half times the anticipated wave height above still water level. This crest height should then be converted to a d/z ratio and the envelop curves extended to Hl/Hs = 0 or El/Es = 0 at this value of d/z. For example, a structure founded in 20 feet of water at high tide with a design wave height of 10 feet, to be completely effective, should have its crest 15 feet above the high water line. In this case, d = 20 feet, z = d + 15 = 35 feet, and Hl/Hs or El/Es = 0 at d/z = 20/35 = 0,57. May 1961 115 a • ' • 1 * ;3\ ’'1 - / IT «(,r _l;' .* _ "’.Hi Vv'"’* i. ■<•• :.’r.iir^ii«^i!*- »r ^• ■'• • ' W'4.’.tMt'.idUit^'^QiT .j^C«?«> '* - .- ;if ff^ 4 ..^ 1. , fl^ V J ^ .a^?>'hU xiu tfCte.4|AY//-. f.; •' ^0 *4^ btiti^JS:\ni3 id fc/sfeJto ?-• :'*?A *v*^^ ^ •(yo^iACWjtf. <'^*\lit^||ij 1 j*g-v^f iw.- «>n • r ^ • ,* ^?iS^l ‘ * * < ”t*J' .■‘i iwyrt *.■ .r'r ^4^ :>. udi nc / f..'.^ dfrtidrir*»d?.>c. .;: ^ ^^ r-: ^ • rAfti^ * • • •; T>Vi^ snip Msr^y f'i i>».iv/ .*»---rxt^ fo ..v.,i r- t^\:y:-’> '^' *' "’■ ‘ ^ : ' ^ ■ -■ ' l|n /.i. '».-7i -* “ fh«-6 % ir;.j^ f», *h^ \ r:».j.'.;- : • -.■.•|^» ^ ii » .'iM.' ..| ^ ' S- IW ■ ‘ ^lvttl%rr* r .'‘t H|f '■’' -» •- *-?i ''7^ Cfl'' juBK^fp ife*-lijfj.V . -ap*^ '-5t' ^ 1 - * T' l 4leAd ad^ rifi^f'^ii c*ii 3o “r*. *>» tWfl'll^'^SiSlBl to tttf, . Coilou oiuit f<»Ui;*r Y.v>'>ryt%^'».y ;-||mi* in ../ .jS i-"vAM*. nciil >- • t©:.# 4W..X-iM ^#U.'.*'«• rt-'-*. i^t. -- an EEl-fa d I Part II Chapter 4 PART II STRUCTURAL DESIGN CHAPTER 4 PHYSICAL FACTORS 4.1 WAVE HEIGHT 4.11 SELECTION OF DESIGN WAVE - If economically feasible, any structure exposed to wave action should be designed to withstand the effects of the highest wave to be expected at the structure’s location. Visual ob¬ servations of storm waves are usually \inreliable. Direct measurement is costly and time consuming. The wave height parameter which will normally be available from statistical analysis of synoptic weather charts will be that of the significant wave, Hs« Expressed in terms of nonnal wave spectra in nature, Hs may be further defined in approximate relation to other parameters in the spectznim as follows: Hi/ 3 ®s ~ Average of highest l/3 of all waves. 1.27Hg- = Average of highest 10% of all waves. 1.67^^H^ = Average of highest 1% of all waves. Prior to the development of laboratory data to establish coeffi¬ cients for use in the rubble-mound design formula it has been customary to use Hs as the design wave height. Since laboratory data on rubble- mound stability were determined employing waves of relatively \iniform height the continued use of Hs in conjunction with these coefficients would result in a structure less stable than indicated in the laboratory tests since 1/6 of the waves wotild be higher than Hg. Damage to a rubble-mound structure is progressive, however, and a long period of destructive wave action is normally required before the structure is damaged to such extent that it ceases to provide adequate protection. It is therefore usually necessary, in the interest of economy, to consider the frequency of occurrence of the waves of damaging size in selecting the design wave. For example, on the Atlantic and Gulf coasts of the United States hurricanes may provide the basis for design conditions, whereas the frequency of occurrence of the project hurriccuie at any specific site may range from once in 20 yeaurs to once in 50 years. On the north Pacific coast of the United States the weather pattern is more uniform and storms of intense severity are likely to occur several times each year. In the former case it would probably be uneconomical May 1961 116 Part II Chapter 4 to build a structure which would be stable under all conditions, and Hg may be a more practical selection. Use of Hs as the design wave height in the latter case would result in extensive annual damage caused by waves higher than Hs in the spectrum, thus necessitating frequent main¬ tenance; in such a case a higher wave height in the order of Hio ^7 advisable. It is accordingly considered generally preferable to select the design wave height between ranges of Hs and Hio on the basis of analysis of the following factors: 1. Effects of structure damage upon protected areas. 2. Frequency of maintenance requirements. 3. Availability of armor materials. 4* Comparative maintenance cost versus amortization of higher first cost. Selection of the design wave height also involves a determination of whether the structure is subject to attack by breaking waves. It is common practice to assume that a structure sited at a depth, d, (measured at design water stage) equivalent to 1.3H (or less), where H is the design wave height, will be s\ibjected to breaking waves. Likewise, it is customary in such cases to disregard larger waves which will break prior to reaching the structure on the assumption that the maximum destructive power will be delivered by the wave which breaks fully upon the structure. Therefore, when d equals or is less than 1.3H, the depth governs in selecting the design wave. When the design depth is within a range between 1.3Hs and 1.3Hi, it is obvious that some waves will break at the structtire but in decreasing frequency as d approaches 1.3Hi, where Hi is the average height of the highest 1 % of the waves. When the design depth is equal to or less than 1.3Hio it is considered preferable to use a breaking wave with H = 0.78d as the design wave. Statistical significemt wave data will normally be available only for deep water conditions (d>Lo/2). It is thus necessary to apply refraction analysis to determine wave characteristics at the structure site. Methods of refraction auialysis are given in Section 1.261. In certain special cases where the continental shelf is relatively broad and shallow it will be advisable to allow for energy loss due to bottom friction. Methods for computing wave height reduction due to friction are given in the section on Hurricane Waves. 4.12 DETERMINATION OF SIGNIFICANT DESIGN WAVE AT STRUCTURE - General procedures for developing the height and direction of the design wave by use of refraction diagrams are described as follows: From une structure's intended location, draw a set of "refraction fans," for the various wave periods in increments of about 2 seconds that might be expected at the site, and determine refraction coefficients by the method given in Chapter I. Tabulate the refraction coefficients so May 1961 116 a Part II Chapter 4 determined for the various wave periods chosen and for each deep water direction of approach. The statistical wave data derived from synoptic weather charts or other sources should then be reviewed to determine if the directions and periods, for which the refraction coefficients are large, maj be expected to recur with reasonable frequency. That combination of deep water wave height and refraction eoeffi> oient which gives the highest significant wave height at the structure's location determines the design wave direction of approach and period. The Inshore height so determined is the significant design wave height. A typical example of such an analysis is shown in Table 9. In the following example, although the highest significant deep water waves approached from directions ranging from U to NW, the refraction study indicated that higher significant waves Inshore may be expected from more southerly directions. It must be remembered that the accuracy with which refraction procedures may be applied in the determination of design wave character¬ istics decreases as the underwater contours over which the refraction diagrams must be drawn become more complex. Under highly complex bottom conditions, direct observations may be required. May 1961 116 b Part II Chapter 4 TABLE 9 DETERMINATION OF SIGNIFICANT DESIGN WAVE HEIGHTS Direction Significant Deep Water Wave Height Period Refraction Coefficient* Refracted Wave Height to Nearest Half Foot (l) (2) (3) (4) (5) (feet) (seconds) (feet) NW 15 8 0.10 1.5 10 0.07 1.0 12 0.04 0.5 WNW 12 8 0.15 2.0 10 0.12 1.5 12 0.09 1.0 W 10 10 0.30 3.0 12 0.31 3.0 14 0.20 2.0 16 0.25 2.5 WSW 10 10 0o60 6.0** 12 0.50 5.0 14 Oc35 3.5 16 0.35 3.5 sw 8 12 0.72 6.0** U 0.59 4.5 16 0o40 3.0 * This refraction coefficient is equal to ^ (h/Hq') ** Adopted as the significant design wave height Columns 1, 2, and 3 are taken from the statistical wave data as deter¬ mined from synoptic weather charts Column 4 is determined from the relative distances between two adjacent orthogonals in deep water and in shallow water Column 5 is the product of columns 2 and 4 4.2 WAVE FORCES 4.21 GENERAL - In an analysis of the forces exerted on structures by waves, a division should be made between the action of non-breaking waves, breaking waves, and broken waves. Pressures due to non-breaking waves will be essentially hydrostatic» Broken and breaking waves on the other hand, exert additional pressures due to the dynamic effects of the turbulent water in motion and the compression of entrapped air pocketso These pressures may be much greater than those due entirely to hydrostatic forces. Therefore structures located in an area in which storm waves may break, should be designed to 'withstand much greater forces and moments than those structures which would be attacked only by non-breaking waves. 117 Part II Chapter 4 4.22 DETERMINATION OF BREAKER DEPTH MD HEIGHTS - Given deep water wave conditions, depths at which waves will break may be found by using the full line curves (those labelled "Iversen") of Figure 38. However, if the depth of breaking as found from this c\irve is less than the anticipated depth of water at a structure, it should not immediately be assumed that these waves would not break on the structure. Storm waves are rarely so regular that the depth of breaking may be precisely determined. Their heights and lengths may be extremely variable, especially if the generating area is not far removed from the structTire. The curve of Figure 38 probably represents the lower limit of a range of breaker depths which may actually be found in nature for any one wave condition. For safety, it should be assumed that waves may break in depths greater than those given by the curve, up to the point at which the actual depth (d) at the structirre is lo5 times the depth of breaking (d-^) as found from Figure 38o For structures located in shallow water, when deep water wave con¬ ditions are not known or when the design wave determined from deep water conditions would break before reaching the structure, the height of the maximum wave which would break on the structure can be found approximately from the relationship d^ = l,3H^o 4.23 NON-BREAKING WAVES - Ordinarily, a shore structure would be so located that storm waves would break in the depth in which the structure js- founded. However, in protected regions or in areas where the available fetch is limited, non-breaking wave conditions may occur. The most commonly used method for the determination of pressure due to these waves in that of M. Sainflou(llO). 4.231 Sainflou Method(llO); Forces Due to Non-breaking Waves - If a wave of length L and height H strikes the vertical face of the wall ^ (Figure 78), a standing wave or clapotis will be set up. The point A is the maximum elevation of the crest, and point G is the minimum elevation of the trough of the clapotis. The mean level or orbit center is above the still water level D a distance, 2 h = ( TT HVL) coth (2 TT d/L) ( 21 ) The hydrostatic pressure and plotted as E. The and M is equal to H / h^, while ^ equals H - hQ. (wd) at the base C of the wall is scaled out from C triangle CDE is the hydrostatic pressure distribution against the wall due to water at still water levelo As the surface of the clapotis moves above or below still water it will increase or decrease the hydrostatic pressure at the base of the wall by the amount Pp, This change in pressure is wH_ 2 TT d (22) cosh Plotting Pj^ in both plus and minus directions from point E gives points B and F as the maximum and minimum pressures, respectively, at the base caused by the clapotis against the sea face of the wall. The solid curved lines labelled maximum wave pressure and minimum wave pressure denote the pressure distributions computed by theoretically exact formulas. These ciorved lines are so close to a straight line that it is permissible, and 118 1 Crest of Clopotis Mean Level (Orbit Center of Clopotis) , coth- ° L L Incident Wave d = Depth From Still Woter Level H = Height of Original Free Wove L = Length of Wove W? Wt. Per Cu. Ft. of Water P = Pressure the Clapotis Adds to or Substrocts From Still Water Pressure ho : Height of Orbit Center tor meon level) above Still Water Level wH cosh ZTTd A CLAPOTIS ON VERTICAL WALL FIGURE 78 May 1961 119 Part II Chapter 4 conservative, to approximate this distribution by use of straight dashed lines connecting A to B and G to F as shown in Figure 78. Figure 79 shows the equation for h^ reduced to graphical form, indicating values for Uiq for different ratios of d/L and for wave heights at 5-foot intervals up to 40 feet. Figure 80 shows the equation for Pj reduced to graphical form, indicating values of Pq for different ratios of d/b for the same wave heights. Assuming the same still water level on each side of the wall, an outward or seaward pressure exists which is equal to the hydrostatic pressures shovm by the triangle ODE in Figure 78. As the two pressures at still water level balance each other, the resultant pressure on the wall when the crest of the clapotis is against it is toward the land and is shown by the area ABED or AD'B’G. When the trough of the clapotis is at the wall the resultant pressure is toward the sea and is represented by the area DEFG or ^’F’G. A diagram of the resultant pressures on a vertical wall is also shown in Figure 78. Should there be no water on the landward side of the wall, then the total resultant pressure would be represented by the triangle ACB when the clapotis crest is at A. If there were wave action on the landward side, then the condition of crest of clapotis on the sea side and trough of the wave on the harbor side would produce maximum pressure from the sea side. The maximum pressure from the harbor side would be produced when the trough of the clapotis on the sea side and the crest of the clapotis on the land side are at the structiire. For a unit length of wall, with ho as the mean level of the clapotis above the still water level and P^ the common length of the segments ^ and ^ the resultant Ri and the moment about the base Mj^ are given respectively, for the maximum crest level (subscript e) and the minimum trough level (subscript i) of the clapotis by the formulas: Re = (d / H / hp ) (wd / Pp)_ wd^ (23) (d / hp / H)^ (wd / Pp) 6 % Mp wd^ (d+hp - H) (wd - Pp) "2 2 wd^ (d / hp - H)^ (wd - Pp) 6 (24) (25) (26) These formulas for pressures created by the clapotis are based upon the assumption that the vertical wall rests upon the natural bottom. If the vertical wall.rests on a stone foundation, the action of the wave depends on the profile of the foundation structure. 4*232 Wall of Low Height - If the height of the wall is less than the 120 0) 0x0 DETERMINATION OF VALUE OF ho IN SAINFLOU'S FORMULA FIGURE 79 121 10,000 9- 8 - 7 — 6 - 5 - — 4 - 3- 2 - _l I 1 I 1 1 1 1 1 1 1 1 I 1 1 1 1 1 T T 1 1 1 1 - n r r 1 1 1 1 'Mil “TT'I' M — — — , in which, W: Weight of Water cosh-^^ H = Height •• Wave d = Depth •' Water L = Length " Wave Example; Given H= 30ft.; d = 52ft.; L= 360ft. .^:.5 L:oi 44 Vertical from 0.144 on horizontal scale intersects curve for H :30ft. at a value for R of 1,330 lbs. per sq. ft. — — p. = — — — — — — E — -^ -—^ — — T — — — =. — j 1 1 1 1 t 1 1 1 1 1 1 1 — 1 — — 1 — 1 ^ — — 1 - t 1 r T 1 I" 1 1 1 1 1 -J-L-L-L- r -J 1 1 1 1 J- 1. .1. .1... 1 1 1 1 1 1 1 1 1 1 1 1 ipoo 9 — 8 — 7— 6 -- 4 — 3 — 2 -- 100 9- 8 - 7- 6 - 5- 4- 3- 2 - 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 d/L DETERMINATION OF VALUE OF P, IN SAINFLOU'S FORMULA FIGURE 80 122 Part II Chapter 4 predicted wave height at the wall, forces may be approximated by drawing the force polygons as if the wall were higher than the impinging wave, then analyzing only that portion below the wall crest. Thus in Figure 81 the force due to a wave crest at the wall is computed from the area AFBSC . Wall Elevation ^ A 1 \ B still Water Level V \ \ \ \ \ \ \ \ \ \ . N \ : s E Po FIGURE 81 PRESSURE ON WALLS OF LOW HEIGHT 4.24 WAVES BREAKING ON A STRUCTURE - Ordinarily bulkheads and seawalls are so located that storm waves may break directly on them. Even those structures which are located landward of the low water shore line may be exposed to the action of breaking waves at times of high water. There have been three major attempts to correlate the high pressures known to exist with measurable wave parameters. In 1934j D.A, Molitor(88) published a suggested solution of this problem, using a semi-theoretical approach and making use of the observations of D. D. Gaillard(42) taken on Lake Superior in 1904 and spring dynamometer readings taken during a storm at Toronto in 1915. Unfortunately since publication of the paper, pressures have been observed far in excess of those predicted by the Molitor equations and structures have failed which according to the Molitor equations were adequately designed. Essentially the Molitor wave pressure solution formed an envelope of the dynamometer readings taken in 1915. The measurements were taken throughout the storm and the maxima at various elevations recorded. Thus, though his equations purport to give a pressure curve for a single impinging wave,they really give a curve representing the maximum pressure recorded from many waves. This would ordinarily lead to conservative results but the range of wave paraimeter variables was too restricted for the results of these measurements to be applied to general wave conditions and the dynamometer equipment used would not measure the impact by pressures in¬ volved. 123 Part II, Chapter 4 In 1939, R. A. Bagnold(5) reported on measurements of shock pressures due to breaking waves recorded under model conditions. Pressures so recorded were greatly in excess of any prior predicted ones. Bagnold found that for these "shock pressures", a correlation could be established between the magnitude of the peak pressure and the thickness of a cushion of air entrapped by waves breaking on a structure. Unfortunately these experiments were interrupted and no further relationship has been established between the thickness of the air cushion and various wave parameters. The last approach to the determination of wave pressures was made in 1946 by R. R. Minikin(87). Although this method has some inconsistencies, it probably represents the closest approach to the actual pressures caused by a breaking wave. With the Minikin methods,failure of structures, other¬ wise unpredictable, may be explained. 4.241 Minikin Method; Forces Due to Breaking Waves - According to Minikin, the total pressure caused by waves breaking on a structure is due to a combination of dynamic and hydrostatic pressures, 'llie dynamic pressure is concentrated at still water level, and in units ordinarily used by American engineers, is given by: 101 H w d P = - X - (D ♦ d) (27) " S “ where - the height of wave just breaking on the structure (in feet) w^ - the unit weight of the water (in pounds per cubic foot) d - the depth oi water at the structure (in feet) D and 1,^ = deeper water depth and wave length respectively (in feet) Based on descriptive passages in his paper, values found for D and by the following method should approximate those he considered. dt should be noted that when plotting d/D versus wave pressures, the wave pressures become very large as d/D approaches 1.0, It is recommended that the application of the Minikin Method for determination of pressures due to waves breaking on a structure, be used with reservation when the slope in front of the structure is flatter than about 1 on 20. ) Given a wave period, the deep water wave length L (in feet) may be found from L = 5.12 (28) o With the ratio d/L , the wave length at the structure L at depth d may be determined from thi d/L value taken from Table D-1 of Appendix D (which in this case is d/L^), Knowing the slope S before the structure, D may be determined from D = d ♦ S (29) d may then be determined, again from Table D-1 of Appendix D, by computing e ratio D/L , case is D/L^), and dividing D by this ratio, the seaward side at still water level (subscript tabulating the corresponding value of d/L (vdiich in this The hydrostatic pressure on "s"), and at the depth d (subscript "d") are given by wH. P = s and = w (d + H^/2) d b (30) February 1957 124 Part II Chapter 4 The pressure curve may be plotted by considering that the dynamic pressure is concentrated at still water level and falls away rapidly along a para¬ bolic curve to zero at a distance H^/2 above and below still water level. To this is added the hydrostatic pressure on the seaward side plotted in a triangular area from H]3/2 above still water level to the bottom. Hydro¬ static pressures on the landward side plotted in a triangular area from still water level to the bottom must be subtracted if such pressures obtain. This construction is shown in Figure 82, the dashed and dotted lines indicating the separate effects of the dynamic and hydrostatic pressures, and the solid line the combined pressures, for the case where hydrostatic pressures are on each side of the wall. The resultant wave thrust or force per linear foot of structure may be determined from the area of this diagram and is given by Pm^b R = -3 - / Ps(d/ ^ ) (31) The resultant overturning moment about the ground line before the wall, the sum of the moments of the individual areas and is given by P P d M = d -/ -f— P Pk K is (32) Similar computations may be made if there is no water on the land side, in which case the thrust per linear foot of the structure is R= -Y (d/ -|) and the moment about the ground line is P PL P^ K 2 M = d / ^ ( d ^ ^) (33) (34) The combined pressures for this case are indicated by a dot-dash line on Figure 82. 4*25 WAVES BREAKING SEAWARD OF A STRUCTURE - Certain protective structures may be so located that even under severe storm and tide conditions waves will break before striking them. For example, walls built landward of the ordinary high water shore line may, because of scour or wind-set-up, have a significant depth of water against them during a storm. To date no studies have been made to relate forces due to broken waves to various wave parameters, and it is necessary to make certain simplifying assumptions to determine approximate design forces. Assume that immediately after breaking, the water mass in a wave moves forward with the velocity of wave propagation just before breaking; that is, upon breaking, the water particle motion changes from oscillatory to translatory. This turbulent slug of water then moves up to and over the 125 126 Part II Chapter 4 shore line. Dividing the inshore area after the wave breaks into two parts, seaward and landward of the shore line, for a conservative estimate of wave forces, assume that neither the wave height nor the wave velocity decreases from the breaking point to the shore line, and that after passing the shore line, the wave will run up to twice its height at breaking, decreas¬ ing linearly in velocity to zero at this point. It has been found from model tests, that upon breaking, approximately 70 percent of the full breaking wave height is above the still water level. Using for the velocity of wave propagation (C),the approximate relationship C = ^jgd^ where g is the acceleration of gravity and d-^ is the breaking wave depth, wave pressures on a wall may be approximated. These pressures will be partly dynamic and partly static. For walls located seaward of the shore line, the dynamic part will be: ■ m (35) where w is the unit weight of water. The static part will vary from zero at a height h^ (the height of that portion of the breaking wave above still water level, h^ * 0.7 H]^) to a maximum at the wall base. This maximum will be given by: P3 = «(d / hp (36) where d is the depth of water at the structure. Assuming that the dynamic pressure is uniformly distributed from the still water level to a height h^ above the still water level, the total wave thrust will be: R = R / R m s d / h = ^ 4 ( 37 ) 2 (d / hp2 The overturning moment about the ground line at the seaward face of the structure will be: M = M / M m s - R (d / -|) / R m s d / h ( —T-^) w d. h b c 2 (d / ^ ) / ^ (d / he) 3 (38) 127 Part II Chapter 4 The pressure diagram for this case is shovm on Figure 83. For walls located landward of the shore line, the velocity v' of the water mass at the structure at any point between the shore line and the point of maximum assumed uprush may be approximated by: v« = C(1 - -i) (39) and the wave height (h*) above the ground surface by: h« = h (1 - ^ ) (40) c x^ where x^ = the distance from the shore line to the structure X 2 ■= the distance from the shore line to the limit of wave uprush; X 2 = 2H, cot a a = the angle of the beach slope with the horizontal. An analysis similar to that for structures located seaward of the shore line gives for dynamic and static pressures: w V t2 Dynamic pressure wd X 2 (1 - —) 2 Xg Static pressure P =wh*zwh(l- — ) s c ^^2 (41) (42) and for the wave thrusts and moments: Thrust R = R / R m s - P h« / ~ m Moment P h» s w d,h X, ^ M = M / M m s R / R m2 s 3 w h X, 2 ^ ( 1 - ^) 2 X2 (43) (44) w d, h ^ X, 4 w h ^ X, 3 -7^ ( 1 - ^) / -5^ (1 - z) It must be the above basis simplified one. the shore line, remembered that pressures, forces and moments computed on will be approximations. The wave behavior assumed is a Especially in the case of structures located landward of where the run-up criterion adopted was a fixed function of 128 still Water Level. WAVE PRESSURES FROM BROKEN WAVES: WALL SEAWARD OF SHORE LINE FIGURE 83 129 Part II Chapter 4 the wave height alone, the preceding equations will not be exact. However, the assumptions used should give conservative values. (b) Stepp«d Face Wall WALL SHAPES FIGURE 85 4.26 EFFECT OF FACE SLOPE ON WAVE PRESSURES - Formulas given in the pre¬ ceding section may be used for all cases of structures with essentially vertical faces. If the face is sloped backwards at an angle with the horizontal (Figure 85a) the horizontal component of the dynamic pressure due to waves breaking either on or seaward of the wall should be reduced to P ’ = P sin ^ 6 where Q « 90® (45) mm The vertical component of this wave force may be neglected in stability computations. Forces on stepped face structures (Figure 85b) may, for design calculations, be computed as if the face were vertical, since it is May 1961 130 Part II Chapter 4 probable that dynamic pressures of the same order as those computed for ver¬ tical walls would exist. Forces on curved non-re-entrant face structures (Figure 85£) may be calculated by using a line from the top to the bottom of the face to determine an average slope. Re-entrant curved face walls (Figure 85d) may be considered as vertical, 4,27 - STABILITY OF RUBBLE-MOUND STRUCTURES - A rubble-mound structure of the type commonly employed to provide wave protection is composed of a mound of random-shaped and random-placed stones, protected with a cover layer of selected stones or specially shaped concrete armor units. Armor units in the primary cover layer may be placed in an orderly manner, to obtain some degree of wedging or interlocking action between individual units, or they may be dumped at random. For either method of placing, it is impossible presently to determine by rigorous analytical methods the forces required to displace individual armor luiits from the cover layers. It is also impossible to pre¬ determine theoretically when waves exert sufficient force on the structure to result in displacement of armor units, whether a large area of the cover layer will be displaced down the slope en masse or whether individual armor units will be lifted and rolled either up or down the slope. When short-period (gravity) waves impinge upon rubble-mound structures they may: (a) break com¬ pletely, projecting a jet of water approximately perpendicular to the slope; (b) break partially with a poorly defined jet; or (c) establish an oscillatory motion of the water particles up or down the structure slope, similar to the motion of a clapotis at a vertical wall. Thus, it can be seen that, when waves attack a rubble-mound structure, the resulting interplay of forces de¬ veloped by the wave-induced water motion and the resisting action of the armor units, is extremely complex. Until recently, the design of rubble-mound struc¬ tures was largely based on experience and general knowledge of a particular site’s conditions. Efforts to rationalize the design of these structures have been made. These entail the observation of failures and the determination of constants to be applied to various parameters in an attempt to explain the failures. Because of the empirical nature of the formulas developed, they are generally expressed in terms of the weight stone required to withstand design wave conditions. These formulas have been partially substantiated in model studies. However, they still are only a guide and cannot be blindly substituted for experience . Attempts to determine by theoretical analysis the stability characteristics of these structures, when under attack by storm waves, have not been successful. Empirical methods have been developed, how¬ ever, which if employed with proper care may be expected to give satisfactory results, 4,271 - Armor Units and Slope of Cover Layer , In 1938 and 1950, Iribarren (58) (59) (164) presented formulas for the design of rubble-mound structures. These formulas permitted calculation of side slopes and weights of individual stones above the water surface. In this equation, however, the coefficient is not dimensionless, and it is not possible to conduct small-scale verification of the formula. In 1952 Hudson (55) modified Iribarren*s Formula, using the same assump¬ tions and force diagrams, to obtain a dimensionless coefficient. In 1958 a May 1961 131 Part II Chapter 4 new rubble-mound stability formula was developed by Hudson of the U. S, Army Engineers Waterways Experiment Station. This new formula resulted from comprehensive investigations which found that the modified Iribarren*s Formula has certain limitations which render it unsatisfactory for use in correlating stability data from test of small-scale rubble-moimd structures. This equa¬ tion is based on the results of over 8 years of model testing as well as some preliitiinary verification of prototype data. The stability formula developed for determining the weight of armor units for rubble-mound structures is W w r K (S - 1)^ D r cot a (46) where W = Weight of armor unit in primary cover layer, lbs, 3 w^ = Unit weight (saturated surface dry) of armor unit, Ibs/ft . H = Design wave height measured at the location of the proposed structure (see section 4,1) S = Specific gravity of armor unit, relative to the water in w which structure is situated (S = ^ ) '^w 3 - Unit weight of water, fresh water 62,4 Ibs/ft , sea water 64.0 Ibs/ft^. a = Angle of breakwater slope, measured from horizontal,degrees, Kp = Coefficient that varies primarily with the shape of the armor units, roughness of the surface, sharpness of edges and degree of interlocking obtained in placement. The slope of the cover layer will partly be determined on the basis of rock sizes economically available at the quarry. However, in general, a cover layer slope steeper than 1 on 1,5 is not recommended, 4,272 - Selection of Kp Factor, The various values of the dimensionless coefficient Kp in equation 46 include all variables other than structure slope, wave height, and unit weight of armor units and the fluid in which they are placed. These variables include shape of the armor units, degree of interlocking, and the wave form at the time it reaches the structure. The equation at present ignores the angle of wave approach. While experience has demonstrated that this may be an important factor, especially important when the waves are breaking directly on the structure, there is no present basis for its evaluation. The Waterways Experiment Station has conducted numerous May 1961 132 Part II Chapter 4 laboratory tests with a view to establishing values of Kp for various con¬ ditions of some of the variables, the results of which are published in references 162 and 173, While these data comprise the most useful basis presently available for selecting Kp, there are a number of limitations in the application of laboratory results of prototype conditions which must be recognized. These are: (a) Laboratory waves were of relatively uniform height and period and thus did not produce the variable conditions inherent in nature, (b) Effects of a possible scale factor are still not adequately determined. Preliminary large-scale tests at the Beach Erosion Board indi¬ cate that the scale effect is unlikely to be important, (c) Placement of armor units in the prototype is unlikely to duplicate the degree of interlocking which may be obtained in the laboratory. In the former, while it is possible to set stones or blocks in such a manner as to achieve a closely knit structure above the water surface, the same quality of construction can rarely, if ever, be attained for that portion of the armor below the water surface,. It is thus considered advisable to use data obtained from random placement in the laboratory as a basis for Kp values (d) While very extensive testing has been accomplished for depth conditions producing non-breaking waves, very limited tests are thus far avail able for breaking waves. The depth and bottom slope conditions capable of producing the most destructive breaking wave condition have not yet been es¬ tablished, It is considered advisable, for the present, to apply a uniform 20 percent reduction in Kp factors determined for non-breaking waves for application to a breaking design wave. Such a reduction is more conservative than the values indicated by laboratory tests thus far completed, (e) It is well established that the head of a breakwater or jetty normally suffers more extensive and more frequent damage than the trunk of the structure. This is generally considered to be the result of several factors. The rounded head of the structure is subject to overtopping of a segment of the slope under all wave conditions; a portion of the head is usually subject to direct wave attack regardless of the direction of wave approach; and a wave trough on the lee side may coincide with maximum run-up on the windward side, creating a high static head for flow through the structure. Laboratory data are considered inadequate at this time (1960) to evaluate all of the factors contributing to this condition, however, suffici¬ ent data have been obtained to establish preliminary Kp values that are considered reasonably adequate. Based upon all available data and the limitations discussed above. Table 9-A presents provisional Kp values proposed for use at this time. Because of the limitations discussed above, values given in the table are considered May 1961 133 Part II Chapter 4 to provide little or no factor of safety. The experience of the field engineer may be used to adjust the Kp value obtained from Table 9A and deviation from these values should be fully substantiated. TABLE 9A Provisional Kp Values for Use in Determining Armor Unit Weight No Damage Criteria Armor Units_n* Smooth Quarrystone 2 ’’ *• >3 Rough Quarrystone 2 •» ” >3 Modified Cube 2 Tetrapod 2 Quadripod 2 Hexapod 2 Tribar 2 Structure Trunk Breaking Wave^^^ Non-Breaking Wave 2.1 2.6 2.6 3.2 2.8 3.5 3.4 4.3 6.0 7.5 6, 6 8.3 6.6 8.3 7.2 9.0 8.0 10.0 Structure Head Breaking Wave^®^ Non-Breaking Wave^^^ 2.0 2.4 2.7 3.2 - 5.0 5.0 6.5 5.0 6.5 5.0 7.0 5.0 7.5 n* is the number of units comprising the thickness of the armor layer, (a) Minor overtopping criteria (b) No overtopping criteria Preliminary test data for rough and angular quarrystone as compared with the Smooth quarrystone used in the original tests (non-breaking waves n>3) indi¬ cate a KD value of 5,3 as compared with 3,2. Although this results in a 1,65 to 1 ratio for Kp between rough and smooth stone, a more conservative ratio of 1.33 to 1 is recommended at this time, 4,273 - Precast Concrete Armor Units . In an effort to provide adequate armor units for breakwaters and jetties, that is, in areas where stone of desired weight and durability could not be economically obtained, many dif¬ ferent types of concrete armor units have been developed. The first of the armor units, of significant value, was the tetrapod, developed and patented in 1950 by the Neyrpic Laboratories of Grenoble, France. The tetrapod is an unreinforced concrete shape with four truncated conical legs projecting radially from a center point (see Plate D-ll), May 1961 133 a Part II Chapter 4 The tetrapod has an international patent, and requires a royalty be paid per cubic yard of concrete in the unit, A more recent concrete shape is the tribar, developed and patented by R, Q, Palmer of the Corps of Engineers, Honolulu District, The tribar is an unreinforced concrete shape consisting of three bars tied together by three radial arms (see Plate D-12), Inasmuch as the U, S, Government assisted Palmer in the development of the tribar, the U, S, Government has royalty free use. Various other shapes have been developed and tested by the Corps of Engineers, such as quadripods, modified cubes, and hexapods (see Plates D-13, D-14, and D-15), Table 9A lists the Kp values for the various shapes but only the tetrapod and tribar have been used in prototype structures, Tetrapods have been used all over the world in many structures. The three most notable applications in this country are Crescent City Breakwater, Calif¬ ornia (207) 1956. Kahului Breakwater, Hawaii (208) 1957, and Rincon Island, California (209) 1958, Since tribars are a relatively recent development their application has been limited. These units were used in repairing the Nawiliwili Breakwater, Hawaii (208) in 1958, Recent inspection (1960) indicates that the tribar is a stable unit, but full evaluation of the structural stability of the tribar should be based on a longer trial period. The design of the cover layers using concrete components is similar to that for quarrystone structures. Equation 46 is used with the applicable value of Kp , however, structural damage to the units may occur if movement is per¬ mitted on the slope. The structure may also require a concrete cap, see section 4,275, See also the cross-section for the outer breakwater at Crescent City, California, on Figure 134, Along with the evaluation of the armor units, the tetrahedron was also tested with the following results: The Kp value for the tetrahedron, uni¬ formly pllced in two layers, was approximately the same as for random placed quarrystone, but uniform placement of this shape in the field is impracticable. Armor units of the tetrahedron shape are not adaptable to random placement in two layers, 4,274 - Crest Elevation and Width, The primary function of breakwaters is to provide adequate protection from wave action in selected harbor areas. Consequently, overtopping usually can be tolerated only if it does not gene¬ rate waves in the protected area which exceed allowable limits as determined by the type of harbor and the use for which different areas in the harbor are designed. Factors that determine whether overtopping will occur, and the extent of overtopping when it does occur, are the elevation of the crest above maximum design water level and the height of wave run-up (R), As the wave run-up depends on the degree of porosity and surface roughness of the cover layer if the cover layer is chinked, or in other ways made smoother or less permeable, the limit of maximum run-up will be greater than the value obtained for rubble-mound slopes and less than the value for smooth slopes. 133 b Part II Chapter 4 See section 3.271 and Figure 61-L The elevation selected for the crest of the structure should be the lowest height that provides the protection re¬ quired. Excessive overtopping of the structure can cause choppiness of the water surface in the lee of the structure to the extent that it may be detrimental to harbor operations. Some operations such as mooring of small craft and some types of commercial unloading require relatively calm waters. The width of the crest to a large extent depends on the degree of allow¬ able overtopping, VJhere there will be no appreciable overtopping, the crest width is not critical with respect to the forces on the structure. The question of the crest width of a rubble-type structure when overtopping is allowed is difficult to answer due to the lack of data on the subject. The practical minimum recommended for rubble-mound structures, for overtopping or non-overtopping conditions, is the crest width corresponding to the combined widths of three cap stones (n = 3), Thus on this basis the crest width may be obtained from the following equation; B V /'W a/3 n k. (—; A Wr (47) where B s Crest width in feet n s: Number of Stones (n = 3 is recommended minimum) k = Layer coefficient (Table 9B) A W = Weight of armor unit in primary cover layer, lbs, 3 w^ = Unit weight of armor unit, lbs/ ft In addition, the crest width should provide sufficient width for operation of any construction and maintenance equipment needed, 4,275 - Concrete Cap for Rubble-Mound Structures . In numerous cases, poured-in-place concrete has been added in various forms to the superstructure of rubble-mound jetties and breakwaters. Such use ranges from filling the interstices of surface layers of armor stone, on the crest and as far down the slopes as wave action will permit, to adding large monolithic blocks, up to several hundred tons in weight. This use of concrete is normally designed to serve any or all of three purposes; (a) to strengthen the structure; (b) in¬ crease the crest height; and (c) provide roadway access along the crest for maintenance purposes. In general, experience indicates that concrete placed in the voids on the structure slopes has little value. By reducing slope, roughness and surface porosity, such use of concrete tends to increase the run-up factor. The effec¬ tive life of the concrete is relatively short as the bond between concrete and stone is quickly broken by structure settlement. For roadway purposes, a concrete cap can usually be justified if frequent maintenance of armored slopes is anticipated. May 1961 133 c Part II Chapter 4 In evaluating the need for a massive concrete cap to increase structural stability against overtopping forces, consideration should be given to the comparative cost of including a cap or increasing the dimensions of the rubble structure. Under overtopping conditions, a massive concrete cap is not neces¬ sary for stability purposes when the difference in elevation between the structure’s crest and the maximum wave run-up on the projected slope of the structure is less than 15% of the design wave height (Ho*). It is advisable to defer construction of the cap for at least a year after completion of the rubble- structure to allow time for initial settlement and consolidation. Unless a very substantial saving would result from the use of concrete for this pur¬ pose an all rubble structure is considered preferable. Experience indicates that maintenance costs for an adequately designed rubble-mound are likely to be lower than for any alternative composite type structure. 4,276 - Additional Design Features , Additional information required to design rubble-mound structures include: (a) - The layer thickness (r), porosity (P), and the number of armor units (Nf) in the cover layer, (b) - The depth below minimum swl to which the primary armor units of weight W should be placed, (c) - The stability requirements of the back slope or leeward side of the rubble structure, (d) - The necessary weights of secondary armor units required for stability of that portion of the cover layer below the bottom position of the armor units in the primary cover layer, (e) - The necessary weight and thickness of the layers of stones under the primary cover layer to insure minimum back pressure on the armor units in the primary cover layer, and to prevent the washing out of underlayer material through the voids in the cover layer, (f) - The need of a bedding or filter layer depending on foundation requirement. ^ - Thickness. Number of Armor Units, and Porosity . The thickness of the cover layers and number of armor units required for construction can also be determined from the following formula: r=nk(l!L)^/^ (47a). A Wj. w 2/3 where r is the thickness, in ft, of n layers of armor units of weight W and unit weight Wj-; Nj. is the required number of individual armor units for a given surface area A, where A is in sq, ft,; is the layer coefficient and P is the average porosity of the cover layer in per cent. Values of the co¬ efficient kfi and P, determined experimentally, are presented for random place¬ ment only in Table 9B, May 1961 133 d TABLE 9B Part II Chapter 4 LAY HI COEFFICIENT AND POROSITY FOR VARIOUS ARMOR UNITS Layer Coefficient Armor Unit n k P, % Quarrystone 2 1.0 38 Quarrystone >3 1.0 40 Modified cube 2 1.1 47 Tetrapod 2 1.0 50 Quadr ipod 2 1.0 50 Hexapod 2 1.15 47 Tribar 2 1.0 54 ^ - Bottom Elevation of Primary Cover Layer . The armor units in the primary cover layer, the weights of which are obtained by equation 46, should be extended down slope to an elevation below the minimum swl equal to one wave height (-H) when the structure is in a depth >1.5H as shown in Figure 85-A. When the structure is in a depth < 1,5H, armor units of weight W should be extended to the bottom as shown in Figure 85B, c - Back Slope or Leeward Cover Layer . The design of the back slope or leeward cover layer of a rubble-mound structure is based on the following parameters: (1) - The extent of overtopping (2) - The wave and surge conditions acting directly on the landward slope. (3) - The porosity of the rubble structure and differential hydro¬ static head resulting in uplift forces dislodging the back slope armor units. If the crest were established at an elevation where no possible over¬ topping could occur, theoretically, the weight of armor units on the back slope cover layer should then depend on the wave and surge action that occurs in the lee and the porosity of the structure. When overtopping does occur (see section 4.275), the primary armor units should be carried over the crest and down the back slope to the minimum swl. When both side slopes are subjected to similar wave action, the side slopes should be of similar design. Also foi a distance of at least 150 feet measured back from the seaward end of the structure, the armoring of the head of a breakwater or jetty should be the same on the leeward slope as on the slope exposed to the sea. ^ - Secondary Cover Layer . The weight of armor units in the secondary cover layer, between -H and -1.5H, should be equal to or greater than one-half the weight of armor units in the primary cover layer (Ws^ W/2) and below -1.5H can be reduced to W/15 for the same slope condition, (see Figure 85-A). When the structure is in shallow water (see Figure 85B), that is, the depth d = H to 1,3H, the armor units in the primary cover layer should be carried the full May 1961 133 e Seaward Leeward THEORETICAL SECTION DEEP WATER RUBBLE MOUND BREAKWATER ©NONBREAKING WAVE CONDITIONS @ DEPTH OF WATER > 1.3 WAVE HEIGHTS Moy 1961 133 f May 1961 I33g Part II Chapter 4 height of the structure. The above ratios between the weights of armor units in the primary and secondary cover layers are strictly applicable only when quarrystone armor units are used in the entire cover layer for the same slope, Tlius, when precast concrete armor units are used in the primary cover layer, the weight of quarrystones in the other layers are not accurate unless based on the equivalent weight (W) of quarrystone armor units as would be used in the primary cover layer equal in stability to the selected concrete armor units. The thickness of the secondary cover layer (see Figures 85A and 85B) from -H to the bottom should be equal to or greater than that of the primary cover layer. Thus, based on the preceding ratios between the armor unit weight (W) in the primary cover layer and the armor unit weight in the secondary cover layer, if n = 2 for the primary cover layer (that is two stones thick) then n = 2,5 for the secondary cover layer from -K to 1,5H, and n = 5 for that portion of the secondary cover layer below -1,5H, e_ - Underlayers , From the bottom of the crest armor stone to -1,5H the first underlayer should be composed of a minimum of two layers (n = 2) of stones of weight W/10 approximately (see Figure 85A), The second underlayer for this portion of the structure (above -1,5H) should be a minimum of two layers of stones of weight W/200 (approximately). The first underlayer, for that portion of the structure below -1,5H can be a minimum of two layers of stones of weight = W/300 (approximately). The second underlayer for that portion of the structure below -1,5H as well as the core material can be as small as W/6000, or quarry-run (see Figure 85A), ^ - Bedding or Filter Layer . Foundation for marine structures deserve as much, if not more, careful study than those for land structures. Wave forces acting against a rubble type structure have been found to attack the natural bottom and the foundation of the structure, even at depths usually supposed to be little affected by such forces, A rubble structure may be protected from settlement resulting from leaching, piping, undermining, or scour by the use of a bedding layer or blanket. Experience indicates that the use of a bedding layer or filter blanket to protect the foundations of rubble-mound structures from undermining is advisable, except (1) where the depths of water are well below twice the maximum wave height, (2) where the anticipated current velocities are smaller than that necessary to move the average size of foundation material, or (3) where the foundation is a hard and durable material, such as bed rock. When the foundation consists of cohesive material (other than rock) a filter blan¬ ket may not be required, however, a layer of quarry spalls or other crushed rock or gravel may be placed as a bedding layer or apron to minimize scour of the bottom and/or settlement of the structure. Foundations which meet filter requirements in themselves do not require a filter blanket. When the rubble structure is placed on a sand foundation, filter blanket material may be pro¬ vided to prevent waves and currents from removing sand through the voids of May 1961 133 h Part II Chapter 4 the rubble and thus causing settlement. When large stones are placed directly on a sand foundation at depths insufficient to avoid wave and current action on the bottom, as in the surf zone, the rubble will settle into the sand until it reaches the depth below which the sand will not be disturbed by the cur¬ rents. In this case, a larger amount of rubble would be required to allow for the loss of rubble due to settlement and thus provide a more stable foundation. The gradation requirements of a bedding layer depend principally upon the littoral characteristics in the area and on the foundation conditions. However, quarry spalls, ranging in size from about 1 pound to about 50 pounds, will generally suffice. The thickness of the layer depends generally on the depth of water in which the material is to be placed and the size of stone used in the bedding layer, but should not be less than 12 inches to assure that bottom irregularities are completely covered. Where wave action and currents might scour the bottom and undermine the toe of the structure the bedding layer should extend 5 feet beyond the toe of the cover stone, 4,277 - Plates and Tables . The following plates and tables have been included in Appendix D to facilitate the solution of the WES Rubble-mound Stability Formula. Plates D-5 through D-8 are curves showing the variation of WK^ from equation 46 for sea water with wave height (H), Values of the functions have been computed for slopes of 1 on 1.25 to 1 on 5 with the unit weight of stone being used as a parameter. plates D-11 through D-l5 give the volume, linear dimensions thickness of layers, and number of units required to construct cover layers as a function of weight and unit weight of armor units for tetrapods, quadripods, tribars, modified cubes, and hexapods, respectively. Table D-10 is the same as Table 9-A and is reproduced in Appendix D for convenience in using Plates D-5 through D-8. May 1961 134 Part II Chapter 4 4,28 WAVE FORCES ON PILES - There have been many approaches to the problem of computing the forces which waves will exert on piles (82), (91), (108), amd (177). Perhaps, the most popular method considers the force, f, divided into two parts, the first fp depending on the horizontal velocity of the water and the second, f^, depending on the change of the horizontal velocity (acceleration) of the water (see Figures 86A and 86B), f = fj) + (48) The two components are given by the following equations: ^T) = i P^d Du I u I C49) |u|= absolute value of u fi = il pCm d2 ^ (50) 4 dt Tidiere p = mass density (p = — or 2 slugs per cubic foot for sea water), u is the horizontal particle-vflocity in ft/sec, du/dt the horizontal particle acceleration in ft/sec^, Cp is the characteristic drag coefficient and the inertial coefficient, both of which must be obtained by experiments, u and du/dt are functions of wave height, period, and water depth (H, T, d) and vary with position (x, z) within the wave. The total resultant force on a section of pile is given by: F s= Fp + Fi (51) where p 4 ^2 Fp = zi fp dz (52) and 4 _ II a, fi dz (53) If the pile is supported rigidly at the sea bottom and extends above the crest of the wave, then zi = -d and Z2 "^he free surface elevation with respect to the Stillwater depth. (See Figure 86-A) The moments M = Mjj ♦ where Md = and Mi = about the bottom of the pile may be obtained from fp (d ♦ z) dz f£ (d * z) dz (54) (55) (56) The above equations represent those used for obtaining wave forces, and 135 Wave Direction -► D z> ”'z H = Wave height L = Wave Iength T = Wove period d : Still-water level z = Depth below still- water level (neg.) t = Time 6: Angular particle position in it's orbit s 27Tt/T ♦ z i k -Pile -► -f X Bottom ^V77777777777777777777777777777;V777777777777777777. V/7, A-SCHEMATIC DIAGRAM OF PARTICLE MOTION Inertia B-FORCC COMPONENTS FIGURE 86-AandB FORCES ON PILES (Morrison, 1951) 136 Part II Chapter 4 require that u, du/dt, Cp and be known. The horizontal par tide-velocities and accelerations are obtained from wave theory, and the drag and inertial coefficients from wave force experiments. Various theories given in the references are available for computing particle velocities and accelerations. These theories can be used when hori¬ zontal and vertical distribution of the forces is required in great detail. However, in many instances it is sufficient to know only the crest elevation, wave length, and maximum total force and overturning moments. A set of generalized graphs, discussed subsequently may be used together with the following formulas; where: >■ 0 ™ ' 1/2 PW2 '■to = 1/2 pd2 HKj„ and ^Dm M- tm ^Dm P. im (56a) (56b) (5 6c) (56d) (56e) Fqj^ is the maximum value of the integrated or total horizontal drag force, and occurs at the crest positi IT) ^foCNJ—o May 1961 138 C FIGURE 86 E. CORRECTION FACTOR FOR WAVE LENGTH DUE TO STEEPNESS. FIGURE 86 F d/T* ) From Reid and Bretschneider 1953 FIGURE 86 G 138 d May 1961 UJO E u. 6 a li. Moy 1961 138 e FIGURE 86H. ^ VS FOR THE BREAKING SOLITARY WAVE AND THE AIRY WAVE. May 1961 138 f FIGURE 86 I RELATIVE LEVER ARM MEASURED FROM BOTTOM vt RELATIVE DEPTH CORRESPONDING TO MAXIMUM INERTIAL FORCE ro tn I( — D fe« — >t STEADY STATE (v'ennord ) ' • • • « • • •• • • * * .* •• S\ r... • • I .*•*1 • • • N. \ yh . ,% •• r ; • • % \ \ * A • * • t » • • V ’\ . t • • • • \ .\ ' • to* • • \ \ • • • • • » • Force = Co Au^ V- — • • • • 4 A : Projec u = Portic D = Pile d V = Kinem P - Water ted area le velocit ameter atic vise density y osi ty i » • • 10 ’ 2 3 4 Reynolds Number, Re = 5 6 uD 7 8 9 10' FIGURE 87-A. COEFFICIENT OF DRAG AS A FUNCTION OF REYNOLDS NUMBER FOR WAVES HIGHER THAN 10 FEET May 1961 138 i DRAG COEFFICIENT VS REYNOLDS NUMBER FOR CIRCULAR CYLINDER SOLID CURVE REPRESENTS STEADY STATE VELOCITY SYMBOLS REPRESENT OSCILLATORY MOTION A BERKELEY WAVE TANK EXPERIMENTS • BERKELEY MONTEREY FIELD DATA ■ TEXAS ABM FIELD DATA BERKELEY WAVE TANK NUMBER AT EACH POINT IS NUMBER OF WAVES CONSIDERED 10 ' 8 lO' 6 8 uD I . 10 ' 4 6 8 1 - 10 * FIGURE 87-B May 1961 138 j Part II Chapter 4 and inertial coefficients, Cp and C]^, There are a number of publications per¬ taining to the determination of Cq and from wave force measurements. Some of the tests were laboratory experiments and others were conducted under field conditions. The laboratory data are associated with low Reynolds numbers, and the field data with higher Reynolds numbers. Certain design waves will be within the range of the field tests, whereas others will be associated with higher waves and larger piles with corresponding higher Reynolds numbers than those associated with the field tests. Based on the laboratory data and linear wave theory, Morison, et al presented relationships for obtaining Cp and , as shown in Figure 87, The phase position p corresponds to that associated with maximum wave force and depends on the factor D^d/H^L, where D is pile diameter, d water depth, H wave height, and L wave length. Drag coefficients (Cp) and inertial coefficients (Q^) have also been de¬ termined from field tests, but there is wide disagreement between the various publications in regard to these coefficients. The main disagreement arises from the fact that different theories and methods were used to evaluate these coef f icients. Two important factors should be taken into account in regard to under¬ standing the results of the various experiments so that the proper application can be made for design purposes. In regard to the analysis of wave force data, one should use the most suitable wave theory or means for computing particle velocities and accelerations, so that one may obtain realistic values for Cp and In regard to application of the results of analyses for design pur¬ poses, once again one should use the most suitable wave theory for computing particle velocities and accelerations. Prom a design point of view there are two sources of prototype experi¬ mental data which can be considered, that of the University of California and that of Texas A & M Research Foundation, The first soturce of data reports an average value of s 2,5 and a large scatter of Cp around a mean value of Cp * 1,0, Figure 87A shows Cp versus Reynolds number based on the California data. The second source reports C^ = 1.47 and Cp = 0,53, Figure 87B shows Cp versus Reynolds number based on the Texas A & M data. The wide discrepancy between the two sets of experiments is only fictitious, since the methods of analysis were vastly different. The analysis of the data from University of California was based on linear wave theory, which is far from applicable for the measured wave conditions. The data from Texas A & M were analyzed rigor¬ ously, and velocities and accelerations were quite accurately computed. The wave height and the pile diameter used for the Texas A & M data were both considerably smaller than those of California, In order to reconcile the apparent discrepancies between the two sources of information, some of the wave data of the University of California were 138 k Part II Chapter 4 re-analyzed using higher order wave theory. In order to insure that the velocities were above critical Reynolds number only waves of 10 feet and higher were considered for the drag force determination. Two sets of testing conditions were considered: (1) mean water depth cf 47,6 feet, test section at 42,5 feet above the bottom and a pile diameter of 1,062 feet; and (2) mean water depth of 47,0 feet, test section at 33,0 feet above the bottom and a pile diameter of 2,0 feet. The first order, second order, and third order wave theories were used to compute the velocities, and drag coefficients were determined for each of the three orders. The results of this analysis are given below: Pile size D, feet 1,062 2.0 Mean water depth d, feet 47,6 47.0 Elevation of test section S, feet 42,5 33.0 Range in wave heights, feet 10 to 20,5 10 to 12.5 Range in wave periods, seconds 13,5 to 15,9 9,1 to 15,4 Number of waves 50 31 First order theory 0,859 0.682 Standard deviation 0,364 0.311 Second order theory 0.475 0.429 standard deviation 0.200 0.212 Third order theory 0.352 0.353 Standard deviation 0.171 0.189 From the above table it is seen that the use of linear wave theory order) gives mean values of the drag coefficient which are different for dif¬ ferent pile sizes, whereas, the third order wave theory gives the same values of Q) for both pile sizes. The position of the test section is higher for the 1,062-foot pile than for the 2,0-foot pile, and it is this factor which re¬ sults in the different computed velocities, and hence different computed drag coefficients when linear wave theory is used. The standard deviation reflects the scatter of the data about a mean value. From normal probability distribu¬ tion 15,87 percent of the data will have a Cp equal to or greater than mean Cp + one standard deviation, 2,275 percent will have Cp + two standard devia¬ tions and 0,135 percent will have Cp + three standard deviations. The results of the second order determination are in fairly good agree¬ ment with the results given by Reid (178), The results of the third order determinations for Cp are in very close agreement with those given for steady flow. For design purposes one might use a mean value of Cp plus one or two standard deviations, depending on the degree of conservatism required. In many May 1961 138 i Part II Chapter 4 cases for preliminary estimates of the wave forces, the first order theory might be adequate, and for final design the third or higher order theory might be required. In view of the above and the results of Cd determination, the following table of Cp values is suggested. _Coefficient of drag_ Plus one Plus two Plus three Standard Standard Standard Theory _ Mean _ Deviation _ Deviations _ Deviations Linear 0,77 1,10 1,44 1,77 Second order 0.40 0,61 0.82 1,03 Third order 0.35 0.53 0.71 0.89 If a higher order wave theory than given above is used to determine par tide-velocities then a re-evaluation of the data might be required or else the drag coefficients corresponding to those given for the third order theory should be used. The use of a higher order wave theory will not change the Cp values given above, since the corresponding waves in many instances if analyzed by use of higher order theory, will not have velocities much dif¬ ferent from those obtained by use of third order theory. The reason for this is that the waves were not as large as they might have been, or, for example, as might be used for design purposes. The design wave may be larger and have velocities greater than predicted by the third order theory, and a higher order theory might be used to compute the velocities for such a wave. The greater the deviation of the design wave from a theory (first, second, third order, etc.) the greater should be the number of standard deviations of the drag coefficient applied to the mean drag coefficient. If the generalized graphs are used to obtain then Cp = 0,53 or 0,71 from the above table should be satisfactory, depending on whether one or two standard deviations is advisable. The inertial coefficient is also required particularly when the pile diameter is large compared with the wave parameter. The theoretical value of the inertial coefficient for a vertical circular pile for potential flow is = 2,0, The field tests mentioned above yield data from which com¬ puted values differ from = 2,0, Some of this difference is due to the actual theory used for determining accelerations, and some is due to the fact that the flow about the pile may be somewhat different from potential flow. If linear theory is used to obtain accelerations, then a value of Cj^ should be greater than = 2,0 according to the University of California data. If the generalized graphs are used to determine K then Qui = 2.0 should be satisfactory, 4,282 Use of the Generalized Graphs : The generalized graphs are given in Figures 86C to 86J, Use of these May 1961 138 m Part II Chapter 4 figures is illustrated in the following examples: Example: Given H = 35 feet, d = 85 feet and T * 12 seconds Compute H/T^ = 35/144 * 0,243 ft/sec^ dA^ = 85/144 = 0.590 ft/sec^ H/d = 35/85 e 0.412 From Figure 86C ^q/H = 0,68 estimated Compute «= 0.68 (35) = 23,8 feet From Figure 86D: * 0,75 From Figure 86B: L/L^ * 1.04, and using equation 56f Compute L = 5.12 t2 (La/Lq) (LAa> « 573 feet From Figure 86F: = 13,0 ft/sec^, estimated Using Cp = 0,53 and D = 8,0 feet, compute F^^ « 1/2 p Cp D Kp *r 1/2 (2) (.53) (8) (13.0) (35)^ s 67,500 lbs. From Figure 861: Sp/d = 0,91 Compute Sp = 0.91 (85) = 77,5 feet Compute Mp = 67,500 x 77,5 = 5,230,000 foot-lbs. From Figure 86G: = 19.5 ft/sec^ Compute F. = 1/2 p C K. H « 1/2 (2) (2.0) (8)^ (19.5) (35) im m im = 87,200 lbs. From Figure 86J: S^/d = 0.78 Compute $£ = 0,78 (85) * 74,8 feet Compute M£j,,= 87,200 (74,8) = 6,550,000 foot-lbs. Compute Pixn/PDm " 87,500/67,500 = 1,3 May 1961 138 n Part II Chapter 4 Prom Figure 8®!; Pm/^Dm ® 1.60 estimated Compute Fm =1.60 (67,500) = 108,000 lbs, maximum total force The maximum total moment will be approximately Mf = 1.60 (5,230,000) = 8,400,000 foot-lbs. If the pile size had been D = 12,0 feet instead of 8 feet, the drag forces and moments would be increased by 12/8 = 1.5 and the inertial forces and moments by (12/8)2 s 2,25 For a 12-foot pile F^ = 1,5 C67,500) * 101,250 lbs. Mp = 77.5 (101,250) = 7,850,000 foot-lbs. F^ = 2,25 (87,200) = 196,000 lbs. and Mi = 74,8 (196,000) » 14,650,000 foot-lbs. Compute Pim/Pniti = » 1,94 im Dm 101,250 From Figure 8^: Pn/Pom = 2,2 Compute P = 2,2 (101,250) *= 223,000 lbs, total force The total maximum moment will be approximately Mp = 2.2 (7,850,000) = 17,300,000 foot-lbs. May 1961 139 Part II Chapter 4 4.3 EARTH FORCES ^^^^ 4.31 ACTIVE FORCES - The horizontal component of the active earth force is evaluated from the general wedge theory formula: P = ^ tan^ I (90 - 0) (57) where P = the horizontal component of the lateral force w = the unit weight of fill 0 = the internal angle of friction of the material. Equation 57 is used only for vertical walls with substantially horizontal backfill. The structure is assiomed to be non-rigid to the extent that an extremely small rotational movement, necessary to produce the internal friction of the backfill, can occur. Values for tan^ (9O-0)/2 for various values of 0 are given in Table D-12, Appendix D. If the wall is vertical but the fill slopes at an angle p to the horizontal, the complete equation is wh P - cos p - ^ 2 2 ^ cos p- cos jS cos p cos cos2 p- cos2 0 (5E) For structures having a uniform back batter and fills with uniform slope , the general wedge theory, equation 59 niay be used to evaluate the magnitude and direction of earth force. This formula takes into account the friction along the surface of the wall (see Figure 88). P = ^ sin (Q - 0) (1 / N) sin 0 wh‘ sin(<^, + 0) in which N - sin (0 / 0q) sin (0 - p) sin (0^ / 0) sin (0 - p) (59) ( 60 ) where P = total force in pounds h = vertical height of fill in feet w = unit weight of fill material 0 = internal friction angle of fill material 0j_ = friction angle between backfill and face of wall p = angle between surface of backfill and horizontal plane 0 = angle between back of retaining wall and a horizontal plane. 4.311 Unit Weights and Internal Friction Angles - The unit weight (w) of typical materials and their internal friction angle (0) are given in Table 10, These are average or normal values of w and 0. 140 EARTH PRESSURE ON AN INCLINED WALL FIGURE 88 —. —- - fp>_ A A /•O . , M — ^ o\ 1 /■ 0 “ m * •« • m * 1 • ' f > /•£) 4 /o . 0 V -. / «* . ■ n H / 4 2 h A. •. H A. ® 0.33 h k t . o.33h y 4 / r ' ' ^ 0 ^ 0 ^ \ o!33h . i (a). Liquid Pressure (b) Negative Slope (c) Back Batter Flatter Backfill Than 2 to 1. APPLICATION OF RESULTANT OF EARTH PRESSURE FIGURE 89 May 1961 141 TABLE 10 Part II Chapter 4 UNIT WEIGHTS AND INTERNAL FRICTION ANGLES Weight Natural Weight Internal Material Wei ght Dry Drained Submersed Friction Ansle (0) lbs. /cu. ft. Ibs./cu. ft. / Ibs./cu. ft. desrees Clay, soft 60 - 90 100 - 120 35 - 55 10 Clay, compact 90 - 115 120 - 135 55 - 70 20 - 25 Gravel 105 - 120 115 - 135 65 - 75 40 - 50 Silt, compact 85 - 105 115 - 130 55 - 65 25 - 40 Silt,loose 65 - 85 105 - 115 40 - 50 20 - 30 Sand 90 - 105 105 - 120 55 - 65 25 - 40 Sand, clay, compact 115 - 130 135 - 140 70 - 80 40 - 50 The resultant pressure is inclined from the normal to the back of the wall by the angle of wall friction Values for can be taken from Table 11, but should never exceed the Internal friction angle of the back¬ fill material. The vertical component of the earth pressure (P) need not be considered in the stability analysis unless it has considerable effect on the structural design. TABLE 11 COEFFICIENTS AND ANGLES OF FRICTION Coefficient of Angle of Kind of Surface Friction f Friction Granite, Limestone, Marble: Soft dressed on soft dressed 0.70 35° 00» Hard dressed on hard dressed 0.55 28° 50’ Hard dressed on soft dressed 0.65 33° 00' Stone, brick, or concrete: Masonry on masonry 0.65 33° 00' Masonry on wood-with grain 0.60 31° 00' Masonry on wood-cross grain 0.50 26° 40 ' Masonry on dry clay 0.50 26° 40 ' Masonry on wet or moist clay 0.33 .18° 20' Masonry on sand 0.40 21° 50' Masonry on gravel 0.60 31° 00' Soft stone on steel or iron 0.40 21° 50' Hard stone on steel or iron 0.30 16° 40' Note:(S6) Angles of friction should be reduced by about 5 degrees if the wall fill will support train or truck traffic. The coefficient of friction f would equal the tangent of the new angle JSj. 142 Part II Chapter 4 Earth pressure against structures of irregular section such as stepped stone blocks or those having two or more back batters may be computed by equation 59 by substituting an approximate average back batter to determine the angle (©). 4.312 Application of the Resultant of Earth Force - For all structures with hydrostatic pressure or negative sloped backfills, the resultant of force should be considered to act at one-third of the height. For structures having batters on the back face equal to or greater than 2 on 1, the earth pressure should also be applied at 0.33 h. For structures with vertical back faces (or batters less than 2 on l) and horizontal or positive sloped backfills the earth pressures should be applied at 0.375 h (see Figures 88 and 89). 4.32 PASSIVE FORCES - Though the passive resistance of an earth mass to movement is very much greater than the active pressure, to develop this passive resistance to movement, the wall must translate or rotate to a significant degree. Therefore, except for sheet pile structures, the passive resistance should not be considered in a stability analysis. For sheet pile structures the passive earth resistance may be computed as R = uh2 tan^ 122^ ' (6l) P 2 4.33 SURCHARGE LOADS - For a uniform surcharge the density of backfill material may be assumed to be increased in the following manner: Let W = the surcharge load in poxands per square foot w = the unit weight of backfill material either drained or saturated dependent on backfill condition h - the vertical height of fill in feet W h* = —, the equivalent surcharge height w then w’ the new value of vinlt backfill weight is given by , (h / 2h') (62) h When this procedure for dealing with surcharge loads is adopted, the point of application of the active thrust for vertical walls with horizontal or positive sloped backfill should be raised from 0.375 h, at the rate of O.Olh for each 10 percent increase in the artificial density, to a limit of 0.475h. 4.34 SUBMERGED MATERIALS - Pressures due to submerged fills may be calcula¬ ted by substituting for w in the preceding equations the unit weight of the material reduced by buoyancy,and adding to the pressures so calculated, the full hydrostatic head of water. Note that for surcharge loads this buoyed unit weight of the material must be increased as shown in the pre¬ ceding section. 143 Part II Chapter 4 4.35 UPLIFT - For design computations, uplift pressures should be con- . sidered as full hydrostatic pressure for walls whose bases are below sea V level or for computations involving saturated backfill. 4.4 ICE FORGES The common forms of ice are usually classified by the use of terms which indicate the manner of formation or the effects produced. Usual classifications include ; sheet ice, shale, slush, frazil ice, anchor ice, and agglomerate ice. The amount of expansion of water in cooling from 39° F. to 32° F. is 1.32 hundredths of 1 percent, whereas in changing from water at 32° F. to ice at 32° F. the amount of expansion is about 9.05 percent or 685 times as great. It has been found that a change of structure to denser form takes place in the ice when, with a temperature lower than -8° F., it is subjected to pressures greater than about 30,000 pounds per square inch. Excessive pressure, with temperatures above -8°F., causes the ice to melt. With the temperature below -8°F., the change to a denser form at high pressure results in shrinkage which relieves pressure. Thus, the probable maximum pressure which can be produced by water freezing in an inclosed space is 30,000 pounds per square inch. Designs for dams include allowances for ice pressures varying from no special allowance to as much as 45,000 to 50,000 pounds per linear foot. The crushing strength of ice has been found to be about 400 pounds per square inch and the thrust per linear foot for various thicknesses of ice as about 28,800 pounds for 6 inches, 57,600 pounds for 12 inches, ( etc. Structures subject to blows from floating ice should be capable of resisting from 10 to 12 tons per square foot 139 to 167 pounds per square inch) on the area exposed to the greatest thickness of floating ice. Ice also expands when warmed from temperatures below freezing to a temperature of 32° F. without melting. Assuming a lake surface to be free from snow, with an average coefficient of expansion of ice between -20° F and 32°F equalling 0.0000284, the total expansion of a sheet of ice a mile long for a rise in temperature of 50° F. would be 3.75 feet. Normally, shore structures are subject to wave forces comparable in magnitude to the maximum probable pressure that might be developed by an ice sheet. As the maximum wave forces and ice thrust cannot occur at the same time, usually no special allowance for overturning stability to resist ice thrust is made. However, where heavy ice, either in the form of solid ice sheet or floating lee fields may occur, adequate precautions must be observed to ins-ure that the structure is secure against sliding on its base. Ice breakers may be required in relatively sheltered water where wave action does not require a heavy type structure. Floating ice fields may exert a major pressure on structures, when driven by a strong wind or ciorrent, by piling up in large ice packs against 144 Part II Chapter 4 the obstructions. This condition must be given special attention in the design of small isolated structures. However, because of the flexibility of the ice field, the pressures exerted probably are not as great as would be caused by a solid ice sheet in a confined area. Ice formations may at times cause considerable damage to shore lines in local areas, but their net effects are largely beneficial. Spray thrown up by wind and wave action during the winter may freeze on the banks and structures along the shore, covering them with a protective layer of ice. Ice piled on shore by wind and wave action does not, in general, cause serious damage to beaches, bulkheads, or protective riprap, and generally provides additional protection against damage from the severe winter storm waves. Ice often has a definite effect on impoundment of littoral drift, Updrift source material is less erodible in the frozen state and wind rowed ice acts as a barrier to shoreward moving wave energy, therefore, the quantity of material reaching an impounding structure is reduced. During the winter in 1951-52 it was estimated that ice caused a reduction in rate of impoundment of 40 to 50 percent at the Port Sheridan, Illinois groin system. Some abrasion of timber or concrete structures may be caused and individual members may be broken or bent due to the weight of the ice mass. Piling has been slowly pulled by the repeated lifting effect of ice freezing to the piles or attached obstructions such as wales, and then being forced upward by a rise in water stage or wave action, 4.5 VELOenY FORCES . ■ ■ ■■■ ,ii^ » I. ■ In the design of entrance jetties, the armor stone along the channel slope should be capable of withstanding anticipated current velocities ^196 ) without being displaced. The maximum velocity of tidal currents in raid-channel through a navigation opening can be approximated by the following formularies); V max s -4-i^ (62a) STS where V = maximum velocity of tidal current, which occurs at the center of the opening,T ** period of tide, A s surface area of harbor basin, S = cross- sectional area of openings, and H = the range of tide. The current in mid¬ channel is about one-third larger than at each side of the channel. If the stable stone weight (W) = JL d^^ Wg 6 ^ where dg is the diameter of a stone sphere of equival#»nt weight, and V ■ y (2g)^ ( ^S , ^W ) (Cos a - Sin a)“ w^^ g (Prom Isbash), 19S2, then combining the two equations for y = 1,20 (embedded stone) yields ,,6 S V w-w„ W ■ -- (62d) 15,2S X 10^ (Wg - w^)^(Cos a - Sin a)^ (62b) (62c) May 1961 145 Part II Chapter 4 • where W- = weight of the stable stone, pounds V = velocity of water acting directly on stone, feet per second Wg ^ Unit weight of stone, pounds per cubic foot w^ ~ Unit weight of water, pounds per cubic foot g = gravity, 32,2 feet per second^ a = angle of repose / dg = equivalent stone diameter, feet = 1,24 'yy W/wg y = Isbash constant 1,20 and 0,86 for embedded and non-embedded stone Figure 89A is a graphical solution of equations (62b) and (62c) for the equivalent stone diameter and stone weight where the unit weight of the stone is 165 pounds per cubic foot and the unit weight of water is 62,4 pounds per cubic foot. In the case of saltwater, the curve would be adjusted accordingly. The curves are considered applicable to conditions where turbulence is not excessive and the stones are either embedded of non- embedded, 4,6 MATERIALS The structural design of shore protective works must take into account the effects of the environmental conditions peculiar to the shore line area on the material used. General modifying criteria which should be applied to materials commonly used are discussed in the following paragraphs, 4.61 CONCRETE - Concrete exposed to sea water, freezing and thawing or other destructive agents or conditions should have an ultimate compressive strength of 3,000 pounds per square inch, A rich, dense, stiff mix is to be preferred where placement is to be done underwater. Care should be taken to cover reinforcing steel adequately, thereby minimizing possible spalling and exposure of the steel. Working stresses for those conditions may be found in the Corps of Engineers Engineering Manual for Civil Works Construction, EM 1110-2-2101, May 1953, 4.62 STEEL - Where exposed to weathering, allowable working stresses must be reduced to take into account corrosive action, abrasion, or a combina¬ tion thereof, which would result in loss of effective steel area. Working stresses for reinforcing and structural steel may be found in the preceding reference, 4.63 TIMBER - Allowable stresses for timber should be those for timbers more or less continuously damp or wet. These working stresses may be found in U, S, Department of Commerce publications dealing with American Lumber standards, 4.64 STONE - Usually the availability of stone sources determines the quality of stone used in waterfront structures. However, care should be taken to avoid use of stone which may decompose more or less rapidly under wave and water action. Where such stone has been used, the effective life of the structure was decreased considerably. May 1961 146 Mov 1961 146 a Equivalent Spherical Stone Diameter (dg ) feet FIGURE 89"A. VELOCITY VS STONE WEIGHT AND EQUIVALENT STONE DIAMETER 'i-, : te'4^' i t - f. , »' ■* VP i'dm 5Tv «ri :> M ra- .ijy ti/ w. ■a rf, d- ^r^, "f" ‘^irl fftf ^4 -v^ :i3 u» \v iM: r-’ss -jii' .‘i”. .d : r’i^ .’t’ ft f ■ ^' u S'^ imr ■ri^ i\>-' -i- i -.-i- ■rri-, i.'iJ'TI 'tt> .f ■ »<■ e».j a »» .fe 3 1 tj^'- ' >v\ V • ^ % ’3 to VW?-v :r,.’V.i ^ -- • * !■. ''^ -ft'- ^ "S?, ^ ‘^-A. ikJi. fel K'i p4" '/iJ't'k ■^~ -yiy*prx^. '■'■■■ "t -i ili -* , : t'scfr ir: -- . T- ;; ...M/'4 V,k/ ■.^«;s:i^^r'.t,£f Iffy * *i2l.’’ '*■ ^ /ifc—/ ' --fl>''4 \ -4 ‘■‘‘V4 *'♦'■<»• ■ ^ >.4 ■ ^4 ^ i yt \ t % «'<•» »<• • <., »‘*<%4 • ».., ttlUv^T -* • J c might be used under these conditions. Second, it should be remembered that the presence of a seawall or bulkhead may change the foundation con¬ ditions so that, unless precautions are taken, a structure might fail. Because of induced bottom scour, a foundation otherwise stable could become unstable. For example, a masonry wall or mass concrete wall must be pro¬ tected from the effects of settlement due to bottom scovir, induced by the wall itself. (See Section 3.2) 5.122 Exposure to Wave Action - This factor is most Important in the structural design of any one wall or bulkhead, and must also be considered in choosing between structural types. For example, in areas exposed to severe wave action, the lighter types of structures (timber crib, light riprap revetment, etc) may not be used. Where waves are high, a curved re-entrant face wall might be considered over a stepped face wall. 5.123 Availability of Materials - This factor would normally be reflected in the cost, as generally, any kind of material can be made available at a price. In times of shortages and restrictions this does not always hold true and more costly structures have been constructed of stone,for example, due to shortages of steel. The price which must be paid for the constituent materials is a major item in first construction and maintenance costs. If these materials are not available near the site of construction, or are in short supply, a particular type of seawall or bulkhead may become economically infeasible to construct. In some instances a compromise may have to be made and a lesser degree of protection provided. February 1957 150 NOTE: A splash apron may be added next to coping chonnel to reduce damage due to overtoping. NOTE : Dimensions and details to be determined by particular site conditions. Coping Chonnel Top of Bulkheod_ Send Fill-^^_Slope I on20 Former ground surfoce^^^, IfTHl ^Water Level Datum Timber Strut^ Timber Wole ... ^ Timber Wole ^Steel Sheor Piling Timber Block J- Rd. Timber Piling SHEET PILE BULKHEAD; STEEL FIGURE 95 February 1957 151 Februory 1957 152 CONCRETE REVETMENT PIONEER POINT, CHESAPEAKE BAY, MARYLAND FIGURE 98 February 1957 153 Part II Chapter 5 5.124 Costs - The analysis of costs must include consideration of the first costs of construction and annual costs over the economic life of the structiare. The annual costs include interest and amortization on the investment, and average maintenance costs. Other things being equal, that structure would be built which would provide the desired degree of pro¬ tection at the lowest annual or total cost. Because of wide variations, in first and maintenance costs, this comparison is usually made by reducing all costs to an annual basis. 5.13 DESIGN 5.131 High Semi-Gravity Type Concrete Wall - a. Design Data .- It is desired to build a protective seawall in an area where there is a 6-foot tide and a possible additional 2-foot wind set-up. An erosion trend analysis indicates that the expected ultimate ground line at the wall's location will be 2 feet below mean low water, with a bottom slope before the wall of approximately 1 on 20. Analysis of refraction diagrams indicates that waves from a certain direction may approach the area without decrease in height due to refraction while waves from other directions will be significantly decreased. Analysis of synoptic weather charts indicates that waves from this direction, under storm conditions, may have deep water heights up to 8 feet and periods from 7 to 11 seconds. The backfill material has a unit weight of 120 pounds per cubic foot, and an angle of internal friction of 25° . The backfill will be subject to a uniform surcharge load of 250 pounds per square foot. The foundation material has a bearing capacity of 2,000 poiinds per square foot. b. General - To reduce water overtopping, the face of the wall will have a re-entrant angle of 15° to the vertical. To reduce beach scour, the face of the wall will be stepped. The wall base is to be at least 2 feet below the ultimate ground line. A sheet pile cut-off wall is to be placed at the toe as a safety factor to prevent damage to the wall by imdercutting of the foundation should the beach lower to a greater depth than estimated. The analysis which follows illustrates the general principles involved in designing for stability against overturning. Sliding stability is not analyzed. The procedures suggested are generally in accord with those presented in the Engineering Manual, Civil Works Construction, Part CXXV, Chapter 2 - Retaining Walls(23), except that in designing against wave forces, the momentary nature of peak forces involved permits the use of wall sections in which the resultant force on the base falls outside the middle third of the base. Though the general features of the wall section adopted conform to accepted criteria for seawalls, the design presented is not to be construed as a restriction on design initiative. c. Wave Forces - Deep water wave lengths may be computed from the relationship Lq = 5.12 1“^. With these the following table should be drawn 154 Part II Chapter 5 up to determine whether waves may break on the structure. BREAKER POSITION 1^0 T Lo Ho/I^ dt/H’Q(curves) ^b/^o(calculated) (feet) (1) (seconds) (2) (feet) '(3) (4) (5) (6) 8 7 251 0.032 1.25 1.25 8 9 415 0.019 1.3 1.25 8 11 620 0.013 1.5 1.25 Column 1 - Hq = deep water design wave height from synoptic charts Column 2 - T = wave period from synoptic charts Column 3t- Lq = wave length from Lq =5.12 Column 4 - ~ coliimn 1 divided by coliamn 3 Coliimn 5 - d^j/H’^ = These values are tabulated from the curve of (curve) Figure 38 Column 6 - d^H>Q = The design depth at the structure divided by (calculated) column 1 = 10/8 = 1.25 For the 7-second period, the fact that columns 5 and 6 are the same, indicates that a 7-second wave may break directly on the wall. The values for the 9-second period are so close that breaking waves may again be assumed. For these two periods, then, wave forces must be determined by Minikin’s criteria. For the 11-second wave, column 5 is larger than column 6 indicating that waves would break seaward of the wall, and the force crit^ia for broken waves apply. If coliamn 5 had been smaller than column 6 the Sainflou criteria would then have been used. d. Breaking Waves - The curve IVERSEN(6l) of Figure 37 may be used to determine the breaking wave height H^. BREAKING WAVE HEIGHTS Period (T) Hq/I^ (seconds) 7 0.032 1.1 9 0.019 1.25 11 0.013 1.4 Hv (feet) 9 10 11 = the design wave height at the stmcture 155 Part II Chapter 5 The Minikin relationships for shock pressures, thrusts, and overturn¬ ing moments due to wave action are; Pm = lOlw(H^Lp) X d(l / d/D) for maximum pressure (2?) Rjn = X H^3 for total thrust per foot of wall (33 first term) Mjjj = RijjX (d/2) for overturning moment (note that the moment arm is d / 2 since the (34 first term) wall base is 2 feet below ultimate ground level) where Lp and D may be determined as follows for the 7 and 9-second wave periods with d - 10 feet. Values for d/L,^ and D/Lp for the various values of d/liQ and D/Lq respectively are taken from Table D-1, Appendix D. DETERMINATION OF D AND Ln Period(T) (seconds) (feet) d/Lo d/Ld (feet) D (feet) D/I« D/Lp (1) (2) (3) (4) (5) (6) (7) (8) (9) 7 251 0.040 0.083 120 16 0.064 0.11 U5 9 415 0.024 0.063 160 18 0.043 0.087 207 Accordingly, the shock pressures, turust, and moments about the base due to the 7 and 9-second waves respectively are given in the following table. Period (T) (seconds) 7 9 SHOCK PRESSURES. THRUST AND MOMENTS Breaker Height (H, ) Pressure (Pm) Thrust (Rm) (feet) (lbs./so. ft.) (lbs./so. ft.) 9 6540 19,620 10 4860 16,200 Moment (J^) (ft. lbs./ft.) 236,000 194,500 Since the wall is to be founded 2 feet below the ultimate groiand line, the static forces on the face of the wall due to wave action may be com¬ puted as that force due to a head of water of height (d / 2 / H^2). Therefore, the maximum static pressure will occur at the wall base (the depth dp) and is given by P^ = w(d / 2 / ^ ) (30) the total thrust per linear foot of wall is given by (d / 2 / Hb ) (33-2nd term) ^d = Pd - — 2 and the overturning moment is given by 156 Part II Chapter 5 M. ( d / 2 / V i) (34-2nd term) For the 7 and 9-second waves, these values are Period (T) (seconds) 7 9 STATIC PRESSURES. THRUSTS AND MOMENTS Breaker Height (H]^) Pressure (P^) Thrust (R^j) (feet) (Ib./ft.^) (lbs./ft.) 9 1,060 8,750 10 1,090 9,260 Moment (M^) (ft.-lbs./ft.) 48,000 52,500 The manner in which these breaking wave pressures are applied is illustrated in Figure 99. e. Broken Waves - To evaluate the dynamic forces due to the 11-second wave, the following relationship should be used: wdb P^ r for maximum pressure (35) R r P (h ) for total thrust per foot of wall m m c ■ ^ h / Q for maximum moment I.'l III ■ iC (37-1st term) (38-1st term) 157 Part II Chapter 5 where dp = the breaker depth for the Il-second wave and equals 8 x 1.5 = 12 feet (see tabulation on breaker position). d = the depth of water at the structure - 10 feet h = the elevation of the breaking wave crest over still water ^ level = 0.7 x = 7.7 feet w = the unit weight of sea water =64.2 pounds per cubic' foot. Therefore, for the 11-second wave, the dynamic pressure = 385 pounds per square foot, the dynamic thrust = 3,000 pounds and the overturning moment = 41,100 foot pounds per lineal foot of wall. The equivalent expressions for static loadings are: P = w(d / h ) for maximum pressure at the wall base (36) S 0 R = P ( ^ ^, , ^^ ) for total thrust per foot of wall (37-2nd term) S 3 ^ d / he M = R (—r—^) for maximum moment (38-2nd term) s s 3 The static broken wave pressures, forces, and moments are approximately 1,140 pounds per square foot at the base, 10,100 pounds per lineal foot of wall, and 59,300 foot-pounds per lineal foot of wall, respectively. f. Wave Forces - The preceding computations may be summarized in the following tabulation: WAVE PRg,SSURES. THRUSTS AND MOMENTS ABOUT WALL BASE Breaker Period (T) Height Pressure Thrust Moments (seconds) (feet) (ibs./sq. ft.) (lbs./ft.) (ft.-lbs./ft) 7 9 Dynamic 6,540 19,620 236,000 Static 1,060 8.750 48,000 TOTAL 28,370 284,000 Dynamic 4,860 16,200 194,500 9 10 Static 1,090 9,260 .^2,5P.Q, TOTAL 25,460 247,000 Dynamic 385 3,000 41,100 11 11 Static 1,140 10.100 59,300 TOTAL 13,100 100,400 Note that in this case the thrusts and moments are highest for the lowest breaking wave heights. 158 Part II Chapter 5 Wall Height - Though the forces due to the 11-second wave would be far less than those due to the 7 and 9-second waves, the wave crest height above sea level would be greater. To prevent significant overtopping, this height would be used for the design wall height. That is, the wall crest would be 8 feet above the still water level. FIGURE 100 A wall section confonning to the previously noted general design criteria and to the above is shown in Figure 100. The weight of concrete in the wall, considering concrete to have a unit weight of 150 pounds per cubic foot, is 31,710 pounds per lineal foot. The wall center of gravity determined by taking moment areas about A is 9.35 feet seaward of A. h. Stability Calculations - (Wall Overturning Landward) - Maximiim landward overturning moments would occur with maximvim wave action on the face of the wall and with the backfill unsaturated. Wave forces and uplift would tend to cause the wall to overturn landward. (From the preceding table the design wave forces are those due to the 7-second period wave). The overturning moment would be resisted by the moments due to the weight of water over the toe of the seawall, the weight of the wall, active earth pressure on the back of the wall, and the weight of earth on the back of the wall. (Wave Forces and Moments)- Total wave forces and moments about A are tabulated below, the vertical water value being computed assuming a maximum water level of 0.7 x H^, = 6.3 feet above storm sea level. 159 Part II Chapter 5 mw FORGES AND MOMENTS Force Arm (A) Item (lbs./ft.) (feet) Wave (horizontal) 28,370 Water (vertical) 11,110 16.38 Moment about A ( ft.-lbs./ ft. ) - 284,000 /181,970 (Earth Forces and Moments) -For stability calculations it will be assumed that the backfill is unsaturated. Then the total earth thrust is given by P = 1 2 2 sin (Q -(ji) 1 (l + N) sin0 J w' sin i(f>+9) (59) where N = 4in {(fi + (fi) sin 0) sin {- p) sin {Q+p) (60) and w w( h I 2h‘ ~h ( 61 ) in which h' = y = the equivalent siucharge height and for this wall W = 250 pounds per square foot = the surcharge load w =• 120 pounds per cubic foot = the backfill unit weight h = 20 ft. = the backfill height w'= 145 pounds per cubic foot = backfill unit weight with sur¬ charge load 0 = 90 / tan ^ {—) - 104 ° 2' = the wall backface angle with 20 the horizontal / = 25 ° = the backfill internal friction angle 02 _ ~ 17° = the backfill wall friction angle = 0° = the angle between surface of backfill and horizontal plane This thrust will act at a distance (0.375 / iQy^ above the base and at an angle (0 - 90 / 0p) with the horizontal. By resolving P into horizontal (Ph) and vertical (Py) components we may find the thrust and moments about A. These are as follows: 160 Part II Chapter 5 EARTH FOBCES MD MOMENTS Item Force Arm (from A) (lbs./ft.) _ (feet )_ Moment ( ft. - lbs./ft.) Earth (horizontal) = 11,850 7.92 Earth (vertical) Py = 7,130 1.98 93,830 14,110 (Uplift Forces and Moments)- Uplift forces may be computed as a straight line triangular variation, assuming a head of 18.3 feet at the toe and none at the heel. UPLIFT FORCES AND MOMENTS Item Force Arm (from A) (lbs./ft.) _(feet)_ Moments ( ft. - lbs./ft. ) Uplift (vertical ) 13,500 15.33 -207,120 (Stability Table) - Summarizing the preceding tables, the following table may be set up: STABILITY TABLE. OVERTURNING LANDWARD Item Force Arm (from A) Moments about A (lbs./ft.) (feet) (ft. - lbs./ft.) Concrete (vertical) 31,710 9.35 /296,450 Earth (horizontal) 11,850 7.92 / 93,830 Earth (vertical) 7,130 1.98 / 14,110 Water (vertical) 11,110 16.38 /181,970 Wave (horizontal) 28,370 -284,000 Uplift (vertical) 13,500 15.33 TOTAL -207,120 /586,360 -491,120 NET TOTAL /95,240 The moments which would 'cause overturning around point A are negative, therefore, the wall is stable under wave forces. It should be remembered that designing against wave forces in this manner is equivalent to assuming their uniform application over the entire length of wall at one time. That this is never so, plus the fact that the impact nature of the computed maximum force makes more effective the high inertia of the wall, are in¬ dications that a wall so designed is far more stable under wave forces than the above computations would indicate. (Wall Overturning Seaward) -Maximum moment causing the wall to overturn seaward would occior with no water in front of the wall, and assuming a 161 Part II Chapter 5 saturated backfill and a surcharge of 250 pounds per square foot. The earth forces and moments may be computed in the same manner as for the unsaturated case. All angles would be the same, but w = 14 . 5.7 = 120 / 0.40 X 64 . 2 , assuming 40^ voids in the material h' = 1.72 so w' =. 171 pounds per cubic foot. since w’ for the unsaturated case was 145 po\inds per cubic foot, the earth thrust will be in the proportion 17l/l45 to the force calculated for an \ansaturated fill, and will be applied at a height of 7.84 feet above the base. On resolving the thrust into horizontal and vertical components, the forces and moments about B may be tabulated. EARTH FORCES AND MOMENTS Item Earth (horizontal) Earth (vertical) Force (lbs./ft.) Arm (from B) (feet) Moment about B (ft.-lbs./ft.) Ph - 13,520 7.84 Py = 8,390 21.04 106,000 176,480 (Uplift Forces and Moments) - Uplift may be calculated for conservative conditions by assuming the water level at the toe to be at the base. Then the uplift force diagram is triangular and the thrust and moment about B are given in the following table. UPLIFT FORCES AND MOMENTS Force Arm (from B) Moment about B Item (lbs./ft. ) (feet) _ (ft.-lbs./ft.) Uplift 14,770 15.33 226,360 (Stability Table) - Summarizing the preceding the following table may be set up: STABILITY. OVERTURNING SEAWARD Force Arm (from B) Moment about B Item (lbs./ft.) (feet) (ft.-lbs/ft.) Concrete (vertical) 31,710 13.65 - 432,840 Earth (vertical) 8,390 21.04 -176,480 Earth (horizontal) 13,520 7.84 /106,000 Uplift (vertical) 14,770 15.33 / 226,360 TOTAL /332,360 - 609,320 RET TOTAL -276,960 162 FORCE RESULTANT, OVERTURNING LANDWARD FIGURE 101 163 Part II Chapter 5 The moments which would cause overturning around point B are positive, therefore the wall is stable under saturated earth forces. The factor of safety here is well over the 1.5 value normally used for earth loads with uplift considered. (internal Stresses. - Overturning l5.ndward)- Although the wall is stable against overturning, calculation of the resultant of the forces in¬ dicates that this resultant would fall outside the middle third of the base. The total vertical force of 36,400 pounds per foot downward would be applied 7.84 feet from A. The total horizontal force, of 16,520 pounds per foot directed landward, would be applied 11.48 feet above the base. From Figure 101, the resultant of these forces would fall 2.6l feet from A inside the base. It would be possible, by adding to the area oi concrete to bring the resultant of the forces within the middle third, or at least to reduce the tension in the base. However, because of the high dynamic pressvires of the breaking waves, to eliminate tension completely (or even to materially reduce the tension) would require an excessively large amount of additional concrete. Accordingly the economics of providing the additional concrete as opposed to the provision of tension steel must be weighed locally. No such analysis will be made for the purpose of design Illustration. It should also be noted that tension steel for the given design will be re¬ quired in the front face of the wall. (Foimdation Pressures. - Overturning Landward) - The bearing pressiire of the loads may be calculated from (63) A ^ Z where P^ = sum of vertical loads = 36,400 A - base area (unit length of wall) = 23 M = the total moment about the base centerUne 36,400 (11.5 - 7.84) = 322,870 Z = the section modulus (unit length of wall) Therefore f = = 1580 / 3660 The pressure distribution is shown by ^ in Figure 102. ( Bearing Piling). - Since the maximum base pressure exceeds the allowable bearing capacity of the foundation material, piles must be used to support the wall. Even if the maximum base pressure did not exceed the bearing capacity of the foundation material, for safety, bearing piling would be used. The piles should be designed for the total bearing pressure for the loads as shown on Figure 102. A network of piles under the entire base should be used to prevent unequal settlement. In a case such as this, where large horizontal thrusts are Involved, it may be economical = 16,520 X 11.48 / 2 = 88.167 C 164 Part II Chapter 5 to use batter piles as well as vertical bearing piles. Allowable pile loading varies with the material in which the pile is driven. If an individual pile rests on a hard stratum its bearing capacity is determined by the ultimate strength in compression of the pile material. Essentially the pile becomes a totally laterally supported column. However, when a pile's bearing load is supported primarily by skin friction with the material through which it is driven, its bearing capacity must be determined by empirical means. The best practice in piling design is to rely upon statically loaded test piles to assess the safe loading for the piling. For most founda¬ tion materials on which seawalls will be built, the resistance to a static load is almost the same as that offered to the dynamic load of driving and for limited piling projects this relationship may be utilized to determine allowable loads. The allowable bearing load per pile can thus be calculated by relating the depth of penetration of a test pile per blow to the hammer weight. Relationships commonly in use for timber and concrete piling are as follows: table 12 - SAFE LOAD PER PILE(86 ) Driver .Drop Hammer Timber P = 2Wh ^1 Steam hammer, single acting P _2Wh_ s / 0.1 P = P = Concrete 2Wh s (1 / Wp/W) 2Wh s / 0.1 Wp/W Steam hammer, double acting _ 2h (W / ap) s / 0.1 2h (W / ap) s / 0.1 Wp/W P = safe bearing load, pounds W = hammer weight, pounds Wp = pile weight, pounds h = free fall of hammer, feet 2 a = effective area of piston, inches p = mean effective steaim pressure, pounds per square inch s = penetration of sinking, inches (taken as average of last 5 to 10 blows for a drop hammer and as average of last 20 blows for a steam hammer.) These formulas will give £fafe loadings of individual piles provided the Individual piles of a group are not so closely spaced that skin friction is materially reduced.(125). (internal Stresses - Overturning Seaward)- On obtaining the resultant 165 FORCE RESULTANT, OVERTURNING SEAWARD FIGURE 103 A Ground Line^- B -in^^iTJ^^^mifmnsTTMnTmTmim o 0 > BEARING PRESSURES, OVERTURNING SEAWARD FIGURE 104 166 Part II Chapter 5 of vertical and horizontal forces it is found that this resultant would fall within the middle third of the base. The total vertical force of 25,330 pounds per foot downward would be applied 15.12 feet from B. The total horizontal thrust of 13,520 pounds per foot directed seaward would be applied 7.84 feet above the base. From Figure 103 the resultant of these forces would fall 10.94 feet from B within the base, or well within the middle third. Therefore, there would be no tension in the base due to the force of the saturated fill. (Foundation Pressures-- Overturning Seaward)- Pv Using f - j — / —— (see equation 63) where P = 25,330 A = 23 M = 13,520 X 7.84 - 25,330 (15.12 - 11.5) = 14,300 Z = 2^ = 88.167 6 f = 25 J ^0 / ]4,300 23 " 88.167 = 1,100 / no The pressure distribution would be as shown in Figure 104 by the line CD . 5.132 Steel Sheet Pile Cellular Seawall - a. Design Data ,- A wall is to be placed in fresh water at a low bluff area, to protect the bliiff line from the action of storm waves. The water depth at the toe of the wall is to be 3 feet below low water datum. There is a maximum expected wind set-up of 2 feet. The slope of the bottom seaward of the site of the wall is approximately 1 on 40. The elevation of the top of the bluff is about 10 feet above low water datum. This would also be the elevation of the top of the backfill. Wave analysis indicates that the maximum deep water wave height expected is 10 feet, and that a wave this high will have a period of 12 seconds. The maximum refraction coefficient is 0.5, at a depth of 50 feet, for this 12-second wave. Refraction coefficients for other period waves are much smaller. The bottom is composed of a 2-foot thickness of gravel and shale overlying bedrock. Because of poor penetration possibilities, a sheet pile, cellular type structure was chosen for design. The backfill material has a natural drained unit weight of 130 pounds per cubic foot and an internal friction angle of 25? The cell fill material (gravel) has a anatural drained unit weight of 120 poiuids per cubic foot and an internal frictibn angle of 45°. The coefficient of friction between the cell fill and the bottom is 0.5* 167 Part II Chapter 5 b. Wave Forces - The deep water wave length of a 12-second wave is Lq = 5.12 (12)^^ = 737 feet. At a depth of 50 feet d/l^ = 50/737 = 0.068. From Table D-I of Appendix D the corresponding d/L = 0.II2 and H/H’q z 0.98 Therefore,, the wave height in this depth of water where the refraction coefficient k is 0.5 for a wave 10 feet high in deep water is H = Hq X k X (H/H'q) = 10 X 0.5 x 0.98 = 5 feet (13) Where k is also equal to (H’q/Hq) and H*q is the deep water wave height which, in the absence of refraction, would give a wave height of H in a depth of 50 feet. This equivalent deep water wave height H’q is 5 feet. A wave with deep water length of 737 feet and deep water height of 5 feet will have a steepness of H’^/Lq = 5/737 = 0.0068. From Figure 38, the ratio dj^/H’^ =2.0, from which d^, the breaking depth, is d]-, = 2.0 x 5 = 10 feet. The maximum anticipated depth at the wall including tidal rise is 5 feet, which indicates that waves will break before reaching the structure, and that the wave force criteria for broken waves must be used for this deep water design wave. c,. Forces Due to Broken Waves - Referring to Figure 37 H,/H'q for a wave whose deep water steepness (H^/Lq) = 0.0068, moving up a 1:40 slope is H^/H’q = 1.5. Therefore, the breaking wave height is 7.5 feet. The dynamic pressures, thrusts and moments are given by: w d^ Pm - ~2 — maximum pressure (35) = P^(h^) for total thrust per foot of wall (37-lst term) h M = R (d / ) for overturning moment per foot (38-1st term) m ni ^ n T of wall where d^^ = the breaker depth referred to the maximum anticipated water level = 10 feet d = the depth of water at the structure = 5 feet h = the elevation of the breaking wave crest above still water ^ level = 0.7 x 7-5 = 5.25 feet w = the unit weight of fresh water =62.4 pounds per cubic foot. The numerical values for these are approximately: Pjjj = 312 pounds per square foot; Rj^ = 1640 pounds per foot of wall; and = 12,500 foot-pounds per foot of wall. The equivalent expressions for static loadings are: 168 FIGURE 105* DISTRIBUTION OF PRESSURES; BROKEN WAVES FIGURE 106.-DISTRIBUTION OF PRESSURES; BREAKING WAVES 169 Part II Chapter 5 R M w(d / h^) for maximum pressure at wall base P ^ ^c ) for total thrust per foot of wall ® 2 d h Rg(_____c) for overturning moment per foot 3 of wall (36) (37-2nd term) (38-2nd term) These give for static loadings approximately: Pg = 64 O pounds per square foot; Rg = 3,280 pounds per foot of wall; and Mg = 11,200 foot-pounds per foot of wall. The manner of application of these pressures is shown in Figure 105. d. Forces Due to Breaking Waves - Though the maximum deep water wave breaks before reaching the wall, some lesser height wave will break right at it. This wave height may be approximated by the relationship d^Hi^ = 1.3 where d-j^ is the maximum water depth at the wall. In this case then, H-j^ = 5/1.3 = 3.8 feet. To use the Minikin relationships for forces, it is necessary to determine D and Lp as indicated in Section 5.131 d. In tabular form, the computations are as follows: DETERIgNATION OF D AND Ln FOR d = 5 FEET L 0 (feet) d/L 0 dAd I'd (feet) D (feet) d/l 0 D/Lj, h (feet) 737 0.0068 0.033 d = 152 0.033 152 / 5 = 8.8 40 0.0119 0.044 D = 200 0.044 The wave shock pressure is given by 101 H, w , P^ = -7—^ X d (1 / ^) (27) m D 101 X 3.8 X 62.4 X 5 /. / 5 >> = --^^ (1 /■ } = 940 pounds per sq. ft. 200 and the maximum hydrostatic pressure at the depth dp (the base) ignoring any water pressure oii the wall backface would be P d w (d^ / ^) = 62.4 (5 / 1.9) (30) = 430 pounds per square foot. The manner of application of these pressures is shown in Figure 106. From this figure, the total thrust is given by 170 Part II Chapter 5 R = R / R. m d - r «b (33) = 940 X ^ / 215 (6.9) = 1190 / 1480 2670 pounds per lineal foot of wall. and the total overturning moment would be R H M = R^d / ^ (d / ^ ) (34) - 1190 X 5 / 1480 (6.9) = 5950 / 3400 = 9350 foot-pounds per lineal foot of wall. e. Tabulation of Design Wave Forces and Moments - The forces and moments exerted on the wall by the deep water design wave breaking in 10 feet of water are greater than those exerted by some lesser height of wave breaking right on the structure. These larger forces will be used for design. HAVE FORCES AND MOMENTS Item Dynamic Static Force (pounds per foot ) 1,640 3.280 TOTAL 4,920 Moment (foot-pounds per foot) 12,500 11.200 23,700 f. Earth Forces - The backfill thrust is given by P= (22^^ ) where h = 13 feet = height of backfill above low water level 0 = 25 ° = internal friction angle of the backfill (57) w - 130 pounds per cubic foot = the unit weight of unsaturated fill or w = 130 / 0.4 X 62.4 = 154.8 pounds per cubic foot, the unit weight of saturated fill assuming the material has 40 ^ voids. Part II Chapter 5 The point of application of this thrust is at a distance 0.375h above the cell base. Therefore, for an unsaturated backfill 130 X 169 p P z -2- (0-637) = 4)460 pounds per lineal foot of wall and for a saturated backfill 154.8 X 169 (0.637)^ ^ " 2 = 5)300 pounds per lineal foot of wall each applied at a distance 4*88 feet hbove the base. The thrust on the sheet pile due to the fill material is similarly- calculated with 0 = 45° w = 120 pounds per cubic foot for iinsaturated fill = 120 / 0.4 X 62.4 = 144.8 pounds per cubic foot for sat-urated fill. Therefore, for an unsaturated cell fill P = — (0.414)^ = 1,730 pounds per lineal foot of wall and for a saturated cell fill P - ^ 169 .(0.414)^ - 2100 pounds per lineal foot of wall. Note especially for the cell fill, that the maximum pressure (not thrust) is given by p = wh tan^ ( 2 ^ ) (64) 172 Part II Chapter 5 and occurs at the cell bottom. For the case being considered with a saturated cell fill, this pressure is p = 1A4.S X 13 ( 0 . 414 )^ = 322 pounds per square foot, and with an unsaturated cell fill p = 120 X 13 ( 0 . 414 )'^ = 267 pounds per square foot. g. General Design - The type wall chosen is the so-called diaphragm type illustrated in Figure 107, The dashed lines indicate the dimensions of an equivalent rectangular cell used for stability calculations. The width (b) of the equivalent rectangle is the average width of the actual cell, the length (L) is the length of one cell section; and the distance (r) is the radius of the outside walls. Note that the crosswalls have a slight arc for stability against differential earth pressures due to partial filling of one cell before filling of the adjacent cell is started. Failure criteria for such walls, if the piling has a significant amount of penetration, are difficult to determine. However, there will be an ad¬ equate factor of safety if they are considered to be open end boxes resting on the bottom. In the example chosen, the rock stratum is only 2 feet below 173 Part II Chapter 5 the lake bottom and the open end box calculations will be fairly accurate. h. Piling Calculations - Pile interlock tension resistance to cell rupture is directly proportional to the cell fill pressure and the radius of the cell wall. Expressing the fill pressure p in pounds per square foot and the radius in feet, the Interlock tension in pounds per linear inch is given by t - ^ ^ - 12 (65) Assuming an M112 section piling would be used, with an allowable interlock tension of 6,000 pounds per inch, the maximum allowable cell wall radius for this problem, with p being the pressiire of the saturated cell fill would be 12t _ 12 X 6.000 - p - 322 = 224 feet and for safety will be taken as 50 feet. (Note that any pressure due to hydrostatic water pressure has been ignored, l.e., that the water level for maximum interlock tension is at the wall base.) i. Stability Calculations -(Overturning)- The circular tension per foot of pile developed in the pile Interlocks of the outside cell wall is directly proportional to the radius of the wall and to the maximum pressure of the cell fill; that is t = pr, where p is the pressure of the fill. The total tension developed in one interlock is T = t x h/2 = Pr, where P is the total thrust due to the cell fill. The tension in the cross wall, if two cells are joined by a standard Y-pile with 120° legs is the same as the tensions in the individual wall arcs, that is T = Pr again. For the cell to overturn, the cross wall piles must slip along the interlocks and the fill material must slip along the piling and shear through a vertical section. Therefore, if f is the coefficient of lock friction, and L is the length of one cell section, the resistance to over- tiirning developed by the cross walls per unit length of structure is S P Prf L ( 66 ) In addition to this resistance, there is the shear resistance developed by the cell fill itself which is given by = P tan iZi (67) a that is, the lateral fill thrust times the coefficient of internal friction. Therefore, the total resistance to overturning per unit length of wall developed by the crosswall and cell fill is S = Sp / = P (^ / tan 0) (68) 174 Part II Chapter 5 The exterior overturning moment (M) per unit length of wall about the center of the cell may be replaced by a couple about the neutral axis (also the center of the cell). If it is assumed that pressures along the wall bottom which may cause this couple are linearly distributed along the ’wall width, the "couple diagram" may be drawn as shown in Figure 108. FIGURE 108-COUPLE REPLACING OVERTURNING MOMENT The couple (F) due to the two pressure diagrams is applied at the centroids of the two triangles, and the moment due to this couple must equal the overturning moment about the neutral axis. Therefore, the overturning forces may be represented by F 3 M = 2 b (69) For stability, these forces should be least equal the resisting forces due to the crosswalls and fill, or F = S. Solving equations 68 and 69 for b, the minimum width of the equivalent rectangular cell section for stability is b 3M _ 2P (^ / tan 0 ) (70) where M = the total overturning moment about the center line P = the total force due to the cell fill r = the radius of the outside cell wall f = the coefficient of interlock friction L = cell section length tan 0 = the coefficient of internal friction of the cell fill. (Overturning Seaward) - The maximum overturning moment due to the backfill will be caused by a saturated backfill with no water in front of the wall. 175 Part II Chapter 5 From the section on earth forces, this backfill force would be 5,340 pounds per lineal foot of wall applied at a distance 4*88 feet above the base, giving a moment of 26,060 foot-pounds per lineal foot of wall. Assuming an interlock friction coefficient of 0.3 , an unsaturated cell fill, and a cell length of 40 feet, the equation for minimum width would become b = _ _ _ = 16.4 feet, say 17 feet 2 X 1730 / 1) the maximum width between outside walls would be b^ = b/O.9 = 18.2 feet, say 19 feet. (Overtiorning Landward)- Similarly, the maximum overturning moment due to wave forces will be that due to the maximum wave Impinging on the structure, with no active backfill pressure opposing it. From the section on wave forces, this moment would be 23,700 foot-pounds per lineal foot of wall. Since this moment is smaller than that due to earth pressures alone, the wall would be stable against wave attack. (stability; Sliding Seaward)- The weight of the filling material per foot of seawall length is W = whb (71) where w = the unsaturated cell fill unit weight h = the fill height b = the average cell width so W = 120 X 13 X 17 = 26,520 pounds per foot. The known forces for seaward sliding are as shown in Figure 109. FIGURE 109-RESULTANT FORCE 176 Part II Chapter 5 For stability against sliding, the horizontal force divided by the total weight must be less than the coefficient of friction multiplied by the factor of safety. Because of the penetration of the piling, a factor of safety of 1 is satisfactory, or P W ^ 0.5 or 5340 r 0.201 which is less than that allowable, and the wall 26,520 is stable against sliding seaward. ( Stability, Sliding Landwarc^ - The numeiator of the above equation is smaller when a wave force of 5j020 foot-pounds per foot is substituted for 5,340. Since the denominator remains the same, the wall is also stable against wave forces. A cell section designed for stability is shown on Figure 110. 5.133 Steel Sheet Pile Bulkhead^ a. Design Assumptions - Sheet pile bulkheads may be constructed where adequate penetration is obtainable. For design computations, the following assumptions for earth forces calculations may be made: (l) A surcharge load on a fill may be considered to be replaced 177 Part II Chapter 5 by an equivalent height of fill which will produce the same pressure as the surcharge at its point of application; (2) Active pressures may be computed by use of relationship: p = w h tan^ ^90 “ (See Equation 58) ( 64 ) 3 . ^ where w = the unit weight of the material h = the height (including the equivalent height due to siarcharge if any) of the fill ^ = the internal friction angle of the fill material. The point of application of the resultant thrust will be through the centroid of the pressure diagram drawn from this relationship; (3) The maximum passive pressure may be computed by use of the relationship: p - w h tan^ ( 90 (see Equation 6 I) (72) P 2 The point of application of the resultant thrust will be through the centroid of the pressure diagram which may be drawn from this relationship. The application of these relationships to the design of sheet pile bulkheads is Illustrated in the following example. b. Design Problem - It is desired to construct a bulkhead in a pro¬ tected anchorage where the maximum tidal variation is 3 feet. The anchorage is to be dredged to a depth of 6 feet below extreme low water. The original ground line is 3 feet below extreme low water, but the area is to be back¬ filled to 10 feet above the original groimd line or 4 feet above maximum high water. A surcharge load of 150 pounds per square foot is anticipated. The fill material has a drained \mit weight of 120 pounds per cubic foot with an internal friction angle ( 0 ) of 30°. Its saturated unit weight is 140 pounds per cubic foot, and its submerged unit weight is 75 pounds per cubic foot, both with an internal friction angle of 25°. The undisturbed ground has a submerged weight of 80 pounds per cubic foot with an internal friction angle of 30°. Figure 111 is a diagram of the desired placement for the proposed sheet pile bulkhead. For design purposes, it will be assumed that the backfill is saturated to the high water line. The heights hq, h 2 , and h^, refer respectively to the backfill height above the high water line and the original ground line. The height h^ refers to the height of the original ground above the dredged bottom. The corresponding unit weights and internal friction angles are w^ and 0q, W 2 and 02> ^3 03 ^and w^ and c. Loading Diagram - A loading diagram may be constructed by drawing horizontal lines at various points of lengths proportional to the pressure intensities (as given by equations 64 and 72) at the points. 178 Surcharge, 150 psf FIGURE ill - BULKHEAD PLACEMENT FIGURE IIZ-LOAPNG DIAGRAM FOR MINIMUM PILE LENGTH 179 Part II Chapter 5 Thus in Figure 112 W 90° - ^ Pi = “l ^ <-2~> W, (64-Pf) 1 1 Pi where — = the equivalent surcharge height = 150 tan 30° =50 pounds per square foot = pressure due to siircharge load alone exerted by the drained fill W-, 2 90° - 0, = “i (‘'i z:'’ <~2—) 2 (64-P2) = (120 X 4 / 150 ) tan 30° = 210 pounds per square foot - pressure due to surcharge and drained backfillo Considering the sxarcharge plus drained backfill to be a new surcharge applied Wi to the saturated fill with a pressure of W = w (h, / — ) = 630 pounds per square foot, then Pq = W w (-p) tan 2 w^ ( 90 ° - 0 . '■) (64-P3) 630 tan 32.5° 256 potinds per square foot pressTore due to s\archarge alone exerted by the satiirated fill W 2 90° - 0 ^ \ = ^2 ^^2 ^ ^ ^ (“2-) = (140 X 3 / 630) tan^ 32.5° ( 64 -p^) = 426 pounds per square foot = pressTire due to surcharge and saturated backfill. Below the low water line, hydrostatic pressTore on the land side of the wall would be balanced by the water in front of the wall. Accordingly, these pressures may be ignored. Again, considering the surcharge plus the drained backfill plus the saturated fill to be a new surcharge. W 2 W_ = w (h„ / —) = 1050 pounds per square foot 3 275 pounds per square foot 4 -? ^2 W. . 90° - 0 . Py = “4 ^ ^ (64-?^) and 1275 tan 30° 425 pounds per square foot pressure due to surcharge W, exerted by original ground, 4 P8 = !4 90-0 w. (h, / “^ ) tan (-:r 4 4 w. 2 4 4' ) (64-Pg) = (80 X 3 / 1275) tan"^ 30° = 505 pounds per square foot = pressure due to surcharge W/ and original ground to the dredged bottom at 6 feet below the low water line. At the point of application of pg, both active pressure on the pile back and passive pressure on the pile face would be applied. The active pressure on the pile back would Increase in the same manner as it increased from py to pg, because there would be no change of material at the level of pg. From the expressions for py and pg, this rate of increase of active pressure with increase in depth would be R = a increase in h^. Similarly, the maximum passive pressure on the pile face tan^ ( ^^ 2 —unit would increase with depth at the rate R P tan2 ( 21414 ) per unit May 1961 181 Part II Chapter 5 increase in h^, since at any point h below the level of the dredged bottom this passive pressure pg = w. h tan 4 2 The rate of increase of passive pressure applied on the wall face would be greater than the increase of active pressure applied at the wall back, and the net rate of change of pressure is given by R = R - R P a = w 4 tan^ (20^) . tan2 (2^^) = 80 (3.00 - 0.33) (73) z 214 pounds per square foot per foot of depth = rate of decrease of outward pressures The point on the loading diagram at which the earth pressures would be zero is labelled G and its distance h^ below the dredged bottom may be found by dividing pg by R. ^ *'5 = Pg/® = 214 = 2.36 feet = depth of zero loading point below the dredged bottom. The magnitude of the concentrated force (T) on the wall due to the deadman is dependent on the depth of penetration of the pile. d. Pile Length - A pile which is just long enough to support the backfill will have a larger cross section than one, somewhat longer, which is designed for minimum bending moment. The choice between the two is a matter of economics. e. Pile Design -(Minimum Pile Length) - For optimum pile design, the point of application of the deadman thrust should be such that the bending moment at this point equals the maximum bending moment in the pile below it. To attain this optimiam design, the maximum moment below an assumed point of application must be found and compared with that at the assumed point. If these moments are not equal, the process must be repeated. For the purpose of this illustrative problem, only the methods of calculating the bending moments will be carried out. Assuming the point of application (C) of the deadman tension (T) to be 1 foot above the low water line, the moment Me of all the forces about this point between the surface and point G is given by 182 Part II Chapter 5 Me = / Pgh^ (1 / hg / h^/ 3^) r P 7 , / 2 : P7h^ (1^3/ 2 3 / h- P/ -P- 2h^-i p^h 3 ( 1 / ^)/ ( ^ 2 / 1 / P 4 Pq , 2 Pc ^ 2 ^ 2 X 1 X — (75) „ 1 / Pc'P? „ 2 ' p X 2 X x X 2 X P/-P 3 where p r p, -( ~~^)= 369 pounds per square foot L/ l\. ^2 h p -p h -1 (p^h^) (2/ ^)/ (-^)h^ (2/ M - c - 4620 / 7720 / 3600 / 209 331 / 1865 = 13,953 foot-pounds per foot of wall For stability, the sum of all moments about G must be zero. Then if H is the point of deepest pile penetration, h. X R D N X -::5 —) M = * c (CG / 3 * **'6 where h ^ x R * p ’9 2 Therefore ( h ^) (CG / f >>6) - or in this case ( hg )^ (9.36 / fi ^) = M 3 R (76) 2U from which h^ "S' 3.4 feet Since point G is. itself 2.36 feet below the dredged bottom, the pile must penetrate a distance 3.4 / 2.4 = 5.8 feet below the dredged bottom, at which point the pressure p^ would be P(-. - R X h, = 214 X 3.4 = 728 pounds per square foot, 9 D (77) The miniminn length of pile for stability with the deadman tie-rod one foot above the low water line would be 18.8 feet. (Maximum Bending Moment for Minimum Length Pile)- The maximum bending moment in the wall would occur at that point above the dredged bottom where the shear passes through zero. The shear Sp at the dredged bottom F is given by the sum of the forces below that point, which sum is the algebraic sum of the areas of the pressure diagrams below that point. Thus this shear 3 £2^!6 ^F - 2 2 183 Part II Chapter 5 728 X 3.A _ 505 x 2.36 - 2 2 = 1235 - 595 = 640 pounds per foot of wall. Above this point, the shear would decrease as pressiire diagram area is Pc> - P7 added. The quantity , -^ = 26.7 pounds per square foot per foot of wall *'4 would be the rate of change of pressure over h^, and calling this rate the equation for finding the point of zero shear when this point is within h, is ^ R Z.2 =F = (Ps - R4 h'>h ^ 2 or \\ -2P8 2i/2S,r = 0 where Zj, is the distance of this zero shear point above the dredge bottom F. Solving for Zq , '8 -/ Ps - 2 R R (79) 4 = 1.31 feet The pressxire at Z would be p^ = Pg - R^Z = 470 poiinds per square foot. The bending moment 1^ at the dredged bottom is the sum of the moments of the individual areas p^ HG and Pg FG Rh. Mp _ ru / 2 , N ^5 P8 2 h^) - ^ Rh, V. 2 N Ps 6 (2h^^3h3)--^ (Note: R is the rate of change of pressure between pg and p^) Mp = 214 X (3.4) (2.36 / 2.26) - (236)^(^0^) (80) Mp = 5230 foot-pounds per foot of wall, Fig\ire 113 shows the loading, shear, and moment diagrams between F and Zq. At any point x above F, the shear S is given by Sp less the area of the loading diagrams between F and x or R, x2 = Sp - (p 8 X - A. ■) ( 81 ) 184 Part II Chapter 5 Integrating the area under this curve between the limits of 0 and Z^, the shear diagram area Ag is found to be A s or since S F A s R Z 2 Psh - "2 - Pz^ = Ps - Vi — (pg/ 2P2^) (82) Since the area of the shear diagram between points F and is the total increase of moment between these points, the bending moment at Z is z/ = M, / -r (Pg / 2P2^) (83) 2 = 5230 / 0 ^^) (505 / 940) = 5230 / 415 “ 5,645 foot-pounds per foot of wall = the maximum bending moment. 185 Part II Chapter 5 (Bending Moment at the Deadman Tie for Minimum Length Pile )- Above the point of application of the deadman tension (T) the shear and moment diagrams are similarly analyzed. Thus in Figure 114 the shear at any point xp below A down to B is given by 2 Po " Pt ® ^ ^ h, “2" from which the shear at B = Sg = (p^ / p^) = 520 pounds per foot of wall. The area of the shear diagram between A and B is A = —^ (2p / p ) r 825 foot-pounds per foot of wall (85) O Jw rfC which is also the moment Mg, at B^ since the bending moment at A is zero. Similarly the shear at any point X 2 below B down to G is given by , P^ -Po S = Sg / P3X2 / ^ ( 86 ) The area of the diagram between B and C is Ag = Sgh / h_ ( 2 p 2 / p^) = 1,627 foot-pounds per foot of wall (87) . P/ .Po (Note: Pq = -— ■ 369 pounds per square foot) 186 Part II Chapter 5 The bending moment at C is ( 88 ) = 825 / 1627 = 2,452 foot-pounds per foot of wall. For this trial, it can be seen that the maximum positive bending moment in the pile is greater than that at the deadman tie. Lowering the point of application of the tie tension would bring these two values closer, but would put the tie under water where increase in construction costs would more than offset the saving in material costs. The tension T an the tie rod is found by summing algebraically the forces due to the various earth pressure polygons. Pi ^ Po . P? ^ Py , P'^ Pa , P7 Pft ( 2 ) ( 2 ^ ^ ^ ^ 2 ^ 2 4 -gxh,; = (520 / 1022 / 14-18 / 1390 / 595 ) - 1240 (89) = 4945 - 1240 = 3,705 pounds per foot of wall F is the seaward (active) earth force, and Fp is the landward (passive) force. Note that the point of application of the force F^ is a distance X below the tie rod given by . l!c _ _ 2 82 feet - F - 4,945 ■ a where Mp is given by equation 75' (90) f. Pile Design Minimum Bending Moment - By driving the pile somewhat deeper, a cantilever would be formed near the pile bottom, which would develop passive pressures in the backfill. The loading diagram would be changed as shown in Figure 115. For optimum pile design, that is for the least length of pile for which the maximum bending moment would be smallest, pressures pq and p^Q must be as large as can be developed by the ground at these points. This may be proven for pq by finding the expression for the maximum negative bending moment. This maximum moment would occur below G, the point of zero pile load. Calling the moment at G, Mq, and the shear at the same point, Sq, the maximum moment would occur at the point of zero shear (say Z 2 ), and would equal Mq plus the area of the shear diagram between G and Z 2 . Referring 187 Part II Chapter 5 to Figure Il6 the loading, shear, and bending moment diagrams for this section of pile, the shear at any point Z below G would be = Sq - RZ /2 where R is the rate of change at pressure between pg and pn. At Z = Zp, this shear would be zero, therefore The moment at Z^ is M (91) M / - G ^ 3 = the maximum negative moment. 188 Part II Chapter 5 Now Mq and Sq are functions of the pressure area pg G F; as this area increases, both Mq and Sq will increase. These would be smallest when the rate of pressure change R is greatest (some lesser rate would appear as the dashed line in Figure 116 and would cause point G to be lowered). But the greatest possible rate of pressure change between pg and p^ would make Pg as large as the ground would permit at that point, which was to be shown. Referring again to Figiire 115, one of the earth forces which cause the pressure pg to be developed is that due to the pressure triangle p^Q I J. Whatever this force is, the least length of pile I J over which it is applied would be that length corresponding to the highest possible point of application of the pressure ppQ. But the magnitude of p^Q at this highest point can be no greater than the pressure which may be generated by the earth. Therefore for minimum pile length, p^^g must be as large as this maximum permissible earth pressure. The maximum rate of pressure change between pg and pg has already been determined as R - w 4 ? 90 / 0. 2 90 - , ; tan (-^ ) - tan (- ) C73) z 214 pounds per square foot per foot of depth and the distance between F and G with this rate was found to be h^ =2.36 (Equation 74)• The maximum positive moment in terms of the moment at F, the dredged bottom, has been shown to be ^ 2 (83) “ 6 ^ '"8 189 Part II Chapter 5 It may be seen from Figxire 115 that in going from F to G, the moments increase negatively. Mq is related to Mp by Mq = Mp - |Ag| where Ag is the area of the shear diagram between F and G. Therefore ^ / A^ / —^— (Pg ^ ) The magnitude of the shear at any point x above G up to F at which point X = h^ , is given by R X therefore since Rh^ z Po the area A is 5 8 s 2 As = S^h^ - (92) (93) The maximiom positive moment in terms of and 3 Lr U “V" / (pg / 2P2^) (94) and the maximum negative moment is «z, = ”g 3 (91) In the above equations, Sq is given by the sum of all forces above G or G a (95) = 4945 - T (see equation gq) and the moment is \ \ = Y ) - CG X T = ^ ' 2^ ■ 2.82)-9.36 T (90) = 32,300 - 9.36 T (see equations 75 and 90) In addition, h^ = 2.36 feet (eqxiation 74) Pg = 505 pounds per square foot R = 214 pounds per square foot per foot of depth (equation 73) 190 Part II Chapter 5 R =26.7 pounds per square foot per foot of depth 4 and = P8 R - 2R,S -^— (equation 79) ■4 Rh, where r = (4945 - T) - 595 = 4350 - T therefore = 505 -7(505)^ - 53.4 (4350 - T ) 26.7 - 18.9 - y 32.3 / 0.075 T . For optimum design the maximum bending moments should be equal. The exact solution of the equation = M is difficult, and it is best solved ^2 by trial and error, assuming for the first trial = 0. Then T = ^ 9 ’ 36 ^ “ 3,450 pounds per foot of wall Sp = 4945 - T,= 1^95 pounds per foot of wall Z^ = 1.8 feet. Substituting these values in the expression for the maximum moments, the maximum positive moment is found to be = 5>355 foot-pounds per foot of wall and the maximum negative moment is found to be IL = 3,680 foot-pounds per foot of wall. ^2 By decreasing the value of T (that is, making Mq negative), these moments may be equated. Through trial and error the correct values of T, Mq, Sq, and Z^ are found to be approximately T = 3,350 pounds per foot of wall - 1,000 foot-poimds per foot of wall = 1,595 pounds per foot of wall Z^ ^ 2.05 feet The bending moments = M_ = 5100 foot-pounds per foot of wall. ^1 ^2 For stability, the forces below G must equal Sq, and the moments below G must equal M^. It has been shown that the pressure p^Q for minimum 191 Part II Chapter 5 pile length must be as great as can be generated by the earth at that point. Now the maximum passive pressure at the depth of p^Q is that due to a "surcharge" consisting of all the earth above point G, applied to an un¬ disturbed ground mass of height h^. This surcharge load is given by W P “4 ^ ^ (97) where W = 1,275 pounds per square foot = the surcharge load of the earth above h^ 4 w, = Wc = w, = 80 pounds per cubic foot = the unit weight of the submerged ground h = 3 feet 4 h^ = 2.36 feet 5 therefore, = 1,705 pounds per square foot. The maximum passive pressure p at the depth of p^^Q is applied at the wall back and is given by ^ Pp = “6 <‘'6 ^ (^^4^) (72) where = 30°, The active pressure at the depth of p^Q at the wall face is due to a surcharge composed of the groTuid above G applied to the earth mass of height h^. The surcharge in this case is hj (9 8) = 80 X 2.36 s= 186.4 pounds per square foot The maximum active pressure p^ at the depth of p^^^ is applied at the wall face and is given by Pa = “6 '*'6 ^ (^2^) (64) The actual pressure p^^ is given by ^10 ■ Pp Pa = ^ 6^6 p 90 / 0 / p tan (- - tan ( 90 - 0, —)/ w tan^ ( 2 ' ' 2 ■ ■ p = Rh^ / 1705 (3) - 186.4 (0.33) = 2U h^ / 5053 = 214 (h^ / 23.6) 90 / 0, (99) 2 90 - 2 “a 192 Part II Chapter 5 which may be written symbolically PlO = '“5 ‘'o’ The loading diagram for this section of piling is shown in Figure 117. The sum of all forces below G must equal the shear Sq at G. Summing the forces, represented by the areas GH’ H, H’H JJ’, and H*J* p^^ equating them to Sq we have Sg = R / R (hQ - n ) Jl -R(b^ - jL ) / R(h^/h^) I and which, when solved for gives Rh.^ - 2S„ ^ ( 100 ) R (h / 2h^) Similarly the sum of all moments below G must equal the moment at G. Using the same areas h, -jL ^ 2(h, -jl) 2 0 Mg = R(-^2-^ - 3- ^ ^ ^ ^^6 ■ P - i ) f (h^ - i) which may be reduced to 6>L „ . 2 = 2h^^ - i (bh^'^ / 3h^h^) / ^ (h^ / 2h^) Substituting the solution for ^ from equation 100 the final expression for ‘'6 Substituting the previously determined values for R, h , , and S^ in this equation we have approximately ° 45.8 h^"^ / l,080h^^ - 2,730h^^ - 50,970h^ = 40,480 which when solved for h^^ gives h^^ ^ 7.4 feet The length of piling to G is 15.36 feet, and the total length of pile of minimum moment is about 22.8 feet. 193 Part II Chapter c: g. Final Pile Design - When designed for minimum length, the maximixm bending moment in the pile was M 2 = 5,645 foot-povuids per foot of wall, and the length needed was 18.8 feet. When designed for minimum bending moment, M 2 was reduced to 5,100 foot-pounds per foot of wall, but the length needed for this reduced moment was 22.8 feet, an Increase of 4 feet. The lightest pile section which can withstand a moment of 5,100 foot-pounds per foot of wall is an MP 115 which with a section modulus of 5.4 is good for a bending moment of 6,300 foot-pounds per foot. Since this is larger than the maximum moment which would occur in the pile when designed for minimum length, no saving is Introduced by designing for minimum moment. Accordingly, the design pile would be an MP 115 (or an MZ 22 which has the same weight per square foot of wall but greatly increased strength) and the total pile length would be 18.8 (19) feet. h. Deadman Design - If the backfill is such that dependable passive resistance would be developed, either an intermittent or a continuous deadman could be used instead of one (say) of sheet pile which would pro¬ bably be more expensive. The deadman may be of reinforced concrete, steel, timber, or any other material which would develop the required tie rod tensions over the life of the structure. The loading diagram for the deadman is shown on Figure 118. The dead¬ man would experience an active pressure on its back (the side farthest from the bulkhead wall) which would Increase at the rate 2 90-1^1 Ra r Wj tan (—5-) and a passive pressure on its front which would increase at the rate of P 90 / 0 . R = w tan (-—) pi ^ assuming the entire deadman to be above the line of permanent saturation. 194 Part II Chapter 5 c D K K It ' • ' 1 7 1 M 'HI / d c d ./ AGRAM R. FIGURE 118 DEADMAN LO ADING D The total rate of pressure increase then is = R - R d p a ( 102 ) = w IL P 90/0^ p 90-0/ tan (-- tan (- 2 ^ ' 2 * 120 ( 3.00 - 0.33 ) R^= 321 pounds per square foot per foot of depth. The total earth pressure on the deadman, given by the area of the trapezoid p/L p^ must equal the tie-rod tension per unit width of wall (T) times the length of wall supported by each deadman, divided by the deadman width. In the case of a continuous deadman this earth force must equal T. Assuming a continuous deadman, T = ( Pk ^ Pl ) X d . (103) where and Therefore X Pl = Pk / R^ X (d^ / d^) T = «d / d,) X d. February 195 7 195 Part II Chapter 5 if we call — = ’*d = T' and solve for dj^ 2T' - d. _£ 2d. (104) In addition, the sum of the moments of these forces about (say) K must equal zero. T (d^ - d^) = Pv-d, d 2d / R^d^ X ^ X ^ (105) ^dVd / d d ^ 3 which, when the value of d^^ and T' above are substituted, gives d^"^ - 12 d, T' d^ / 12 (T')^ = 0 d t d an equation in d^ which must be solved by trial and error. ( 106 ) This quartic equation may have as many as two positive root s but no more. One of these roots will be slightly smaller than d^ = yi2d^T' but there may be another root smaller than this. In the.case of the pile designed for minimum length where T = 3,705 pounds per foot of wall, the quantity Vl2dTT' = 9.A feet and a positive root does exist between d^ = 8 and 9. If this value for d^j is substituted in equation 104, d^^ is found to be negative, which is impossible. However investigation shows that there is another root very close to d(j = 2. This root is the applicable solution. Substituting in the equation for d^^ 2 "k = 2T' - d. 2d, 23.05 - 4 4 = 4.76 feet. The deadman dimensions and placement are shown on Figure 119. i. Sheet Pile Deadman - If the backfill is not dependable for passive resistance, a sheet pile deadman, or one which depends on active as well as passive pressures must be used. The analysis for a deadman of this type is similar to the analysis of the lower part of the pile wall designed for minimum moment. (See Figure 120). 196 Part II Chapter 5 By sTimming forces and moments about K as done previously, d^ Is given by 2 _ P'n Ih ^ (-r^ _ 7^ w L ' " d T ^ •a 'ti ■' !■«> - “ L d (107) d and d are given by mo d = m if) o a. o Q UJ O li- ui - 3 O o < o 0. >- UJ o if) UJ QC O c C « « i I « § •s. H I Moy 1961 210a Part II Chapter 5 Finally, the choice of structure would depend on either its first cost or its annual cost. All of the factors discussed affect the cost in one way or another. Generally the structures selected would be that one which would accomplish the desired purpose at the lowest average annual cost over the life of the project. Under some circumstances a structure with a some¬ what higher annual cost might be selected to secure a substantial reduction in the first cost, 5,24 DESIGN PROBLEMS : 5,241 Caisson Type Breakwater (28) - It is desired to build a caisson type breakwater on a rubble riprap base in a Great Lakes area. Still water is assumed on the shoreward side of the breakwater. See Figure 135, By application of methods for forecasting and determining the character¬ istics of the design wave, described in the section on wave action, the fol¬ lowing data have been found: H = 13 feet = design wave height at the structure L = 150 feet = wave length at the structure T = 5,4 seconds = wave period d = 26 feet = depth of water below datum at the structure The wave direction is normal to the breakwater. The depth of water being more than twice the wave height, the waves will not break on or before the structure and the Sainflou method for computing wave pressures from non¬ breaking waves can be used. The clapotis formed (see section 4,23) will be influenced by the depth, d = 26 feet, at the foot of the wall; the elements h^ and Pi will be computed for this depth. Between the top of the riprap at minus 26 feet and the bottom of the wall at minus 32 feet the riprap does not eliminate the pressure and it must be taken into account, which is done by using the pressure. Pi, computed at the top of the riprap and the depth of the floor of the wall, d = 32 feet in the formulas giving Rg and Mg, The pressure diagram developed by this method is shown in Figure 135, The height (hg) of the mean level of clapotis (orbit center) above the still water level, is taken from the graph. Figure 79, using the value of d/L = . = 0,173, In like manner the value of Pj^ is taken from Figure 80, hg = 4,6 feet Pi = 490 pounds = the pressure the clapotis adds or substracts from the still water pressure. The upper and lower limits reached by the clapotis are hg / H = 4,6 / 13 = 17,6 feet above still water level hg - H = 4,6 - 13 = 8,4 feet below still water level May 1961 210 b Part II Chapter 5 Accordingly, to obstruct the oscillating wave completely, the breakwater should rise to not less than 17.6 feet above still water level. With d = 32 feet, and may be found by „ (d / H / ho) (wd / Pi) _ wd^ ^e = 2 2 M (d / h^ / H)^ (wd / P^) wd' T so R = e (32 / 17.6) r62.5 (32) / 490l _ 62.3 (32)‘ 2 • R = 29>750 pounds „ (^9.6)^ [62.5 (32) / 496 ] 62.5 (32)^ = G 6 M = 679 >630 foot-pounds 6 (23) (24) Moy 1961 211 Part II Chapter 5 The width of the concrete structure is found by setting up the weight of the breakwater in terms of the unknown width, X , using submerged weights below still water level. If 150 pounds per cubic foot is the weight of the concrete in air and (150-62.4) = 87.5 pounds per cubic foot the weight of concrete in water, the equation is weight = 18X (150) / 32X (150 - 62.4) = 5,500X If, for stability, it is required that the resultant of wave pressiore and weight must fall within the middle third of the base, then assuming uplift in a triangular distribution, ^ = 679,630 / ^ (113) X = 28.6 feet (moment about point where the resultant cuts the base) and the weight - 5500X = 157,300 pounds As the resultant cuts the base at the edge of the middle third, the factor of safety against overturning is 3 and the entire structure will be in com¬ pression. To determine the factor of safety against sliding, multiply the effective downward force acting on the structiare by a coefficient of static friction and divide by the horizontal thrust of the wave force. If the foundation is not dressed smooth, a friction coefficient of 0.5 or 0.6 is generally considered adequate. The factor of safety should not be less than 2. wt X 0.5 157,300 X 0.5 ^ . R = 29,750 2.64 e which exceeds two and is satisfactory The distance Z from the inner edge to the point where the resultant cuts the base, is 28.6/3 =9.53 feet since equation 113 was derived assuming that the total moment about this point is zero. The maximum pressure against the foundation would be at the inside edge with maximum wave conditions. If Z is the distance between the inside edge of the base and the point where all the resultant forces intersect the base, V is the effective downward force per unit length of wall, G is the pressiare on one square unit of foundation, and A is the width of the breakwater, then the ground pressure at the shoreward edge of the breakwater a = f (2 -^) (114) a . (» - %?>) . U.050 212 Part II Chapter 5 This pressure would be large for most foundation conditions. Should this ground pressure be too great for the bearing power of the bed at the site, the base width of the structure must be increased until the pressure is within the allowable load, 5,242 - Rubble-Mound Breakwaters , The design of a rubble-mound break¬ water is best illustrated by an example problem. For this example it is assumed that: (a) a rubble-mound breakwater is to be constructed in a sheltered sea coast area (w^^, = 64,0 lb, per cu. ft,) where the water depth is 40 ft, below MLW with a tide of 5 feet; (b) no appreciable overtopping of the structure can be allowed; (c) the average of the highest 10% of all deep water waves was found to have a height of 15 feet and period of 7 seconds, and this wave was selected as the design wave (see section 4,11); (d) the refraction coefficient (K^) from deep water to the structure is 0,87; (e) the quarrystone armor units have a unit weight (Wj.) of 165 lb, per cu, ft, and Sr = ^ = 2,58; and (f) the largest quarrystones that can be obtained from a selected quarry, economically, vary in weight from 8 to 10 tons, A value of 3,5 for KD is obtained from Table 9A for breakwater trunk with a non-breaking wave condition; however, experience indicates that the rock from the local quarries is quite angular and some degree of inter¬ locking has been obtained and the Kp factor may be adjusted (see section 4,272), therefore, Kp = 3,7 is selected. Wave Height at Structure (H) The wave height at the breakwater, based on the above assumptions, is calculated as follows: = 15,0 ft, (deep-water design wave height) T = 7,0 sec, (wave period) d = 40 feet below MLW +5 ft, tide = 45 feet Lq = 5,12 T^ = 251 ft, (deep-water wave length) d/Lo = 45/251 = 0,179 then H/Hq* = 0,914 (shoaling coefficient, for d/L^ = 0,179 Table D-1, Appendix D) with Kj = 0,87 (refraction coefficient) H = Hq (H/Hq*) Kr (48) H = 15 (0,914) (0,87) = 11,9 feet Armor Unit Weight and Slope of Primary Cover Layer Several approaches may be taken to obtain an economic design for a rubble-mound structure. The following is a suggested method of approach. May 1961 213 Part II Chapter 5 After having obtained the values for H, Kp and Sr, the only values left are slope (Cot a) and armor stone weight (W), These may be determined by substituting a value for one in the equation and solving for the other. In either case, a table as shown below should be calculated (see Plate D-7b Appendix D), The relation of slope to armor stone weight can quickly be seen. In order to determine an economic structure, the wave run-up and the resulting crest elevation must be also considered, inasmuch as any change in elevation of the crest for a fixed crest width, will make a change in the volume of material at the bottom of the structure rather than at the top. Stone Wt, Run-up Slope ([tons) (ft) 1 on 1,2 8.0 1 on 1.5 6.3 10.9 1 on 2 4.6 9.9 1 on 2.5 3.8 9.4 1 on 3 3.2 a.5 Looking at this table it can quickly be seen that the 8-ton stone (largest stone economically available) results in a steeper slope than the maximum recommended. Therefore, an evaluation must be made by comparing the volume of the structure considering the side slope and crest elevation, and the availability of lesser weight stone. Let us say for example that an economical comparision of volume versus side slope and crest elevation found that a 1 on 2 slope was justified and 5 to 6-ton stone was readily available at the selected quarry. Continuing the example, the following computations are made to show some of the various steps involved in designing a rubble-mound structure. The resultant break¬ water cross-section is illustrated in Figure 136, Af ter step is to Values for having selected a side slope for the structure trunk the next determine the minimum stone weight that is stable on this slope, Wf, H, Sj., Cot a and Kp are substituted in the following equation. W = Wj. H Kp (Sr - l)”^ Cot a 165 (11,9)^ (46) W = 9,500 lbs. Thus 4.7-ton stone is the minimum weight required for the primary armor cover stone. Blevation of Crest, In order to prevent, all except minor, overtopping of storm waves, the elevation of the crest should be established at or above the maximum limit May 1961 214 May 1961 215 Part II Chapter 5 of wave run-up, see section 4,274. The relative wave run-up (R/H©*) (see section 3,271) is determined for non-breaking wave conditions as follows: H = 11.9 ft. H/Hq* = 0.914 (Shoaling coefficient) T = 7 secs. Ho' =H/(H/«o') = 13.0 feet Hq* = 13.0/49 = 0.265 from Figure 61-L (see section 3.271) 2 for 1 on 2 slope and = 0,265 R/Hq* = 0,76 (rubble-mound slope) R = 0.76 (13) = 9.9 feet The maximum wave run-up (R) would be 9.9 feet for a rubble-mound slope. If the voids of the armor stone are reduced by chinking and the roughness and porosity decreased, the run-up would be increased. The value for wave run-up would then approach the wave run-up value for a smooth slope. The crest ele¬ vation would be +9.9 feet above design water surface or 14,9 feet above MLW, Width of Crest There will be no appreciable overtopping of this structure. Therefore, the crest width is not critical with respect to the forces of overtopping water, and a top width corresponding to the combined widths of three cap stones is selected. This is, perhaps, the minimum practical width of a rubble-mound structure of this type. Thus, using equation 47 for values of k^ = 1.0, and n = 3 from Table 9B (see section 4,274) ® B = 3 X 1.0 165 B = 3 X 3.93 = 11.8 ft. Thickness of primary cover layer r = n k. (47a) “ Wj. r = 2 X 1.0 ^ 2 (60.6)^^^^ 165 r = 2 X 3.9 = 7.8 ft. May 1961 215a Part II Chapter 5 Bottom Elevation of Primary Cover Layer Armor units in the primary cover layer are extended, for practical pur¬ poses, dovmslope to elevation -1,5H (below MLW). Thus, the armor units of weight W = 4,7 tons and greater are extended to elevation -17,9 ft, below MLW, Secondary Cover Layer Below elevation -1,5H the approximate weight of the armor units can range W/10 to W/15, or, from elevation -17,9 to -40 ft, armor units ranging from 600 to 2000 lbs, can be used. The thickness of the secondary cover layer should not be less than that of the primary cover layer or 7.8 feet. Underlayers The first underlayer for the primary cover layer, from the bottom eleva¬ tion of the crest armor units to -1.5H or +7,0 to -17,9 ft, should be composed of stones of approximate weight W/10 to W/15 or the same material as the secondary cover layer. Remaining underlayers should be composed of stones of approximate weight ranging from W/200 to W/6000, Quarry-run material will generally meet these requirements with the larger stone placed on the outer surface of the core. Bedding or Filter Layer When large stones are placed directly on a sand bottom at depths insuf¬ ficient to avoid wave action and currents on the bottom, the rubble will settle into the sand until it reaches a depth below which the sand will not be disturbed by the currents. In this case, the amount of rubble deposited should be sufficient to provide protection after settlement, or more rubble should be placed after settlement has taken place. To prevent waves and currents from removing foundation materials through the voids in the rubble structures or protective aprons and destroying their support, all materials having large voids should be placed on a blanket of crushed stone. The blanket material should be sufficiently graded to prevent the removal of the foundation material through the bedding layer (see section 4,276). Quarry-run material will generally meet these conditions. May 1961 215 b Part II Chapter 5 5.3 SAND BYPASSING 5,31 GENERAL - A coastal inlet may be, for the purpose of this section, considered as any relatively narrow waterway connecting the sea or major lake with interior waters. Such inlets, either in their natural state or after improvement to meet navigation requirements, tend to interrupt normal littoral transport along the shore. In the case of natural inlets, which have a well-defined bar formation on the seaward side of the inlet, a major proportion of the littoral drift ordinarily moves across the inlet by way of the outer bar, but the supply reaching the downdrift shore is usually intermittent rather than regular, with the result that the shore downdrift from the inlet is normally unstable for a considerable distance. If the strength of tidal flow through the inlet into the interior body of water is appreciable, part of the available littoral drift is permanently stored in this body of water in the form of an inner bar, reducing the supply avail¬ able to nourish downdrift shores. In the case of migrating inlets the outer bar normally migrates with the inlet but the inner bar does not. The volume of interior storage in such cases constantly increases and does not tend to stabilize until inlet migration is halted, When the natural depth of an inlet is increased by dredging, either through the outer or inner bars or the gorge, additional storage area is created to trap the available littoral dr ift,thereby reducing the quantity which would naturally pass the inlet. If the material dredged (either initially or for channel maintenance) is deposited beyond the limits of the littoral zone, as in the case of disposal in deep water at sea, the supply to the downdrift shore may be virtually eliminated with consequent erosion of the downdrift shore by the littoral forces at a rate equivalent to the reduction in supply. The normal method of inlet improvement has been to provide jetties flanking the inlet channel. The purpose of jetties may include any or all of the following purposes: block the entry of littoral drift into the channel; serve as training walls to increase the velocity of tidal currents and thereby flush sediments from the channel; serve as breakwaters to reduce wave action in the channel; and serve to prevent further inlet migration. In the rare cases where there is no predominant direction of littoral trans¬ port jetties also serve to stabilize the adjoining coastal shores. In the more common cases where littoral drift in one direction predominates, jetties cause accretion of the updrift shore and equivalent erosion of the downdrift shore. Stability of the shore downdrift from inlets with or without jetties may be improved by artificial nourishment to make up the deficiency in supply. When such nourishment is accomplished by using the available lit¬ toral supply from updrift sources the process is called "sand bypassing", A number of mechanical methods of sand bypassing have been actually employed, however, this is still a relatively recent engineering development and additional methods will no doubt be developed as experience is gained. May 1961 216 Part II Chapter 5 5.32 MEIHODS - Several techniques have been (and are presently) employed for mechanically bypassing littoral materials at barriers. Sometimes a combination of techniques has proved to be the most practicable and economical. The basic techniques and equipment which have been used are as follows; (a,) Land-based dredging plants (b,) Floating dredges Cc,) Mobile land-based vehicles, 5.33 LAND-BASED CTIEDGING PLANTS - This type of operation usually involves a dredging plant at a fixed position in the littoral zone from which the plant more or less continuously extracts or intercepts the littoral materials as they move within reach of the plant. All presently installed plants are of the pump type and operate basically as an ordinary suction dredge. The plant may be positioned on an existing structure, such as a jetty, or on an independent foundation constructed to support the plant. There are certain components of shore processes at a littoral barrier that must be given cri¬ tical study in order to design and position a fixed bypassing plant. Know¬ ledge of the average annual rate of littoral drift moving to the barrier is one of the most important factors, as this annual rate will normally be the controlling criterion for determining the capacity of the pumping plant. The average annual impoundment of littoral materials by the littoral barrier is equal to the minimum quantity that must be supplied to the downdrift shores to achieve stability of these shores. Short-term fluctuations of the actual rate of littoral material movement to the barrier as on an hourly, daily or weekly basis may be many times greater or less than the estimated average annual rate reduced arithmetically to an hourly, daily, or weekly basis. Therefore, even though a bypassing plant may be designed to handle the total drift reaching a barrier on an annual basis, there will be occa¬ sions during the year when the quantity of sand reaching the barrier will greatly exceed the pumping capacity of the plant and occasions when the plant may operate well below capacity due to an insufficiency of material reaching the barrier. No land-based plant thus far build in this country (1961) has succeeded in bypassing more than about 50 percent of the available littoral drift. Two fixed bypassing plants abroad were discontinued because of this defici¬ ency and none is known to be operating outside this country at present. These limitations must be considered in evaluating the effectiveness of a land-based bypassing plant. In order to establish design criteria for this type of installation, detailed study must be made of the beach profile around and updrift of the littoral barrier as optimum location of the plant along the profile must be determined, A comparison of foreshore profiles over a period of time will aid in predicting the future position of the foreshore and allow a determi¬ nation of the optimum position of the plant. In general, location of the plant too far landward may result in a "land-locked” plant when the rate May 1961 217 Part II Chapter 5 of drift reaching the barrier in a short interval of time exceeds the plant’s pumping capacity. Such location may also result in large losses of material around the barrier, A location too far seaward may result in ineffective operation until sufficient materials have been impounded by the barrier and are within reach of the plant’s intake mechanism. The disadvantage of the fixed position of the plant has led to consideration of a movable dredging unit on a pier. This would permit dredging of a reservoir for littoral drift and avoid land-locking of the suction. Mobility of land-based plant may overcome the deficiencies of a fixed plant as described above, but this still remains to be demonstrated in an actual field test. The optimum alignment of the discharge line from the fixed plant to the downdrift side of the littoral barrier is generally controlled by local conditions. In most cases the discharge line must traverse a channel which is maintained for vessel traffic and unfavorable conditions exist for a floating discharge line. If the line is positioned on the channel bottom, allowance must be made for protecting the line during periods of maintenance dredging of the channel. Also, a submerged line may necessitate a special flushing system designed for the purpose of keeping the discharge line from clogging. The point of discharge on the downdrift side of the littoral barrier may or may not be of critical importance. Obviously, the discharge point would not be critical if it be unimportant that some of the material at the point of discharge will be transported back toward the littoral barrier during periods of drift reversal. However, if it is desired that the minimum amount of material move back toward the littoral barrier, then a detailed study must be made of the distribution of littoral forces down- drift of the littoral barrier. In this case the optimum discharge point will be that point on the shore where there is minimum influence by the littoral barrier on littoral forces tending to move material in a downcoast direction. Establishment of this point requires the use of statistical wave data, wave refraction and diffraction diagrams, and data on nearshore tidal currents if such currents are present and have significant influence on the littoral processes immediately below the littoral barrier. Alterna¬ tive points of discharge nearer to the littoral barrier may also be con¬ sidered, using auxiliary structures, such as groins, to impede updrift movement of material at the discharge point. Such alternative considerations are of value in determining the most economical discharge point. 5,331 Fixed Bypassing Plants - The following paragraphs present a sum¬ marized description of several existing fixed bypassing plants. South Lake Worth Inlet, Florida - South Lake Worth Inlet is located about 12 miles south of Palm Beach, Florida, After the construction of jetties at this inlet, erosion of the downcoast beach occurred. Construction of a seawall and groin field failed to stabilize the shore line, A sand bypass¬ ing plant was placed in operation in 1937^^^^\see Figures 137 and 137A). The basis of design of the ptsBping plant as initially installed was not re¬ lated to the rate of littoral transport along the shore, but rather to by¬ passing a desired quantity of sand over a period of 2 years, to fill the groins and give adequate protection to the seawall. The pimping plant May 1961 218 rfMSr \ 1 ^-^' FIGURE 137. SAND TRANSFER PLANT, SOUTH LAKE WORTH INLET Moy 1961 219 N Main Pumping Sfotion 4 ^ FIGURE I37B.- FIXED BYPASSING PLANT AT LAKE WORTH INLET, FLORIDA. May 1961 220 Part II Chapter 5 consisted essentially of an 8-inch suction line, a 6-inch, 65-h,p, diesel- driven centrifugal pump, and about 1,200 feet of 6-inch discharge line. The discharge line was carried across the inlet on a highway bridge and the outfall was on the beach immediately south of the south jetty. The installation had a capacity of about 55 cubic yards of sand an hour and pumped an average of 48,000 cubic yards of sand a year during 4 years of operation. The normal rate of littoral transport during the same period was estimated to be on the order of 225,000 cubic yards a year. At the end of 5 years, the beach was partially restored for a distance of over a mile downcoast. During the 3-year period 1942-1945 pumping was discontinued with the result that severe erosion of the beaches south of the inlet again occurred. In 1945, the plant was again placed in operation, and shortly thereafter the shore immediately south of the inlet was stabi¬ lized, In an attempt to reduce shoaling in the inlet channel, the size of the bypassing plant was increased to an 8-inch pump with 275-h,p, diesel motor with a capacity of about 80 cubic yards of saivi per hour. Under present conditions it is estimated that the plant bypasses about one-third of the available drift. The remainder of the drift, or about 150,000 cubic yards, is transported by wave and tidal currents to the ocean and bay shoal zones and to the shores south of the inlet. Lake Worth Inlet, Florida - Lake Worth Inlet is located at the northerly limit of the Town of Palm Beach, Florida, The Lake Worth Inlet fixed by¬ passing plant^^^^»^^2) housed in a two-story reinforced concrete struc¬ ture located near the end of the north jetty (Figure 137B), The lower floor of the structure is at elevation - 1 foot MLW and contains a 400-h,p, centrifugal dredge pump, electric motor, and a power transformer. The pump has a 12-inch suction and 10-inch discharge and is designed to handle 15 percent solids at more than 60 percent efficiency. This gives the plant a design capacity of approximately 170 cubic yards per hour. The upper floor of the structure contains the controls and ventilating equipment. The suction line is carried by a 30-foot movable boom that can rotate through an arc of about 140 degrees. The discharge line is about 1,750 feet in length. The line is made of steel pipes of |-inch wall thickness, with the exception of an 800-foot section of submerged line crossing the navigation channel. This section of line is made of wire-reinforced rubber hose in 50-foot sections with couplings which allow it to be removed for channel maintenance. The following safety features have been, or are planned to be, installed into this plant to minimize the possibility of clogging the submerged discharge line; a. An alarm device to warn the operator when the pressure is abnormal. He can then raise the suction line to exclude sand from the intake. b, A device to admit clear water in the suction line just ahead of the pump when pressure in the discharge line drops below a certain value. This is a precaution in case the suction intake becomes clogged. May 1961 220a Part II Chapter 5 c, A tank with a volume twice as large as that of the discharge line to feed water into the line automatically in event of a power failure and an air compressor to provide sufficient pressure to produce a velocity of flow of 16 feet per second in the line, d, An emergency gasoline-driven 12-inch centrifugal pump, to pump clear water into the discharge line by the time the flush tank has emptied, e, A self-propelling nozzle of the type used to open blocked sewers, and sufficient hose to feed it through the pipe. Operation of the plant began in mid-August of 1958, It was estimated that approximately 71,400 cubic yards of sand were bypassed in 451 hours of operation, during a period of 8 months. This indicates that the plant can bypass in the order of 100,000 cubic yards of sand per year, somewhat less than half of the estimated available littoral drift. The plant has had to curtail operations on occasion due to high seas which prevented personnel from reaching the pumping station, A cable suspended from a steel post on top of the plant to carry personnel to and from the plant in a boatswain's chair is expected to eliminate this problem. The final contract cost for construction of the plant approached $514,000, Based on the operation and maintenance costs for a period of 5 months, it is estimated that all costs (including maintenance, operation, interest, debt retirement, and deprecia¬ tion spread over a period of 30 years at an average pumping rate of 100,000 cubic yards per year) would be in the order of $0,49 per cubic yard, Durban, Natal, South Africa - The operation of a fixed bypassing plant was started in 1950 at Durban located on the southeast coast of Africa^The plant, located on the southeast side of the harbor entrance is shown in Figure 138 and consisted of a 16-inch pump and discharge line. The discharge line ran under the harbor entrance, A boom supporting the suction line could swing through an arc of about 180 degrees on a 36-foot radius and allow dredg¬ ing to a depth of 20 feet below low water. Prom 1950 to 1954 the plant by¬ passed approximately 200,000 cubic yards of sand per year. The quantity of material bypassed decreased each year after 1954 and authorities reported that the lack of drift reaching the plant intake necessitated its removal in 1959, It was also reported there was no apparent reduction in maintenance dredging of the entrance channel to the harbor during the 10 years of bypassing opera¬ tions, Starting in 1960 the material dredged from the channel was pumped to the beaches to the north by a pump-out arrangement from the dredge, and booster pumps located along the beach. This plan of by-passing the material dredged from the entrance channel is similar to that used prior to 1949, 5,34 FLOATING DRBDGES - The operation of floating dredges may be classified in two general categories, namely, hydraulic and mechanical. Hydraulic dredges include the suction pipeline dredges with plain suction or with cutterhead for digging in hard bottom, and the self-propelled hopper dredge. The mechanical types include the dipper and bucket dredges. The pipeline dredges utilize a discharge pipeline (including booster pumps in this line, if required by distance to discharge point) to transport the dredged material May 1961 220 b Moy I96> 220c Part II Chapter 5 to the desired point or area of placement. The hopper dredge, although hy¬ draulically operated to fill its bins, usually discharges by dumping the dredged material out of the bottom of its bins; thus this type dredge re¬ quires disposal in offshore areas of sufficient depth of water. Unless the standard hopper dredge discharges its bins in an area where the material may be handled by another type of dredge or is equipped to pump the material ashore, hopper dredges are not suitable for bypassing operations. Mechanical type dredges require auxiliary equipment (such as dump scows, conveyors, eductors, etc,) to transport the material to the point of placement. This rehandling requirement generally limits the use of the mechanical type dredge in bypassing operations and may add considerable cost to the operation. Therefore, in considering a floating dredge for a bypassing operation at a littoral barrier, each type of dredge plant must be carefully evaluated. This evaluation would include, first, the feasibility of utilizing various types of floating dredges, secondly the operational details and finally the econom¬ ics or determination of that floating plant which will transfer the material at the least unit cost. Local site conditions will vary and factors to be considered relative to each type of floating plant cannot be standardized, however, the following are some of the more important factors that generally must be evaluated: a. Exposure of plant to wave action - Wave action limits the effective operation of a floating dredge, the exact limitation being depend¬ ent on plant type and size, and intensity of wave action. This factor is particularly critical if the dredge will be exposed to open waters wherein sizable waves may be generated. No standard criteria are available to establish the maximum permissible wave action for operation of various types of dredges. Such data must be obtained from dredge operators who are famil¬ iar with the area in question. Hopper dredges may be operated in conditions of higher wave action than the other types of floating dredge plants; however, as previously pointed out, to use the hopper dredge for bypassing operations a rehandling arrangement must be employed or the dredge must be equipped with adequate pump-out features. Pipeline dredges exposed to appreciable wave action are subject to breakage of the ladder carrying the suction line, breakage of spuds, and damage of the pontoon line carrying the discharge pipe. Thus estimates must be made of the operational time with and without man-made structures or natural ground features to protect the dredge and auxiliary equipment. Determination of the time of year when the least wave action will prevail will allow estimates to be made for plant operation under the most favorable conditions. Also, consideration must be given to appropriate pro¬ tection of the plant in the event of a severe storm passing through the area or project site, b. Capacity of Plant, - The use of a floating dredge of a specific capacity is generally controlled by economic consideration. If the impounding zone of a littoral barrier is large and some temporary recession of the down- drift shore may be allowed, consideration may then be given to a periodic bypassing operation whereby a large plant is utilized for a short period of time. An alternative consideration would be the use of a small capacity plant for longer periods of time, however, if pumping distances to discharge point are such to require too many booster pumps, a larger plant may provide the most economical operation. May 1961 220 d Part II Chapter 5 5.341 Bypassing Operations by Floating Plant - The following paragraphs present a summarized description of several bypassing operations by float¬ ing plant, Santa Barbara, California - The Santa Barbara sand bypassing operation^ was necessitated by the construction of a 2,800-foot breakwater, completed in 1928, to protect the harbor, (See Figure 139), Its construction resulted in accretion on the updrift side (west) and erosion on the downdrift side (east). Bypassing was initially accomplished in 1935 by hopper dredges which placed in the order of 202,000 cubic yards offshore at approximately the 20- foot depth contour. Later surveys revealed that sand from this deposition area was not moved to the beaches. The next bypassing operation was accom¬ plished in 1938 by pipeline dredge, A total of 584,700 cubic yards of sand was deposited in the feeder beach area shown on Figure 139. This feeder beach was successful in alleviating erosion downdrift of the harbor and the operation was continued by placing 697,700 cubic yards in 1940; 558,600 cubic yards in 1942; 717,800 cubic yards in 1945; 675,044 in 1947; 755,733 in 1949; and 1,070,000 in 1952. Officials of the city of Santa Barbara decided in 1957 not to remove the shoal at the seaward end of the breakwater, as it provides protection May 1961 220 e Part II Chapter 5 to the inner harbor, A channel is being maintained around the north end of the shoal by continuous dredging by a small floating dredge. Although the operation is described as continuous, wave and weather conditions limit the actual pumping to about 72 percent of the time. With a capacity in the order of 1,600 cubic yards per 8-hour shift, the dredging is adequate on a yearly basis, but inadequate to prevent some shoaling of the channel during some of the shorter storm periods, Hillsboro Inlet, Florida - Hillsboro Inlet is located about 36 miles north of Miami Beach, Florida, Sand bypassing operations at this inlet have been by a pipeline dredge. This method is practical at this location as the littoral material moving to the south is impounded in an area sheltered by a rock reef and rubble-mound jettv. The rock reef and jetty form what has been termed a "sand spillway"(170), Dredging the sand at the spillway and depositing it on the downdrift shore has assisted in keeping the adjacent inlet open for vessel traffic and checked erosion of the downdrift shore. Dredging experience to 1955 has indicated that approximately 75,000 cubic yards of sand should be bypassed each fall for this purpose, A barge-mounted bypassing plant is being considered to transfer the impounded littoral ma¬ terials. Port Hueneme, California - Port Hueneme is located approximately 7 miles south-southeast of the mouth of the Santa Clara River, Figure 139 A, and the direction of littoral material movement in this area is predominantly toward the southeast (171), Port Hueneme Harbor was constructed in 1940 and ac¬ quired by the U, S, Navy in 1942, The 35-foot depth entrance channel to this harbor is protected by two converging rubble-mound jetties. The west jetty has impounded a substantial quantity of littoral drift (the drift moves to the southeast at an estimated rate of 800,000 cubic yards a year in this shore sector) but its greatest effect has been to divert the lit¬ toral drift into the deep waters of the Hueneme submarine canyon, thus preventing this beach material from reaching the shores to the southeast of Port Hueneme through natural littoral processes. Prior to the construction of this harbor the shore southeasterly thereof was in an exceptionally stable condition. The rate of erosion of the shore downcoast of Port Hueneme has been about 1,200,000 cubic yards per year, since 1940, This is about 50% greater than the estimated normal littoral drift rate, the increase being attributed to the exposure of less resistant sub-surface material following erosion of the original sand beach surface. An emergency bypassing operation was undertaken at this harbor in 1953 to alleviate this condition and nourish the downcoast beaches. The sand trapped by the west jetty was pumped to the eroding downdrift shore by floating pipeline dredges. The dredging procedure used at Port Hueneme was unique in that the outer strip of the impounded beach was used as protection to the dredge from wave action during the operation. The initial phase (Figure 139 B) was accomplished by digging a hole in the beach with land equipment, A small pipeline dredge removed enough material from May 1961 220f Moy 1961 220 g FIGURE I39A LOCATION MAP PORT HUENEME, CALIFORNIA Moy 1961 220h FIGURE 139B. PORT HUENEME, CALIFORNIA. APPROXIMATE SHORE LINE DURING DREDGING IN PHASE Part II Chapter 5 that hole to permit entrance of a larger dredge from the open sea. The larger dredge then completed the dredging leaving a protective strip of beach for the final operation. In dredging the barrier strip of beach (See phase I & II areas in Figure 139B) cuts were made from the phase one area to the mean lower low water line at an angle of approximately 60 degrees to the shoreline. The diagonal cuts gave the dredge more protection from wave action than perpendicular cuts. One of the problems encountered in the Port Hueneme sand bypassing operation was water supply for the dredging. This had to be done by pump¬ ing as it was necessary to close the dredge’s entrance route to prevent erosion of the protective barrier. Therefore, there was no channel to supply make-up water as the phase one dredging proceeded. This problem could likely have been avoided had the proposed entry route from inside the harbor (Figure 139 B) been used instead of the optional entry route from the open sea. During the period of sand bypassing (August 1953 to June 1954), 2,032,700 cubic yards of sand were bypassed to two downdrift feeder beaches as shown on Figure 139 B, A survey taken in June 1956 indicated an erosion rate downdrift from the harbor in the order of 2,000,000 cubic yards for the period June 1955 to June 1956, Ventura County Harbor, California - A small-boat harbor is presently under construction about 1 mile upcoast, or northwest, of the entrance channel to Port Hueneme, which was previously described. The sand bypassing plan for Ventura County Harbor^will also transfer the material past Port Hueneme and provide artificial nourishment of the eroded shores downcoast of that harbor. The general plan of Ventura County Harbor is shown on Figure 139 C, This project has three basic objectives; (a) to provide a trap which will impound essentially all of the southward moving littoral drift, thus pre¬ venting shoaling of the harbor entrance and eliminating present losses into the submarine canyon; (b) to provide protection to floating plant during dredging of the sand from the trap area and the transfer of the sand to the shores downcoast of Port Hueneme; and (c) to provide shelter for about 500 civilian small craft. The plan is thus designed to effectively bypass the littoral drift past two harbor entrances in one operation. The littoral drift is impounded by a rubble-mound offshore breakwater and the breakwater serves as protection to floating plant during dredging operations. The breakwater, 2,300 feet in length and located at the 30-foot depth contour, has a crest elevation of 14 feet above mean lower low water. The breakwater position, orientation, length, and crest height were all determined by de¬ tailed wave refraction and diffraction analysis. This analysis not only served to predict the location of the sand trap and wave conditions in the lee of the breakwater, but to locate the entrance channel, entrance jetties, and to predict wave conditions therein. The project depth of the entrance May 1961 220 i May 1961 220 j FIGURE 139-C GENERAL PLAN-VENTURA COUNTY HARBOR, CALIFORNIA Part II Oiapter 5 channel is 20 feet. The crest elevation of the rubble-mound entrance jetties is 14 feet above mean lower low water, and the jetties terminate at about the 14-foot depth contour. The average annual rate of littoral drift at the Ventura County Harbor is estimated to be about 800,000 cubic yards. It is planned to bypass 1,600,000 cubic yards every 2 years to the shore southeast of Port Hueneme, Initial project dredging of the entrance channel, harbor, and sand trap will provide over 5,000,000 cubic yards of material to the downcoast shore. This initial fill will serve to stabilize the downcoast shore until periodic by¬ passing operations are effected. However, about 20,000,000 cubic yards of material has been eroded from these shores since 1940 and although the by¬ passing operation is expected to stabilize these shores; the shore will be positioned substantially landward of its 1939 position. Although this project is not completed it has progressed sufficiently to indicate the validity of basic design assumptions. At this time (1961) this general method of bypass¬ ing is considered to provide greater assurance of complete effectiveness than any other thus far considered. May 1961 220 k Part II Chapter 5 5.35 MOBILE LAND BASED VEHICLES - Local site conditions may favor the use of wheeled vehicles for by-passing operations. This technique requires that a number of factors be evaluated, the importance of each being dependent on local conditions. Typical factors to be considered would be the existence or provision of adequate roadways and bridges (if necessary), accessibility to the impounding zone by land-based equipment, and the volume of material to be by-passed (and length of time required to transport the material, if this is of importance to the overall operation). The following case illustrates the comparative estimated costs of a by¬ passing operation, with the use of land-based vehicular equipment being the most economical procedure. The site was Shark River Inlet, New Jersey, ^210) The by-passing project consisted of removing 250,000 cubic yards of sand from an area 225 feet south of the south jetty at the inlet and the placing of this material along 2,500 feet of beach front on the north side of the inlet. Contractor’s bids were 88 cents per cubic yard for loading by crane and placing by truck, $1.30 per cubic yard for loading by dragline and plac¬ ing by pump and pipeline, and $1,42 per cubic yard for excavating by hydrau¬ lic suction dredge and placement by pipeline. The contract was awarded for loading by crane and placement by truck. The contractor built a trestle to a point beyond the low water line in the borrow area to allow trucks access from the highway to a crane with a 2|-yard bucket. Three other shorter trestles were built north of the inlet from which trucked material was dumped to be distributed by natural forces to downdrift beaches, 5.4 SAND DUNES (53) (71) (84) (85) (161) Material following in this section was abstracted extensively from Davis (reference 161), The problem of dune control resolves itself into two fundamental ob¬ jectives: The stabilization and maintenance of sand dunes at locations where they exist naturally, and the inducement and stabilization of pro¬ tective dunes where they do not exist naturally. The formation of dunes occurs in areas where a substantial supply of sand is available and meteorological conditions are proper. These conditions are fulfilled on many of the wide sandy beaches along the coasts of the United States, Winds with sufficient velocity to move the sand particles erode the dry portions of a beach and transport the sand in three ways: a. Suspension, whereby the samll or low density grains are lifted high into the air stream and carried for appreciable distances; b. Saltation, whereby the wind carries an individual particle by a series of short jumps along the surfacej and May 1961 220 i Part II Chapter 5 c. Surface creep, whereby a particle rolls or bounces along the ground as a result of wind pressure or the impact of a descending saltating particle. The three modes of transportation effectively sort the original beach material. The smaller particles may be completely removed from the dune area while the larger particles remain; therefore, the dunes are composed of material of essentially one size. This uniformity of size materially aids the development of a vegetative cover, 5,41 DUNE BUILDING - The building of a dune begins where an obstruction causes deposition of the suspended particles and stops saltating or creeping particles. As the incipient dune builds, its slope becomes sufficiently steep so that saltating or creeping particles come to rest on its windward slope. With high wind velocities these particles will crawl up the face of the dune, settle in the lee of the dune, and result in the migration of the dune in the direction the wind is blowing. With favorable conditions, dunes build to heights considerably above the limits of storm wave action, and in addition several dune ridges may be formed. Dunes can be formed by the use of Sand fences, brush barriers, mechanical or hydraulic placement of sand, crude oil, and vegetation, (See Figures 139 D and 139 E.) Sand fences may be constructed in movable or permanent sections that are extended as they fill, or may be made of individual pickets driven into the sand. The width and length of the pickets may vary but the spacing of the pickets is important. Standard snow fencing with the space between the pickets equal to the width of the pickets has shown good results in the past. In order to widen the crest of the dune and facilitate establishment of vege¬ tation, two lines of fence about 30 feet apart should be used. The use of a single fence tends to build dunes with sharp crests which are unfavorable to establishment of cover. As the dune builds up on the fence, the fence can be raised until the desired height is attained. It is very important, however, not to remove any of the sand fence that is covered or partially covered but rather extend or raise the fence by adding new fencing. The belt can be broadened by shifting the second fence windward as the dune grown or by the addition of a third fence. Figure 139 P shows various examples of the fence designs used on the Outer Banks in North Carolina. Sand fences should be constructed normal to the prevailing wind direction unless it is desired to cause the sand to move longitudinally along the fence to fill in low gaps. In this case the fences may be built slightly quarter¬ ing to the general shifting direction of the sand movement, or panelling may be utilized. Panelling, to divert or stop sand, consists of curved or flat barriers in a single slant or in V arrangement (71)^ as the line of the slant or V approaches the normal to the wind direction, there is more and more stoppage and less and less diversion of sand. These barriers require frequent chang¬ ing or cleaning. The dune development behind diversion fences is shown in Figure 140 and 140A, May I96< 221 Dune reconstruction employing brush type sand fencing. Dune reconstruction employing commercial snow type sand fencing. Phofogrophy courtesy of Notional Pork Service FIGURE 139-D DUNES READY FOR PIONEER PLANTING AT CAPE HATTERAS NATIONAL SEASHORE RECREATIONAL AREA, NORTH CAROLINA. May 1961 222 Brush type sand fencing shortly after installation. Mechanically constructed dune ridge stabilized by plantings situated in wide gaps through dunes. Lower photo courtesy of Notional Pork Service (Both photos reproduced from Ref. 161 ) FIGURE I39E. WORK UNDER CURRENT PROGRAM AT CAPE HATTERAS NATIONAL SEASHORE RECREATIONAL AREA, NORTH CAROLINA. May 1961 223 STRAIGHT FENCE Reproduced from Ref. 161 FIGURE I39F-FENCING DESIGNS USED ON OUTER BANKS, N.C. May 1961 224 Moy 1961 225 FIGURE 140 DEVELOPMENT AROUND FENCE PANELS FIGURE 140-A DEVELOPMENT AROUND FENCE PANELS EFFECT OF OIL STABILIZATION ON ENTIRE SURFACE OF DUNE Continuous supply of sand drifting down upon the stabilized stationary dune. A before oil stabilization Dune is slowly migrating down wind and is generaly slowly increasing in size. ^ DUNE STABILIZATION WITH OIL Dune is now permanently fixed. It will not migrate , change size or shope. It is now a very effective sand trap. C_ SAND ACCUMULATING BEHIND STABILIZED DUNE The fixed dune is still acting as 0 sand trap. .. but has lost it’s typical crescent shape. D TRAPPED SAND HAS REACHED STABLE PROFILE CONDITION The fixed dune has naw trappd the full amount of sand it is capable of holding. It has now assumed a streamlined shape and no longer acts as a barrier to new sand. E. SIDE VIEW OF ULTIMATE STABLE CONDITION The trapped sand is permanently impounded and will not move; however the pile is no longer capable of stopping new sand drifting down with the wind. Reproduced from Reference 71 FIGURE 141 DEVELOPMENT OF AN OILED DUNE May 1961 225 a Barrier dune built by sand fences and plantings during 1934 — 40. Area well-stabilized by vegetation. Remnants of eroded dunes requiring reconstruction and stabilization through use of sand fencing and plantings. Lower photo courtesy of Notional Pork Service IBoth photos reproduced from Ref. 161 ) FIGURE 142. STABILIZED AND ERODED DUNES AT CAPE HATTERAS NATIONAL SEASHORE RECREATIONAL AREA, NORTH CAROLINA. May 1961 225b Part II Chapter 5 Coats of hot crude oil(71) may be used to stabilize the face of a dune and cause the dune to enlarge in the direction of prevailing winds. Oiling is expensive and special equipment is necessary for highest efficiency. It is effective for only a few years and must be repeated frequently. This method increases the saltation coefficient of material flowing over the sur¬ face. As the sand blows up and over the top it is deposited on the slip face or leeward side in successive advancing fill increments. Enlargement of the dune continues until a streamlined body is built downwind from the oiled face. The entire mass is permanently stored for as long as the oiled surface re¬ mains intact or the dune is stabilized by vegetation. Figure 141 shows the successive steps of dune development behind an oiled surface, 5.42 DUNE STABILIZATION - After protective dunes have been formed they should be stabilized with vegetation (see Figure 142). This is expensive in the beginning but minimizes future difficulty. The most satisfactory plants are long-lived perennials, with extensive root systems that spread rapidly either vegetatively or by seed, or both, and maintain surface growth even though sand is accumulating around them to increasing depths. Such plants are not numerous, but in practically every section a few satisfactory ones are obtainable at reasonable cost. Vegetation in the form of single plants, groups of plants, and a sparse to dense cover of herbs, shrubs, and trees aids in development and stabiliza¬ tion of the coastline. The vegetation is secondary in the development of the dune topography, but it is most important in the stabilization and retention of the dunes after they have been developed. Plants are passive agents that alter wind action but they are also active agents in holding dunes in place. They also aid in the processes that change dune materials to soils. Natural processes would neither approach nor attain equilibriaa without plants to arrest wind, stop or retard shifting of materials, and add organic soil components. Plants influence the wind, erosion, and deposition by two major methods: their aerial parts act as obstructions to increase the roughness of the sur¬ face and deflect or screen wind movement; and the underground parts tend to bind and hold the materials in place. In addition, plants affect the water content of the soil and their decomposition adds humus to the soil. Groups of plants forming a vegetative cover may be sufficiently dense to alter wind action enough to prevent all but very minor erosion. In this manner the vegetation stabilizes the dunes against all but the most violent actions of wind or rain, 5.421 Types of Vegetation - The aggregation of plants in a coastal zone form, in general, three types of vegetative cover: (1) a pioneer type com¬ posed mostly of herbaceous plants; (2) a scrub type of woody shrubs and vines and dwarf form trees with a few associated herbs; and (3) a forest type dominated by trees. Many coasts have some succulent, semi-woody, and fibrous-leaved plants and palms in the first two types of vegetation. Nearly all of the plants of the first two types are tolerant to salt spray and various degrees of salinity of the soil. May 1961 225 c Part II Chapter 5 The three types of vegetation usually occur in narrow zones over the broad coast. The pioneer herbaceous vegetation is on the dune fields nearest the beach. The scrub vegetation, is found on the partly to fully stabilized older dunes, in some of the valley swales, and on some of the flats. The forest vegetation is the climax of vegetal development and is toward the in¬ terior on the oldest and most stable dune fields. It is not always present, but when present it covers the dunes, swales, and flats that have been sta¬ bilized for a long time and have soils with several distinctly different layers that were partly developed by the pioneer and scrub vegetation. Some coastal areas have only pioneer type of vegetation which is poorly developed and temporary, especially in the extremely cold and dry climates. Other areas have a scrub zone near the upper beach because erosion has re¬ moved the foredunes and pioneer vegetation. Many regions lack the forest zone, or the forest present is not a maritime forest but rather the seaward extension of an interior forest, 5,422 Climatic Regions of Dune Vegetation - The climatic regions in which distinctive types of vegetation and certain important plants occur are not the same as the coastal regions based upon physiographic, geographic, and geologic features. Plants and vegetation conform mainly to conditions of seasonal temperatures and rainfall or other precipitation, and the humidity or dryness of their coastal habitats. The climatic conditions of both temperature and rainfall during the growing season are most important along the northern and central parts of the Atlantic and Pacific coasts and around the shores of the Great Lakes, But in southern regions the higher winter temperatures and the drought con¬ ditions are usually more important. Winds, and the salt spray they often carry, are also important but these cannot be well classified on a regional basis. The incidence of storms, the prevalence of ice, snow and fog, and other atmospheric conditions are locally very important, but these cannot be used as general criteria for coastal regions. Therefore, on the general basis of temperature and precipitation the following eight climatic regions are recognized as regions of distinctive plants and vegetation. The regions are delineated as follows: a. North Atlantic Region , This region has a shore line about 570 miles long, extending along all the New England coast from the Canadian border to the border of Connecticut and New York, but not including Long Island, It has arctic and cold temperatures, frozen soils, ice, and snow over a long winter season and short, mild summers. Generally abundant precipitation and usually high humidity prevail, b. Central Atlantic Region , This region, with a shore length of about 520 miles, extending from the New York-Connecticut border and including Long Island to Cape Hatteras, North Carolina, has cold temperatures, some ice, snow and frozen soils but the winters are not severe. This area generally May 1961 225 d Part II Chapter 5 has abundant precipitation and humidity, c. Southern Atlantic Region . This region extends from Cape Hatteras, North Carolina to Cape Canaveral, Florida, a distance of about 650 miles* It has cool temperatures, rare frosts, and very little ice, snow, and frozen soils. The winters are mild and short, and there are long periods of summer heat. The region has abundant precipitation, is generally humid, but seasonally dry, d. Floridian Region . This region extends from Cape Canaveral, Florida around the southern Florida Keys area, and northward along the Gulf of Mexico to Cedar Key, Florida, a total length of about 750 miles, some sections of which are marshy shores without beaches. It has tropical, subtropical, and warm conditions, with very rare to no frosts and long warm to hot summers. Rainfall is irregular, varying from abundant to long droughts, with humidity varying from moist to dry, e. Gulf of Mexico Region , This subtropical region about 1,600 miles in length extends from Cedar Key, Florida northwest, then west and south to the Mexican border. About 350 miles of this coast, mostly in Louisiana, is marsh land without beaches. The region has warm temperature conditions with frosts occasionally in the northern areas. In southern Texas the winters are short; the summers are long, hot, and often quite dry. Rainfall is irregular, varying from frequent in Louisiana to rare in southern Texas, Humidity varies from moist to arid. f. Southern Pacific Region . The California coast from the Mexican border to the Monterey Peninsula, a length of 400 miles, is subtropical with warm temperate conditions. Frosts are rare and summers are long, hot and dry. Rainfall occurs most often in winter and is seldom abund¬ ant, The area is dry to very arid, g. Northern Pacific Region , From Monterey Peninsula, California to the Canadian border, a distance of 900 miles, the climate is warm to cool temperate with some frost, ice, and snow. Winters are generally long and mild. Precipitation varies from abundant to moderate and the humidity is generally high, especially in the northern part of the area, h. Great Lakes Region . This region, extending along some parts of the Great Lakes in the United States, has cold temperature conditions, with long cold winters, ice and snow, and cool to seasonally hot summers. Here precipitation is moderate and varied and humidity is moist. Most of the shores of the Pacific coast region are steep and there are less dune areas than along the Atlantic, Floridian, and Gulf of Mexico coasts. The Great Lakes region has very extensive shores, but those with dunes are mainly along the Lake Michigan shoreline of Michigan and Indiana, Dunes are present along only about 500 miles of the shore line of the Great Lakes in the United States, May 1961 225e Part II Chapter 5 5.423 Available Plants in the Different Climatic Regions - Natural vege¬ tation in each of the eight regions is listed in Table 13, Many of the plants are still available for transplanting for dune stabilization, particu¬ larly for restoration of the native vegetation in the scrub and forest zones. Reforestation of well-stabilized areas is usually successful if the native species are used. The stock for plantings can be obtained from the surviving forests by obtaining small trees, seedlings, and seeds. Many trees and shrubs can be raised in nurseries, or obtained from commercial sources and from Federal and State agencies. The abundant and dominant species in the scrub and forest zones are generally the most available and are usually the best for permanent stabilization of the inland areas. In contrast, the pioneer type plants suitable for dune formation and the first stages of stabilization of the dune fields are not often abundant in the present general vegetation because certain climatic regions lack plants with good dune-building and dune-stabilizing characteristics. Such areas occur along the dry coasts of Texas and California and parts of Florida, Few species occur in some of these areas and may not grow robustly. The use of plants imported from other regions is then advisable, especially if the exotic plants grow where the soil and climate are similar to those of the region where they are to be used. Many of the dune areas of most of the three Atlantic coastal regions, the humid parts of the Gulf coast, the Northern Pacific coast, and the Great Lakes shores have numerous grass and shrub species that are good dune builders and stabilizers. Nearly all of them have some favorable characteristic for particular sites, and some can be transplanted successfully if properly dug, handled, planted, and maintained, 5.424 Methods of Establishing Plantings - The effectiveness of establishing plantings for the various purposes of dune building and stabilization usually depends upon the following: (1) the selection and procurement of suitable plants which may be obtained without too much difficulty or expense; (2) the arrangement of the plantings which will aid and maintain the development of suitable dunes; (3) the methods of planting and care of the plantings which will insure their establishment; and (4) the replenishment of the plantings to promote the final type of vegetation and forms of dunes desired. Three of the phases involved in planting should be planned in conjunction with the overall engineering plans for the coastal region, especially if these plans involve creating or raising dune ridges, artificial nourishment of the beach, and structures on the upper beach or in the dune fields. Vegetation is often useful in conjunction with wind screens, such as sand fences. The protection of valuable property from migration of sand dunes may be accomplished in this manner. An arrangement that will decrease wind erosion and maintain the stability of dune formations is desirable. The design of plantings should be related, as much as possible, to the final stabilization of the dunes and the other features present, so that the final forms will hold against all but the most violent storms and most persistent erosion. May 1961 225f TABLE 13 PLA^f^S OP THE BIGHT COASTAL RBGIOrC The foilwring lltt of plints Is not intended to be t complete list of all plants found along the coasts of the United States. The Intent, rather. Is to Hat all plants which are either maerlcally iaportant or which are greatly involved In coastal changes and stability. Many of the species listed are exotic and nay become important after trial in a coastal region. All species of some genera are not listed because the genera have so many species which are appropriate. Information should be obtained from local horticulturists and botanists to determine those species best suited to a specific area. The table is an alphabetical list of the species by scientific name. The local or c<»on names are included for convenience. All eight climatic regions are not equally represented, however, because more species grow in the warn moist climates and because plants used along the Pacific coast are not as well known to the writer. The remainder of the table indicates the proper region and zone for each species. SYMBOLS AND ABBRgVIATIONS Climatic Regions NA - North Atlantic CA • Central Atlantic SA • South Atlantic P - Ploridian G - Gulf of Mexico SP - South Pacific MP - North Pacific GL o Great Lakes A - Species present in given zone and region and recommended for use in that zone and region. B • Exotic species which is reconaended for use in a given zone and region. Nos, 1 through 4 indicate prevalence of a species in a given zone and coastal region such that 1 is the most abundant or dominant species present in that zone and region. * - Recmmended for use on experimental basis. Zone and region of use not determined at present. SCIENTIFIC NAME COMMON NAME PIONEER ZONE SCRUB ZONE FOREST ZONE NA CA SA F G SP NP OL NA CA SA F G SP NP gl' NA CA SA F G SP NP GL Abronia Latifolia yellow sand verbena A-1 Abronla maritima red sand verbena A-1 Acacia latifolia wattle B Adenostema fasciculatum chamlse B Agave americana century plant B B B B B B Agave spp. sisal century plant B B B B B Agropyroa dasystscbyui hairy wild wheat s Ar-2 B Agreorren repeos witch grass-couch grass B Agropyron spp. wild wheats B B B Alnus glutinosus alder B Aloe arborescens bush aloe B B Aonophila arenaria marram grass B Amnephila brevlligulata beach grass A-1 A-1 B B A-1 Arbutus menziesi arbutus A-1 Arctostaphylus manzanita manzanita B Arctostaphylus uva ursi madrona A-2 B A-2 Arenaria peploides Sea sandwort B ' 1 Arenaria spp. sandworts B B B i Artemisia caudata wormwood B 1 1 Artmesia spp. wormwood B B 1 Baccharis hallmifolia sea*mivrtle. fingerlinc I A-4 A-3 Baccharls pilularis fingerling B iH . , 1 : 1 Bambusa spp. bamboos ^ B ' B ! B ^ Buaelia angustifolia buckthorn 1 ' 1 B B j i 1 Cakile endentula sea rocket A-3 B A-2 • 1 [ Calaaovilfa longifolia dune grass A-3 1 1 1 Calochortus venustus butterfly msriposa A-2 ! I Carex kobomugi Japanese sedge B i i 1 1 I Ceanothus dentatus dwarf ceanothus 1 1 1 A-1 i Cocos nucifera coconut palm B 1 1 ' B _J i * Extracted from Reference 161 May 1961 225g SCIENTIFIC NAME COMMON NAME PIONEER ZONE SCRUB ZONE FOREST ZONE NA CA SA F G SP NP GL NA CA SA F G SP NP GL NA CA SA F G SP NP GL B Cupressus spp. cypresses B B Cynodon dactylon Bermuda grass B Cystisus spp. brooms B B B Dalbergia brownel rosewood B Digitarla filifomis finger grass B B Bleagnus angustifolia B Bleagnus argentata B B B B , — Blyaus areoarias sea lyuegrass B 1 Bncella eallfornica bush Sunflower B Brlca spp. heaths B B B B ■ — Bryngiun maritlnum sea holly 6 B Eucalyptus spp. B B B Eucalyptus umbellata horn cap B B Bugenia buxlfolia box leaf stopper B Bupborbia buxlfolia shrub spurge B _ Patsla japonlca aratla ■ B B Pestuca axenanla sand fescue B Pestuca rubra red fescue B i Porcstlera porulosa Florida privet B 1 Pragaria chlloensls sand strawberry B .-3 Gaultherla shallon salal LI 1 B Holcus lanatus velvet grass B B 1 i i 1 1 Hydrangia aboresccns shrub hydrangia 1 B i ‘ ! 1 1 Kydrocotyle bonariensls seaside pennywort B i , ' i ^ 1 Iva ijnbrlcata sarch*elder A.4 A-3 A-3 1 1 ! i 1 Juniperus canara juniper ! i Junlperus conferta shore juniper 1 B B ^ ! Jualperus procumbens creeping juniper 1 B ' i Junlperus vlrginiana red cedar 1 1 A-4 Kalanchet spp. bryophyllun i B B B 1 Lantana camara cannon lantana i B B B Larlx spp. larches 1 B B B Larrea spp. creosotebushes 1 B Lathyrus llttoralls beach pea B Leptosperaian scopar lun tea tree i B B B Llgustrum luciduo glossy privet 1 6 B B B B Llgustrm ovallfoliuB privet B Llflonlun carollnianum sea lavender B B Lonlcera Involucrita bearberry B B Lonlcera morrowl honeysuckle B B Melaleuca spp. melaleuca B B Metroslderos spp. Iron trees B B Myrlca cerlfera wax^eayrtle A-l A-4 A-2 Myrlca gale sweet gale B 1 Myrlca pennsylvanlca bayberry A-1 A-l 1 Neriuffl oleander oleander B B B B Nollna tuberculata nollna 'b B B Oenothera hunlfusa evening prinrose B O^aanthus anericana wild olive B Pandanus baptitsi screwpine B B Pandanus spp. screwplnes B Paniciao anarulun coastal panic grass A-4 Panlcun anarian dune panic grass A-2 A-2 A-2 Paspalum vaglnatum salt joint-grass B PhormiiBD tenax New Zealand flax B B Plcea sltchensis sltka spruce I A-4 Pious clausa sand pine A-2 Pious cootorta lodgepole pine 1 _ A-3 May 1961 225h SCIENTIFIC NAME COMMON NAME PIONEER ZONE SCRUB ZONE FOREST ZONE NA CA SA F 6 SP NP 6L NA 1 CA SA F 6 SP NP GL NA CA SA F G SP NP GL Pinus elliotti stash pine . B A-l Pinus pinaster cluster pine B B B Pinus radiata Monterey pine r ■ 1_ A^t Pinus riglda pitch pine , B Pithecellobium spp. B Pittosporum crassifoliun karo _ B Poa coopressa Canada bluegrass B I 8 Poa macrantha seashore bluegrass B B Poinsiana gillies bird of Paradise shrub 1 1 B Populus spp. aspens and poplars B B B B Portulacaria afra speckboom B Pninus serrulate flowering cherry B Prunus naritima beach plum B B A-2 A-3 Pmnus serotina black cherry B B A-3 Prunus virginiana choke cherry B B Pteridium spp. brachen fern B B (“ Pyrecantha spp. firethorns B B B B B B B Pyrus arbutlfolia red chokeberry B Rbamnus californica coffee berry 1 A-2 B Rosa rugosa rugose rose 1 B 1 1 Rosa spp. roses 1 B B 1 B B 1 Sabal palmetto cabbage palm 1 ^ A-4'A-2 |A-2 1 A-l Saliz cordata busby willow A-4 1 B ' 1 ! ' Sallx discolor pussywillow A-3 1 B ! 1 1 ! Sallx hookeriana coast willow 1 i i 1 A-l Saliz pentandra bay-leaved willow B B ' B 1 1 1 Sallx syrticola dune willow ! '' A-3 Salsola kali saltwort A-3 1 1 1 Sanseverla spp. B B i i ' 1 Schinus terebinthefelius peppertree B ' j Serenoa repens saw palmetto 1 !a-i 1 1 Sesuviuo portulacastciD dune sea-purslaoe B A-2 1 B B B 1 Severlnla buxifolia box orange 1 B 8 1 Solanum dulcamara bitter nightshade 1 B B Spartina patens salt-neadow grass A-4 A-3 A-4 Spartina towaendi cord grass B B 1 1 StenophoruB secundatum St. Augustine grass B B B 1 Sueda linearis sea-bllte B B B Tamarlx spp. tamarlx B B B B B Tanasetum canpboratun dune tansy A-4 Teminalia spp. terminalias B Trifoliun procumbens hop clover B B Trifolitan spp* clovers B “ i Ulex europeans gorse 1 8 B Uniola paniculata sea oats A-2 A-1 A-l A-l i Washingtonia filifera fan palm i B B Wedelia trilobate wedgegrass B B B -1- Yucca filanentosa Adam* 9 needle B B B Yucca spp. yuccas A-4 B B B Zoysia tenuifolia mascarin grass B B Mesembryanthemum crystallinvm ice plant B - 1 May 1961 225 i Part II Chapter 5 5.5 GROINS 5.51 GENERAL - In the past, groins generally have been built, not designed. As a result, many have been too light to withstand the wave forces and have failed. Others have been overbuilt to such an extent as to be uneconomical. Because of the many unknowns involved, this section on design is an attempt at standardization on a safe basis rather than any presentation of an exact design analysis. As described in section 3.45, based on functional considerations, groins are classified principally as to permeability and height. Groins built of common construction materials can be made permeable or impermeable, high or low profile. The materials commonly used are concrete, steel, stone and timber. Asphalt also has been used to a limited extent for groin construc¬ tion. 5.52 CDNCRETE GROINS 5.521 Concrete Permeable Groins - A part of the theory of permeable groins is that the permeability may be varied in order to provide uniform shore alignment and permit passage of some of the drift. However, as discussed in paragraph 3.451, permeable groins are unlikely to prove as satisfactory as properly designed impermeable groins. Of the numerous types of patented permeable concrete groins, that patented by Sydney M. Wood probably has been the most widely used. Accordingly, it has been taken as typical of this type of groin. The typical section of these groins consists of pre¬ cast concrete beams laced on master piles. The piles contribute materially to the stability of the groin and may be used to support erecting equipment, 5.522 Concrete Impermeable Groins - Impermeable concrete groins may be either articulated or solid throughout their entire length. In general, the articulated type appears to be the most practicable as it will permit unequal settlement. Also, being handled in Smaller units, it is simpler to construct as costly forming is avoided. Figures 143, 144, and 145 show three types of impermeable concrete groins. Figure 143 shows a combination May 1961 225 j May 1961 226 227 Part II Chapter 5 concrete and stone semi-articulated type. Figure 144 shows an all concrete articulated type. Figure 145 shows an all concrete type originally articulated and later solidified with a concrete cap. In Fi.gure 143, the precast reinforced concrete groin is constructed of vertical transverse blocks shaped like a capital "I", 4 feet, 8 inches wide and 6 feet long. The height of the unit may vary with location. Horizontal panels 2 feet high, 8 inches thick, and 10 feet long, with notches on either end, are placed in between the transverse blocks. Rock fill is then placed in between the horizontal panels and covered with a 6-inch concrete slab cast in place. The concrete slab can be used to tie the entire structure into a single unit. In Figure 144, the groin is composed of concrete precast trapezoidal blocks, each 3 feet 3 inches high and weighing 5 tons. Two types of blocks are used. Type A, shown in Figure 144 has a length of 5 feet, a top width of 2 feet and bottom width of 6 feet. Type B, not shown but similar to type A, has a length of 3 feet 10 inches, a top width of 3 feet 2 inches, and a bottom width of 7 feet 2 incheso The block at the offshore end is the same as type A but with a sloping offshore face. The steel joints and reinforcing, shown in Figure 144, are the same for both type A and type B, A patent on the joint is held by Harrison Weber, The reinforcing bars are welded to the I-beam and the channels on opposite ends of the block, and are placed as one unit in the form for casting the concrete block. The blocks are placed on the beach surface, the flexible nature of the joints enabling the blocks to conform with the ground slope and to iindergo a measure of settlemento The height of groin can be increased by maintain¬ ing the relationship between height and base width. As some settlement occurred where this type of groin was originally placed on a sandy beach a mat of crushed stone was used as a foundation in later Installations, Figure 145 shows another type of concrete block groin which is simple to cast and place, and which has served satisfactorily in some locations. The height of groin can be changed by changing the length and base width of the concrete blocks. Concrete channel-ways are cast in the block as a guide in placing and to tie the blocks together. No special type of end block is required. After the blocks have settled for a period of time, the concrete cap can be poTired, This cap will increase the height of the groin somewhat, and tie component parts together into a monolithic groin, 5.53 STEEL SHEET PILE GROINS - Satisfactory steel sheet pile groins have been constructed with straight web, arch,web, M or Z sections. Some have b:^en made permeable by the cutting of openings in the piles. All of these are made with interlocking joints that do not pull apart when subjected to wave forces and provide positive sand-tight connections between adjacent piles. The type of pile selected is governed by the wave action to be encountered. The section modulus increases as the depth of the arch increases in the arch web, M, and Z sections. The steel sheet pile groin is often constructed with horizontal timber or steel wales along the top 228 PLAN rrin I ^TIMBER BLOCK o G.I.BOLT WATER LEVEL DATUM SECTION A-A NOTE = Dimensions and details to be determined by particular site conditions. TYPICAL STEEL SHEET PILE GROIN FIGURE 146 February 1957 229 Part II Chapter 5 of the steel pile and in some cases, vertical round timber piles or brace piles are bolted to the outside of the wales for added support. The round piles are not always required with the M and Z sections but would ordinarily be used with the flat or arch web sections. A typical design for a steel sheet pile groin is shown in Figure I 46 . The round pile and timbers should be creosoted to maximum treatment for use in waters infested with marine borers. This would not be as necessary in the Great Lakes. In some instances the life of the steel sheet pile has been indefinitely pro¬ longed by pouring concrete slabs on either side of the sheet pile after holes have been scoured through the piling by moving sand. The cellular type of steel sfieet-pile groin is being used more exten¬ sively, especially on the Great Lakes where foundation conditions are a problem and adequate pile penetration cannot be obtained. A typical cellular type groin is shown in Figure 147. This groin is comprised of cells of varying sizes, each consisting of semicircular walls connected by cross diaphragms. Each cell is filled with sand or stone to increase stability. A concrete slab may or may not be poured over the top to provide a walkway or platform and to retain the fill material. 5.54 STONE GROINS - Stone groins have been successfully constructed of either rubble or of cut stone blocks. Either type can be made permeable or impermeable. The impermeable rubble-stone groin is constructed with a core of quarry run stone including sufficient fine material to make it sand tight, and with caps of stone sufficiently heavy to protect the structure from anticipated wave damage. The random or rubble-stone mound is usually constructed with 1 on 1 l/2 side and end slopes and a top width of 5 feet or more. A typical stone groin is shown in Figure 148. The size of stone used in the core may vary depending upon the source. The layer of cover stone should be a minimum of 3 feet thick with individual stones weighing from 1 to 4 tons or more depending on the wave action to be resisted; averaging approximately 3 tons. Several variations to the all-stone groin have been used. In some instances, to increase the impermeability of the structure, a diaphragm of timber Wakefield piling or steel sheet piling has been included. In other cases stone filled timber crib groins nave been built especially on the Great Lakes. Except for details these are similar to the crib breakwaters illustrated in Figure 130 and 131. Also in many instances the shore end of the structure has be^n constructed either of timber or steel sheet piling, thus reducing the overall cost of the structure without reducing its economic life or its ability to resist severe wave attack or erosion. Generally the timber or steel section does not extend seaward of the crest of berm. Where it is not exposed to the action of marine borers, untreated timber may give a considerable length of service. One satisfactory method of sealing an all-stone groin to make it impermeable is to fill the voids between the stones with concrete grout. This also increases the stability of the structure to resist wave action. The grout should be placed over the entire exposed surface of the groin 230 February 1957 231 Part II Chapter 5 and forced into the voids to the sand line. A rich mix should be used (7 sacks of cement to the cubic yard). If exposed to wave action while setting, an admixture of calcium chloride may be added to expedite the seto Where building sand and fresh water are not available, beach sand and sea water have been used satisfactorily. Permeable stone groins differ from impermeable stone groins in several ways. The permeable groin is constimeted with core stone large enough to keep it from being sand and water tight. The side slopes are usually 1 on 1 1 / 2 , but the top width is about 3 feet, or the minimum required for stability. The section would be similar to that shown in Figure I 48 . Stone block groins are usually more or less permeable although they have been made impermeable by placing a steel sheet pile diaphragm down the center of the stone blocks. The individual blocks range in weight from 2 to 6 tons but must be of sufficient size to withstand warve action. The height of groin along the profile is easily varied by adding another row of the stone blocks. Typical sections are shown in Figure 14-9. 5.55 TIMBER GROINS - The most common type of timber groin is an impermeable structure composed of sheet piling supported by wales and round piles. Some permeable timber groins have been built, being made permeable by leaving spaces between the sheeting. All timbers and piles should be given February 1957 232 May 1961 233 Part II Chapter 5 the maximum recommended pressure treatment of coal tar creosote. All holes should preferably be drilled before treatment, A typical timber groin is shown in Figure 150. The round timber piles forming the primary support of the groin should be a minimum of 12 inches in diameter at the butt. Stringers or wales, which are bolted to the piling horizontally, should be at least 8 inches by 10 inches, preferably cut and drilled before creosoting. The sheet piling is usually either of the Wakefield or the splined type, supported between the wales in a vertical position and secured to the wales with bolts. The plane of the sheeting is vertical. Although usually verti¬ cal the piling may be driven at an angle in this plane, 5,56 ASPHALT GROINS - A number of attempts have and are being made to use asphaltic material for groin construction. Although there have been some notable instances where the asphalt in shore structures has served the de¬ sired purpose reasonably well, experience for groin construction indicates limited success. Experimentation in the United States with asphalt groins began at Wrightsville Beach, N, C, The first groin was built in 1948 with five ad¬ ditional groins in 1949, The second experimental groins were at Fernandina Beach, Fla,, where eight groins were built in November and December 1953, The third and most extensive installation is the asphalt groin field at Ocean City, Md, During the period August to December 1954, thirty-three groins were constructed, and the following summer (1955) ten additional groins were built. These groins were designed to protect approximately 7 miles of shore line between Ocean City and the Maryland-Delaware State line. The fourth installation of this type of groin was located near Nags Head, N, C, In the spring of 1956 three groins and connecting seawall were con¬ structed of asphaltic material to protect a section of highway in a critical eroding area. The latest installation consists of three experimental as¬ phalt groins at Harvey Cedars, Long Beach Island, N, J,, built in December 1958 and May 1959, In general, the groins at Fernandina Beach, Fla,, Ocean City, Md,, and Nags Head, N, C, were patterned after the asphalt groins constructed at Wrightsville Beach, N, C, The structures consist of an apron or foundation and a crown or mound, and extend from the mean low water line to the dunes. The apron varied in width from 20 to 46 feet and in thickness from 2 inches to 12 inches. The crown, centered on the apron, was built with a general trapezoidal cross section 3 to 4 feet high and 6 to 8 feet wide. At Harvey Cedars the cross section of the asphalt groins was modified by maintaining a 20-foot apron width and widening the crown to 10 feet. The groins at the first two locations, Wrightsville Beach, N, C, and Fernandina, Fla,, had creosoted timber piles 8 to 10 feet long driven at 8 foot centers along the centerline extending about 4 feet above the foundation. Groins at the other three locations had no center piles. The profiles of the groins in general were built to the beach profile with about 50 percent of the groin height buried below the average grade of the beach. Dunes extended generally from the dune line approximately to the low water line. May 1961 234 Part II Chapter 5 The behavior of sand-asphalt groins of the type used to date demon¬ strates definite limitations of effectiveness, partly due to inability to extend the structures beyond the low water line, and early failure seaward of the beach berm crest. This is due to normal seasonal variability of the shore face and consequent undermining of the structure fotndation. Modi¬ fication of the design as to mix, dimensions, and sequence of construction may reveal a different behavior. Use of asphaltic material for groin con¬ struction should be given consideration only if foundation stability can be assured. 5,57 SELECTION OF TYPE - Every beach has its inherent physical character¬ istics, These include the range of tide, the rate, direction and amount of littoral transport, beach composition, wave characteristics, tidal currents, marine borers, beach dimensions and slope, and seasonal variations in the beach profile. Other factors to be considered include beach and upland topography, the character of the foundation, the availability and cost of construction materials, maintenance costs, economic life, and the use of upland areas. After planning considerations have indicated the use of groins is practicable, the selection of groin type is affected in overlapping and varying degrees by the foregoing inter-related factors. No universal plan of protection can be prescribed because of the wide variation in conditions at each location. However, the information required in a problem and the general factors to be considered to determine the groin type are generally the same and depend on the location and characteristscs of the area for detail. A thorough consideration of foundation materials is essential to the selection of groin type. Borings and probings should be taken to determine the subsurface conditions for penetration 6f piling. Where the foundations are poor or where little penetration is possible, considera¬ tion should be given to employing a gravity type structure such as cellular steel sheet pile, rubble-mound, stone block, or pre-cast concrete unit type groin. When good penetration is indicated a cantilever type structure such as concrete, timber, or steel sheet piles should be considered. Although the foundation for the structure is an important factor, it shouldn’t over¬ balance other factors in the selection of types. Availability of materials affects the selection of the type of groin because of the economic aspects. It may happen that the material, which would normally be the most economic with full consideration given to life of the material and maintenance costs, is not available except at a cost that would make some other material or type of construction more economical. This involves the question of the economic life of the material together with the annual cost of maintenance to attain that economic life. The first costs of timber groins and of steel sheet pile groins, in that order, are often less than for other types of construction. The concrete groin is considerably more expensive, but often costs less than does the massive stone groin. However, concrete and stone groins require less maintenance and have much longer economic life than do the timber or steel sheet pile groins. These factors, the amount of funds available for initial construction, the May 1961 235 Part II Chapter 5 annual charges, and the period during which protection will be required must all be studied before deciding on a particular type, 5.58 DESIGN 5,581 Concrete Block Groins - a. Maximum Forces - The forces acting on the structure would be either earth pressures or wave forces or combination thereof. No combination of these forces would exceed the maximum that would occur by application of either singly. The maximum earth pressure would occur with the groin full on one side and unsupported on the other. The maximum wave pressure would occur at the seaward end of the groin where there would be no supporting fill on either side, and where the groin would be sub¬ ject to attack by the largest wave. The end block would be subject to the full impact of the waves. Landward blocks would be subject to reduced wave forces by reason of the angle of wave approach. b. Design Problem - A concrete block groin of the type shown on Figure 144 is to be built in an area in which the beach slope is 1 on 20, and the design wave period is 10 seconds. The sand composing the beach has a unit weight, w, (moist condition) of 110 pounds per cubic foot, an internal friction angle, 0, of 25®, and a friction angle with concrete 01, of 22®, The block to be used has, as the angle © between its face and the horizontal plane, 110®, and a height of 5 feet, (See section 4,3 Earth Forces.) £, Earth Pressures - Maximum earth pressures would occur with one side of the groin full and the other empty. For 0 = 25®, 0 = 110®, 0i = 22°, h s 5 feet and the angle between the surface of the accretion and the hori¬ zontal plane being p = 0®, the total force in pounds P due to the updrift accretion is given by P = 1/2 (59) where N sin (0 / 0j) sin (0 - p) ( 60 ) sin (01 / 0) sin (0 - p) by substitution N s ,731 X ,423 ,743 X .940 and = 750 pounds The maximum moment of horizontal pressure would be: Mh * 750 X 5/3 « 1250 foot-pounds May 1961 236 Part II Chapter 5 If the vertical component of earth pressure is neglected, which may be done unless it has a large effect on stability, the vertical load of the wall passes through the centerline of the base, and the above moment becomes the moment around the centerline of the base. Assuming a block with a 7-foot base, a 3-foot top width, and 5 feet high, its weight per linear foot in air would be 5 C7 3) X 150 = 3,750 pounds. 2 With this block, the stress in the base would be f s P + 6M s 3750 ± 6(1250) s 536 ± 153 (63) A ^2 1x7 (7)2 where P * the sum of vertical loads A s the base area (unit length of groin) M = the total moment about the base centerline bd^s the section modulus of the base (unit length of groin) ss 536 ♦ 153 a 689 pounds per square foot a 536 - 153 a 383 pounds per square foot These pressures are satisfactory both for concrete and for the foundations. To check the possibility of sliding: P ^ 750 W 3750 0,20 The coefficient of friction for masonry on sand equals 0,40, and therefore this value (0,20) is considered safe. d. Wave Pressures - It is assumed that the maximum wave pressures occur on the block groin when the crest of the breaking wave is level with the top of the groin,therefore, the depth of water plus ^ the wave height must equal 5 feet (height of block), Hb/2 ♦ di, = 5 ft, and d^, = 1.28 then ^b ♦ 1,28 H a 5; so = 2,8 ft, and d^, « 3,6 ft, 2 Using a 10-second wave and a beach slope of 1 on 20 dAo « 3,6/512 (0.00704) and d/L a 0.034 May 1961 237 Part II Chapter 5 L D D L Therefore d/„034 - 3 . 7/0 034 - 109 feet 3.7 / IP = 9.2 feet %2 512 = .018 and .055 Ljj = 9. 2/.055 - 167 feeto The dynamic pressure concentrated at still water level if the groin were normal to the direction of wave approach would be determined from equation 27 or p TTg 2.8 X 64.2 3.7 (9.2 / 3.7) m = 167 9.2 Pjjj = 109 X 5.2 = 567 pounds per square foot, and the static pressure is (Equation 30) ^ wH 64.2 X 2.8 . Pg = M —2 - = 90 pounds per square foot. These pressures would be applied in a manner similar to that shown in Figure 151. The resultant wave thrust if the groin were normal to wave approach, by equation 31, would be R = (567 X / 90 (3.7 / ) m j 4(. - 530 / 396 SE. 926 pounds As the dynamic pressure acts at an angle ( a ) with the structure, the total force normal to the structure per unit length of structure would be, with a *30 degrees, (see Figure 152) R = 530 sin^a / 396 n = 133 / 396 = 529 pounds per foot of groin. The moments around the base of the block by equation 32 would be (with sin'^a as a factor in the first term) M = 133 X 3.7 / 2 ^ 4 ^ M - 492 / 616 / 263 = 1371 foot-pounds 238 I = Intercept on groin of wove of width R I = R/sin«^ Rx = Component of dynamic wove force (R) normal to groin. Rn = Component of wave force normal to groin per unit length of groin. Rx = R since Rn = Rx/I Rp = R since sintK=Rsin^cx width of wave approach Note: Not applicable to rubble structures. FIGURE 152- ANGLE OF WAVE APPROACH ( Ref.88'\ ^ Molifor ) May 1961 239 Part II Chapter 5 then f = 12^ ± 6 X 1371 ^ 536 ± 168 7 49 = 704 lbs. per sq, ft, or 368 lbs, per sq. ft, and ^ _ 529 „ q W 3750 The assumed design is also safe for wave forces. 5,582. Rubble-Mound Groins - Assume a rubble-groin to end in 4 feet of water with a 5-foot tide. From the wave studies, the wave with the greatest energy is a 10-second wave with a wave height of 10 feet in deep water. Maximum Design Wave Height - The highest wave that can strike the seaward end of the groin is approximately H s 4 (depth of water) ♦ 5 (tide) _ 7 1.3 Any larger waves will break before reaching the structure and only a reformed wave or wave uprush will be propagated forward. Inspection of the orthogonal pattern indicates little divergence so that 7-foot waves at the structure can occur if the deep water design wave is 10 feet, b. Design - The rubble groin is designed without regard to differ¬ ential ground lines except to be certain that the shore end is carried far enough inshore to insure that the structure will not be flanked. Should one side scour, the stones in the groin will settle and readjust themselves. Should settlement prove excessive, some maintenance may be required. In some instances the width of the groin may be determined by construction methods, and in other instances may be determined by the size of capstone required, c. Stone Weight - Assumed in this case, that (1) the unit weight of available stone is 160 Ibs/ft^; (2) Minor overtopping is allowable; (3) Width 2 layers of rough, angular quarrystone and breaking wave condition, Ko = 2,8 trunk of groin and Kp s 2,7 head of groin (Table 9A), Using Plate D-7A, Appendix D, the following table can be made for H = 7 feet. Slope_WxKp_Trunk (Kf, = 2,8)_Head (Kj^ = 2,7) 1 on 1.5 1.1 X 10^ 3930 lbs 4070 lbs 1 on 2 8.2 X 10^ 2930 3040 1 on 3 5.4 X 10^ 1930 2000 The stone determined for a 1 on 1.5 slope would seem reasonable to obtain from most quarries. However, if stone as large as 2 tons are not available, a flatter slope should be used. The underlayer of W/10 or 400-lbs. rubble should be used between the quarry-run (sand tight) core material and the armor stone, A typical groin of this type is shown in Figure 148. 5.583 Vertical Sheet Pile Groins - This type of groin may be construct¬ ed of timber, concrete, or steel, depending on life expectancy, cost. May 1961 240 Part II Chapter 5 availability of materials, etc. The design of a groin of this type for wave forces is generally the same as that for a concrete block groin; for earth loading, the design follows that for a sheet pile bulkhead. Piling sections designed in this manner may be reduced somewhat if the customary practice is followed of using wales at the top of the sheet pile with secondary support furnished by round timber or other piling driven outside the wales. 5c6 MISCELLANEOUS DESIGN PRACTICES The more important lessons learned from experience on the deterioration of concrete and steel and timber waterfront structures may be summarized as follows; a - The elimination of bracing within the tidal zone to the maximum practicable extent is desirable, since maximum deterioration occurs in that zone; b - Round members, because of their smaller area and better flow characteristics for wave action, generally have a longer life than other shapes; _c - It is imperative that all steel or concrete deck framing be located above normal spray level; d - Untreated timber piles should never be used in waterfront structures unless located below the permanent wet line and protected from marine borer attack; e - The most effective injected preservative appears to be creosote oil having a high phenolic content. For piles subject to marine borer attack a maximum penetration of creosote-coal tar solution is recommended; f - Salt-treated timber gives satisfactory service when protected from the weather; £ - Boring and cutting of piles after treatment should be avoided, and where unavoidablejcut surfaces require field treatment; h - Single timber caps have a longer life than pairs of cap timbers dapped into the piles; i - Untreated timber piles when encased in a gunite armor and properly sealed at the top will give economical service; j, - Concrete to last in the tidal zone must have a high cement content; a minimimi of 6-1/2 bags per cubic yard is recommended; k - The lower the water-cement ratio, the more durable concrete will be in salt water; 241 Part II Chapter 5 1 - Care must be exercised in the selection of coarse and fine aggregates both for density of grading and to avoid unfavorable chemical reaction with the cement, m - Maintenance of specified clear cover over all reinforcing steel is of the greatest importance; n - Smooth formwork and rounded corners improve the durability of con¬ crete structures,, o - All steelwork in and above the tidal range will last longer if protectedo A good method is to provide a concrete envelope. 24 2 APPENDIX A GLOSSARY OF TERMS This glossarj has been compiled and reviewed tsj the steiff of the Beach Elrosion Board, Terms from the following publications were included: (1) Waves, Tides, and Beaches: Glossary of Terms and List of Standard Symbols, R. L. Wiegel, University of California. (2) The hydrographic Manual, K, T, Adams, U, S, Coast and Geodetic Survey, ( 3 ) Webster *s Unabridged Dictionary, 2nd Edition, ( 4 ) Tides and Currents Glossary, U, S. Coast and Geodetic Survey, ( 5 ) U. S. Naval Photographic Interpretation Center Glossary. A ,1 • .-v/ v.nj ■i . . ! “ */>tf "^l-i •, fi't'* I. -.1 ,•> OUT.. . "X • f d 4 *« '..*<0 •fr % . a'; If ^ ■ V APPENDIX A GLOSSARY OP TERMS ACCRETION - May be either NATURAL or ARTIFICIAL. Natural accretion is the gradual build-up of land over a long period of time solely by the action of the forces of nature, on a BEACH by deposition of water- or air-borne material. Artificial accretion is a similar build-up of land by reason of an act of man, such as the accretion formed by a groin, breakwater, or beach fill deposited by mechani¬ cal means. Also AGO^ADATION. ADVANCE (OF A BEACH) - (1) a continuing seaward movement of the shore line; (2) a net seaward movement of the shoreline over a specified time. Also PROGRESSION. A3E, WAVE - The ratio of wave velocity to wind velocity (in wave fore¬ casting theory), ALLUVIUM - Soil (sand, mud, or similar detrital material) deposited by flowing water, or the deposits formed thereby, ALONGSHORE - Same as LONGSHORE. AMPLITUDE, WAVE - (1) in hydrodynamics, one-half the wave height; (2) in engineering usage, loosely, the wave height from crest to trough, ANTINODE - See LOOP. ARTIFICIAL NOURISHMENT - The process of replenishing a beach by artificial means, e.g, by the deposition of dredged materials, ATOLL - A ring-like "coral" island or islands encircling or nearly encir¬ cling a lagoon. It should be noted that the term "coral" island for most of these tropical islands is incorrect as calcareous algae (Lithothamnion) often forms much more than 50% of them. ATOLL REEF - A ring-shaped, coral reef, often carrying low sand islands, enclosing a body of water. AWASH - (1) (Nautical) Condition of an object which is nearly flush with the water level; (2) (Common usage) Condition of being tossed about or washe-^ by waves or tide. BACKBEACH - See BACKSHOIE BA2KRUSH - The seaward return of the water following the uprush of the waves. For any given tide stage the point of farthest return sea¬ ward of the backrush is known as the LIMIT of BACKRUSH or LIMIT of BACKWASH. (See Figure A-2) A-1 Moy 1961 BACKSHORE - That zone of the shore or beach lying between the foreshore and the coast line and acted upon by waves only during severe storms, especially when combined with exceptionally high water. • Also BACKBEACHo (See Figure A-l) BACKWASH - (l) See BAGKRUSH; (2) Water or waves thrown back by an obstruction such as a ship, breakwater, cliff, etc, BANK - (l) The rising ground bordering a lake, river, or sea, on a river designated as right or left as it would appear facing downstream; ( 2 ) An elevation of the sea floor of large area, surrounded by deeper water, but safe for surface navigation; a submerged plateau or shelf, a shoal, or shallow. BAR - An offshore ridge or mound of sand, gravel, or other unconsolidated fnaterial submerged at least at high tide, especially at the mouth of a river or estuary, or lying a short distance from and usually parallel to, the beach. (See Figure A-2 and A-9) BAR, BAYMDUTH - A bar extending partially or entirely across the mouth of a bay. (See Figure A-9) BAR, CUSPATE - A crescent shaped bar uniting with shore at each end, It may be formed by a single spit growing from shore turning back to again meet the shore, or by two spits growing from shore uniting to form a bar of sharply cuspate form. (See Figure A-9) BARRIER BEACH - A bar essentially parallel to the shore, the crest of which is above high water. Also OFFSHORE BARRIER. (See Figure A-9) BARRIER REEF - A reef which roughly parallels land but is some distance offshore, with deeper water intervening. BASIN, BOAT - A naturally or artificially enclosed or nearly enclosed body of water where small craft may lie. BAY - A recess in the shore or an inlet of a sea or lake between two capes or headlands, not as large as a gulf but larger than a cove. See also BIGHT, EMBAYMENT. (See Figure A-9) BAYMOUTH BAR - A bar extending partially or entirely across the mouth of a bay. (See Figure A-9) BAYOU - A minor sluggish waterway or estuarial creek, tributary to, or connecting, other streams or bodies of water. Its course is usually through lowlands or swamps. BEACH (n.) (l) The zone of unconsolidated material that extends landward from the low water line to the place where there is marked change in material or physiographic form,,,or to the line of permanent vegeta¬ tion (usually the effective limit of storm waves). The seaward limit of the beach - unless otherwise specified - is the mean low water line, A beach includes FORESHORE and BACKSHORE; (2) Sometimes, the material which is in more or less active transport, alongshore or on-and-off shore, rather than the zone, (See Figure A-l) BEACH ACCRETION - See ACCRETION, BEACH, BARRIER - A bar essentially parallel to the shore, the crest of which is above high water level. Also OFFSHORE BARRIEIR, BEACH BERM - A nearly horizontal portion of the beach or backshore formed by the deposit of material by wave action. Some beaches have no berms, others have one or several, BEACH CUSP - One of a series of low mounds of beach material separated by crescent shaped troughs spaced at more or less regular intervals along the beach face. Also CUSP, BEIACH EROSION - The carrying away of beach materials by wave action, tidal currents, or littoral currents, or by wind, BEACH FACE - The section of the beach normally exposed to the action of the wave uprush. The FORESHORE zone of a BEACH, (Not synonymous with SHOREFACE), (See Figure A-2) BEACH, FEEDER - An artificially widened beach serving to nourish downdrift beaches by natiiral littoral currents or forces, BEACH RIDGE - An essentially continuous mound of beach material behind the beach that has been heaped up by wave or other action. Ridges may occur singly or as a series of approximately parallel deposits. In England they are called FULLS, BEACH SCARP - An almost vertical slope along the beach caused by erosion by wave action. It may vary in height from a few Inches to several feet, depending on wave action and the nature and composition of the beach, (See Figure A-l) BEACH VfIDTH - The horizontal dimension of the beach as measured normal to the shore line, BENCH - (l) A level or gently sloping erosion plane inclined seawardj (2) A nearly horizontal area at about the level of maximum high water on the sea side of a dike. A-3 BENCH MAJIKS (B.Mo) - A fi^d point used as a reference for elevations. BERM , BEACH - A nearly horizontal portion of a beach formed by the deposit of material by wave action. Some beaches have no berrs, others have one or several. (See Figure A-l) BERM CREST - The seaward limit of a berm. Also BERM EDGE. (See Figure A-l) BIGHT - A slight indentation in the shore line of an open coast or of a bay, usually crescent shaped. (See Figure A-8) BLIND ROLLERS - Long, high swells which have increased in height, almost to the breaking point, as they pass over shoals or run in shoaling water. BLUFF - A high steep bank or cliff. BOLD COAST - A prominent land mass that rises steeply from the sea. BORE - A tidal flood with a high, abrupt front, (e.g. such as occurs in the Amazon in South America, the Hugli in India, and in the Bay of Fundy). Also EAGER. BOTTOM - The ground or bed under any body of water; the bottom of the sea. (See Figure A-l) BOTTOM (nature OF) - The composition or character of the bed of an ocean or other body of water; (e.g. clay, coral, gravel, mud, ooze, pebbles, rock, shell, shingle, hard, or soft). BOULDER - A rounded rock more than 12 inches in diameter; larger than a cobble stone. BREAKER - A wave breaking on the shore, over a reef, etc. Breakers may be (rovighly) classified into three kinds although there is much overlapping; Sni llinp’ breakers break gradually over quite a distance; Plunging breakers tend to curl over and break with a crash; and Surging breakers peak up, but then instead of spilling or plung¬ ing they surge up the beach face. (See Figure A-4) BREAKER DEPTH - The still water depth at the point where the wave breaks. Also BREAKING DEPTH. (See Figure A-2) BREAKWATER - A structure protecting a shore area, harbor, anchorage, or basin from waves. BULKHEAD - A structure separating land and water areas, primarily designed to resist earth pressures. See also SEAWALL. A-4 BUOY - A floatj especially a floating object moored to the bottom, to mark a channel, anchor, shoal rock, etc. Some common types; A nun or nut buoy is conical in shape; A can buoy is squat, and cylindrical or nearly cylindrical above water and conical belowwwater; A spar buoy is a vertical, slender spar anchored at one end; A bell buoy is one having a bell operated mechanically or by the action of waves, usually marking shoals or rocks; A whistling buoy is similarly operated, marking shoals or channel entrances; A dan buoy carries a pole with a flag or light on ito BUOYANCY - The resultant of upward forces, exerted by the water on a submerged or floating body, equal to the weight of the water displaced by this bodyo CANAL - An artificial water color se cut through a land area for use in navigation, irrigation, etc, CANYON - (l) (Oceanographical) A deep submarine depression of valley fonn with relatively steep sides; (2) (Geographical) A deep gorge or ravine with steep sides, often with a river flowing at the bottom of it. CAPE - A relatively extensive land area jutting seaward from a continent or large island which prominently marks a change in, or inteirupts notably, the coastal trend; a prominent feature, (See Figure A-8) CAPILLARY WAVE - A wave whose velocity of propagation is controlled primarily by the surface tension of the liquid in which the wave is travellingo Water waves of length less than one inch are considered to be capillary waves, CAUSEWAY - A raised road, across wet or marshy ground or across water, CAUSTIC - In refraction of waves, the name given to the ciirve to which ad¬ jacent orthogonals of waves, refracted by a bottom whose contovir lines are curved, are tangents. The occ\irrence of a caustic always marks a region of crossed orthogonals and high wave convergence, CAY - See KEY, CHANNEL - (1) A natural or artificial waterway of perceptible extent which either periodically or continuously contains moving water, or which forms a connecting link between two bodies of water; (2) The part of a body of water deep enough to be used for navigation through an area otherwise too shallow for navigation; (3) A large strait, as the English Channel; (4) The deepest portion of a stream, bay, or strait through which the main voliame or current of water flows, CHARACTERISTIC WAVE HEIGHT - See SIGNIFICANT WAVE HEIGHT, A-5 CHART DATUM - The plane or level to which soundings on a chart are referred, usually taken to correspond to a low water stage of the tide. See also DATUM PLANE. CHOP - The short-crested waves that may spring up quickly in a fairly moderate breeze, and break easily at the crest. Also WIND CHOP, CUPOTIS - (1) The French equivalent for a type of STANDING WAVE; (2) In American usage it is usually associated with the standing wave phenomenon caused by the reflection of a wave train from a breakwater, bulkhead, or steep beach. CLAY - See SOIL CLASSIFICATION. CLIFF - A high, steep face of rock; a precipice. See also SEA CLIFF. COAST - A strip of land of indefinite width (may be several miles) that extends from the seashore inland to the first major change in terrain features, (See Figure A-l) COASTAL AREA - The land and sea area bordering the shore line. (See Figure A-l) COASTAL PLAIN - The plain composed of horizontal or gently sloping strata of clastic materials fronting the coast and generally representing a strip of recently emerged sea bottom. COAST LINE - (l) Technically, the line that forms the boundary between the COAST and the SHORE; (2) Commonly, the line that forms the boundary between the land and the water. COBBLES (COBBLESTONE) - See SOIL CLASSIFICATION. COMBER - (l) A deep water wave whose crest is pushed forward by a strong wind, much larger than a whitecap; (2) A long-period spilling breaker, CONTINENTAL SHELF - The zone bordering a continent extending from the line of permanent immersion to the depth (usually about 100 fathoms) where there is a marked or rather steep descent toward the great depths. CONTOUR - (1) A line connecting the points, on a land or submarine surface, that have the same elevation; (2) In topographic or hydrographic work, a line connecting all points of equal elevation above or below a datum plane, CONTROLLING DEPTH - The least depth of water in the navigable parts of a waterway, which limits the allowable draft of vessels. CONVERGENCE - (l) In refraction phenomena, the decreasing of the distance between orthogonals in the direction of wave travel. This denotes an area of increasing wave height and energy concentration; (2) In wind A-6 set-up phenomena, the increase in set-up observed over that which would occur in an equivalent rectangular basin of iiniform depth, caused by changes in planform or depth; also the decrease in basin width or depth causing such increase in set-up, C0RA.L - The calcareous skeletons of various anthozoans and a few hydrozoans; also these skeletons when solidified into a stony mass. Many tropical islands, reefs, and atolls are formed of coral, COVE - A small sheltered recess in a shore or coast, often inside a larger embayment, (See Figure A-8), CREST LENGTH, WAVE - The length of a wave along its crest. Sometimes called CREST WIDTH, CREST OF BERM - The seaward limit of a berm. Also'BERM EDGE, (See Figure A-1) CREST OF WAVE - (l) The highest part of a wave: (2) That part of the wave above still water level, (See Figure A-3) CREST WIDTH, WAVE - See CREST LENGTH, WAVE, CURRENT - A flow of water, CURRENT, COASTAL - One of the offshore currents flowing generally parallel to the shore line with a relatively uniform velocity (as compared to the littoral currents), They are not related genetically to waves and resulting surf but may be composed of currents related to dis¬ tribution of mass in ocean waters (or local eddies), wind-driven CTirrents and/or tidal currents, CURRENT, DRIFT - A broad, shallow, slow-moving ocean or lake current, CURRENT, EBB - The movement of the tidal current away from shore or down a tidal stream, CURRENT, EDDY - A circular movement of water of comparatively limited area formed on the side of a main current, Eddies may be created at points where the main stream passes projecting obstructions, CURRENT, FEEDER - The current which flows parallel to shore before converg¬ ing and forming the neck of a rip c\arrent. See also RIP, CURRENT, FLOOD - The movement of the tidal current toward the shore or up a tidal stream, CURRENT, INSHORE - An current inside the breaker zone, CURRENT, LITTORAL - The nearshore currents primarily due to wave action, e,g. Longshore currents and Rip currents. See also CURRENT, NEARSHORE, A-7 CURRENT, LONGSHORE - The inshore ciorrent moving essentially parallel to the shore, usually generated by waves breaking at an ang]e to the shore line. CURRENT SYSTEM, NEARSHORE - The current system caused primarily by wave action in and near the breaker zone and which consists of four parts: the shoreward mass transport of water; longshore currents; seaward retiarn flow, including rip currents; and the longshore movement of the expanding heads of rip currentso CURRENT, OFFSHORE - (l) Any current in the offshore zone; (2) Any current flowing away from shore. CURRENT, PERIODIC - A current, caused by the tide-producing forces of the moon and the sun, which is a part of the same general movement of the sea manifested in the vertical rise and fall of the tides. Also CURRENT, TIDAL. CURRENT, PERMANENT - A current that lams continuously Independent of the tides and temporary causes. Permanent ciirrents include the fresh water discharge of a river and the currents that form the general circulatory systems of the oceans. CURRENT, RIP - A narrow cxirrent of water flowing seaward thro\agh the breaker zone. A rip current consists of three parts: (l) The "feeder currents" flowing parallel to the shore inside the breakers; (2) the "neck" - where the feeder currents converge and flow through the breakers in a narrow band or "rip"; and (3) The "head" - where the current widens and slackens outside the breaker line. Also RIP SURFV (See Figure A-7) CURRENT, STREAM - A narrow, deep, and fast-moving ocean current. CURRENT, TIDAL - A current, caused by the tide-producing forces of the moon and the sun, which is a part of the same general movement of the sea manifested in the vertical rise and fall of the tides. Also CURRENT, PERIODIC. See also CURRENT, FLOOD AND CURRENT, EBB. CUSP - One of a series of natiirally formed low mounds of beach material separated by crescent-shaped troughs spaced at more or less regular intervals along the beach face. Also BEACH CUSP. (See Figure A-7) CUSPATE BAR - A crescent-shaped bar uniting with shore at each end. It may be formed by a single spit growing from shore turning back to again meet the shore, or by two spits growing from shore iiniting to form a bar of sharply cuspate form. CYCLOIDAL WAVE - A very steep, symmetrical wave whose crest forms an angle of 120°. The wave form is that of a cycloid. A trochoidal wave of maximum steepness. See also WAVE, TROCHOIDAL. DAILY RETARDATION (OF TIDES) - The amount of time by which corresponding tidal phases grow later day by day (averages approximately 50 minutes), DATUM, CHART - See CHART DATUMc DATUM PLANE - The horizontal plane to which soundings, ground elevations, or water surface elevations are referredo Also REFERENCE PLANE. The plane is called a TIDAL DATUM when defined by a certain phase of the tide. The following datums are ordinarily used on hydrographic charts: MEAN LOW WATER - Atlantic Coast (U. S,), Argentina, Sweden and Norway; MEAN LOWER LOW WATER - Pacific Coast (U. S.); MEAN LOW WATER SPRINGS - Great Britain, Germany, Italy, Brazil, and Chileo LOW WATEIR DATUM - Great Lakes (U. S. and Canada); LOWEST LOW WATER SPRINGS - Portugal; LOW WATER INDIAl^" SPRINGS - India and Japan (See INDIAN TIDE PLANE) LOWEST LOW WATER - France, Spain, and Greece, A common datum used on topographic maps is based upon MEAN SEA LEVELo See also BENCH MARK. DEBRIS LINE - A line near the limit of storm wave uprush marking the land¬ ward limit of debris deposits, DECAY DISTANCE - The distance through which waves travel after leaving the generating area. DECAY OF WAVES - The change that waves undergo after they leave a generating area (fetch) and pass through a calm, or region of lighter winds. In the process of decay, the significant wave height decreases and the significant wave length increases. DEEP WATER - Water of depth such that surface waves are little affected by conditions on the ocean bottom. It is customary to consider water deeper than one-half the surface wave length as deep water. DEFLATION - The removal of material from a beach or other land surface by wind action, DELTA - An alluvial deposit, usually triangular, at the mouth of a river. DEPTH - The vertical distance from the still water level (or datum as specified) to the bottom, DEPTH OF BREAKING - The still water depth at the point where the wave breaks. Also BREAKER DEPTH. DEPTH CONTOUR - See CONTOUR. DEPTH, CONTROLLING - The least depth of water in the navigable parts of a waterway, which limits the allowable draft of vessels. A-9 DEPTH FACTOR - See SHOALING COEFFICIENT DERRICK STONE - Stone of a sufficient size as to require handling in individual pieces by mechanical means, generally 1 ton up, DIFFRACTION OF WATER WAVES - The phenomenon by which energy is transmitted laterally along a wave crest. When a portion of a train of waves is interrupted by a barrier such as a breakwater, the effect of diffraction is manifested by propagation of waves into the sheltered region within the barrier’s geometric shadow, DIKE (DYKE) - A wall or mound built around a low-lying area to prevent flooding, DIURNAL - Daily, reciarring once each day, (e,g, lunar day or solar day), DIURNAL TIDE - A tide with one high water and one low water in a tidal day, (See Figure A-IO) DIVERGENCE - (l) In refraction phenomena, the spreading of orthogonals in the direction of wave travel. This denotes an area of decreasing wave height and energy concentration; (2) In wind set-up phenomena, the decrease in set-up observed under that which would occur in an equivalent rectangular basin of uniform depth, caused by changes in planform or depth. Also the increase in basin width or depth causing such decrease DOWNCOAST - In United States usage, the coastal direction generally trend¬ ing towards the south, DOWNDRIFT - The direction of predominant movement of littoral materials. DRIFT (noun) - (l) The speed at which a current runs; (2) Also, floating material deposited on a beach (driftwood); (3) A deposit of a continental ice sheet, as a DRUp^LIN; (4) Sometimes used as an abbrevia¬ tion of LITTORAL DRIFT*’ DRIFT CURRENT - A broad, shallow, slow moving ocean or lake ciorrent, DUKW -(Pronounced duck) Amphibian Truck, 2-l/2 ton, 6x6, DUNES - Ridges or mounds of loose, wind-blown material, usually sand. (See Figure A-7), DURATION - In wave forecasting, the length of time the wind blows in essentially the same direction over the FETCH (generating area), DURATION, MINIMUM - The time necessary for steady state wave conditions to develop for a given wind velocity over a given fetch length, EAGER - See BORE. EBB CURRENT - The movement of the tidal current away from shore or down a tidal stream. A-IO EBB TIDE - A non-technical term referring to that period of tide between a high water and the succeeding low water; falling tide. (See Figure A-IO) ECHO SOUNDER - A survey Instrument that determines the depth of water by measuring the time required for a sound signal to travel to the bottom and return. It may be either "sonic" or "supersonic" depending on the frequency of soiind wave, the "sonic" being generally within the audible ranges (under 15,000 cycles per second)o EDDY - A circular movement of water formed on the side of a main current. Eddies may be created at points where the main stream passes pro¬ jecting obstructionso EDDY CURRENT - See EDDY. EELGRASS - A submerged marine plant with very long, narrow leaves, abiindant along the North Atlantic Coast. See also KELP and SEAWEED,, EMBANKMENT - An artificial bank, moimd, dike or the like, built to hold back water, carry a roadway, etc. EMBAYED - Formed into a bay or bays, as an embayed shore, EMBAYMENT - An indentation in a shore line forming an open bay, ENERGY COEFFICIENT - The ratio of the energy in a wave per unit crest length transmitted forward with the wave at a point in shallow water to the energy in a wave per unit crest length transmitted forward with the wave in deep water. On refraction diagrams this is equal to the ratio of the distance between a pair of orthogonals at a selected point to the distance between the same pair of orthogonals in deep water. Also the square of the RFT’RACTION COEFFICIENT. ENTRANCE - The avenue of access or opening to a navigable channelo EROSION - The wearing away of land by the action of natural forces. (See also SCOUR), On a'BEACH, by carrying away of beach material by wave action, tidal currents, or littoral currents or by the action of the wind (See DEFLATION). ESCARPMENT - A more or less continuous line of cliffs or steep slopes facing in one general direction which are caused by erosion or fault¬ ing, Also SCARP. (See Figure A-l) ESTUARY - (l) That portion of a stream influenced by the tide of the body of water into which it flows; (2) A bay, as the mouth of a river, where the tide meets the river current. FAIRWAY - The parts of a waterway kept open and unobstructed for navigation. A-ll FATHOM - A unit of measurement used for soundings. It is equal to 6 feet (lo83 meters)o FATHOMETER - The copyrighted trade name for a type of echo sounder. FEEDER BEACH - An artificially widened beach serving to nourish downdrift beaches by natural littoral currents or forces. FEEDER CURRENT - The current which flows parallel to shore before converging and forming the neck of a rip current. See also RIPV FETCH - (1) In wave forecasting, the continuous area of water over which the wind blows in essentially a constant direction. Sometimes used synonymously with FETCH LENGTH, Also GENERATING AREA; (2) In wind set¬ up phenomena, for inclosed bodies of water, the distance between the points of maximum and minimum water, surface elevations. This would usually coincide with the longest axis in the general wind direction, FETCH LENGTH - In wave forecasting, the horizontal distance (in the direction of the wind) over which the wind blows. FIRTH - A narrow arm of the sea; also the opening of a river into the sea, FJORD (fiord) - A long narrow arm of the sea between highlands. FLOOD CURRENT - The movement of the tidal ciirrent toward the shore or up a tidal stream, FLOOD TIDE - A non-technical term referring to that period of tide between low water and the succeeding high water; a rising tide, (See Figure A-IO) FOAM LINE - The front of a wave as it advances shoreward, after it has broken. (See Figure A-4) FOLLOWING WIND - In wave forecasting, wind blowing in the same direction that waves are travelling, FORESHORE - The part of the shore, lying between the crest of the seaward berm (or the upper limit of wave wash at high tide) and the ordinary low water mark, that is ordinarily traversed by the uprush and backrush of the waves as the tides rise and fall. (See Figure A-l) FREEBOARD - The additional height of a structure above design high water level to prevent overflow. Also, at a given time the vertical distance between the water level and the top of the structure. On a ship, the distance from the water line to main deck or gunwale, FRESHET - A rapidly rising flood in a stream resulting from snow melt or rainfall, FRINGING REEF - A reef attached to an insular or continental shore. A-12 FRONT OF THE FETCH - In wave forecasting it is that end of the generating area toward which the wind is blowingo GENERATING AREA - In wave forecasting, the continuous area of water surface over which the wind blows in essentially a constant direction. Some¬ times used synonymously vrith FETCH LENGTH, Also FETCH, GENERATION OF WAVES - (l) The creation of waves by natural or mechanical means; (2) In wave forecasting, the creation and growth of waves caused by a wind blowing over a water surface for a certain period of time. The area involved is called the GENERATING AREA or FETCH, GEOMETRIC MEAN DIAMETER - The diameter equivalent of the arithmetic mean of the logarithmic frequency distribution. In the analysis of beach sands it is taken as that grain diameter determined graphically by the intersection of a straight line through selected boundary sizes (generally points on the distribution curve where 16 and 84 percent of the sample by weight is coarser) and a vertical line through the median diameter of the sample. GEOMETRIC SHADOW - In wave diffraction theory, the area outlined by drawing straight lines paralleling the direction of wave approach through the extremities of the protective structure. It differs from the actual protected area to the extent that the diffraction and refraction effects modify the wave pattem, GEOMORPHOLOGY - That branch of both physiography and geology which deals with the form of the earth, the general configuration of its svirface, and the changes that take place in the evolution of land forms. GRADIENT (GRADE) - See SLOPE. With reference to winds or currents, the rate of increase or decrease in speed, usually in the vertical, or the curve which represents this rate. GRAVEL - See SOIL CUSSIFICATIONo GRAVITY WAVE - A wave whose velocity of propagation is controlled primarily by gravity.. Water waves of a length greater than 2 inches are con¬ sidered gravity waves. GROIN (BRITo groyne) - A shore protective structure (built usually per¬ pendicular to the shore line) to trap littoral drift or retard erosion of the shore. It is narrow in width (measured parallel to the shore line), and its length may vary from less than one hundred to several hundred feet (extending from a point landward of the shore line out into the water). Groins may be classified as permeable or impermeable; impermeable groins having a solid or nearly solid structure, permeable groins having openings through them of sufficient size to permit passage of appreciable quantities of littoral drift. GROUND SWELL - A long high ocean swell; also, this swell as it rises to prominent height in shallow water, however, usually so high or dangerous as BLIND ROLLERS. GROUND WATER - Subsiirface water occupying the zone of saturationo In a strict sense the term is applied only to water below the WATER TABLEo GROUP VELOCITY - The velocity at which a wave group travels. In deep water, it is equal to one-half the velocity of the individual waves within the group. GULF - A relatively large portion of sea, partly enclosed by land. GUT - (l) A narrow passage such as a strait or inlet, (2) A channel in otherwise shallower water, generally formed by water in motion, HARBOR (BRITo HARBOUR) - A protected part of a sea, lake, or other body of water used by vessels as a place of safety and/or the transfer of passengers and cargo between water and land carriers. See also PORT. HEAD (HEADLAND) - A point or portion of land jutting out into the sea, a lake, or other body of water; a cape or promontory; now, usually specifically, a promontory especially bold and cliff-like, HEAD OF RIP - The section of a rip current that has widened out seaward of the breakers. See also CURRENT, RIP; CURRENT; FEEDER; and NECK (RIP). HEIGHT OF WAVE - The vertical distance between a crest and the preceding trough. (See Figure A-3) See also SIGNIFICANT WAVE HEIGHT. HIGH TIDE; HIGH WATER (HW) - The maximum height reached by each rising tide. See TIDE. (See Figure A-IO) HIGH WATER OF ORDINARY SPRING TIDES (HWOST) - A tidal datum appearing in some British publications, based on high water of ordinary spring tides, HIGHER HIGH WATER (HHW) - The higher of the two high waters of any tidal day. The single high water occurring daily during periods when the tide is diurnal is considered to be a higher high water. (See Figure A-10). HIGHER LOW WATER (HLW) - The higher of two low waters of any tidal day, (See Figure A-IO) HIGH WATER - See HIGH TIDE. HIGH WATER LINE - In strictness, the intersection of the plane of mean high water with the shore. The shore line delineated on the nautical charts of the Coast and Geodetic Survey is an approximation of the mean high water line, HINDCASTING, WAVE - The calculation from historic synoptic wind charts of the wave characteristics that probably occurred at some past time. A-14 HINTERLAND - The region inland from the coast, HOOK - A spit or narrow cape, turned landward at the outer end, resembling a hook in form. HYDRAULIC JUMP - In fluid flow, a change in flow conditions accompanied by a stationary, abrupt turbulent rise in water level in the direction of flow. A type of STATIONARY WAVE. HYDROGRAPHY - (l) A configuration of an underwater surface including its relief, bottom materials, coastal structures, etc® and (2) The description and study of sea, lakes, rivers, and other waters® IMPERMEABLE GROIN - See under Groin, INDIAN SPRING LOW WATER - The approximate level of the mean of lower low waters at spring tides, used principally in the Indian Ocean and along the east coast of Asia® Also INDIAN TIDE PLANE. INDIAN TIDE PLANE - The datum of INDIAN SPRING LOW WATER® INLET - A short, narrow waterway connecting a bay, lagoon, or similar body of water with a large parent body of water. An arm of the sea (or other body of water), that is long compared to its width, and that may extend a considerable distance inland® See also TIDAL INLET, INSHORE (zone) - In beach terminology, the zone of variable width extend¬ ing from the shore face through the breaker zone. (See Figure A-l) INSHORE CURRENT - Any current in or landward of the breaker zone, INSULAR SHELF - The zone surrounding an island extending from the line of permanent immersion to the depth (usually about 100 fathoms) where there is a marked or rather steep descent toward the great depths. INTERNAL WAVES - Waves that occur within a fluid whose density changes with depth, either abruptly at a sharp surface of discontinuity (an inter¬ face) or gradually. Their amplitude is greatest at the density dis¬ continuity or, in the case of a gradual density change, somewhere in the interior of the fluid and not at the free upper surface where the surface waves have their maximum amplitude, ISTHMUS - A narrow strip of land, bordered on both sides by water, that connects two larger bodies of land, JETTY - (l) (U. S. usage) On open seacoasts, a structure extending into a body of water, and designed to prevent shoaling of a channel by littoral materials, and to direct and confine the stream or tidal flow® Jetties are built at the mouth of a river or tidal inlet to help deepen and stabilize a channel® (2) (British usage) Jetty is synonymous with "wharf" or "pier"® A-15 kelp - The general name for several species of large seaweeds. A mass or growth of large seaweed or any of various large brown seaweeds. KEY - A low insular bank of sand, coral, etc., as one of the islets off the southern coast of Florida, also CAY. KINETIC ENERGY (OF WAVES) - In a progressive oscillatory wave, a summation of the energy of motion of the particles within the wave. This energy does not advance with the wave form. KNOLL - (l) A submerged elevation of rounded shape rising from the ocean floor, but less prominent than a seamoiinto (2) A small rounded hill. KNOT - (Abbreviation kt. or kts.) The unit of speed used in navigation. It is equal to 1 nautical mile (6,080.20 feet), per hour. LAGGING - See DAILY RETARDATION (OF TIDES) LAGOON - A shallow body of water, as a pond or lake, which usually has a shallow, restricted outlet to the sea. (See Figures A-8 and A-9). LAND BREEZE - A light wind blowing from the land caused by unequal cooling of land and water masses. LAND-SEA BREEZE - The oombination of a land breeze and a sea breeze as a divirnal phenomenon. LANDLOCKED - An area of water enclosed, or nearly enclosed, by land, as a bay, a harbor, etc. (thus, protected from the sea). LANDMARK - A conspicuous object - natiiral or artificial - located near or on land which aids in fixing the position of an observer. LEADLINE - A line, wire, or cord used in sounding. It is weighted at one end with a plummet (soiinding lead). Also SOUNDING LINE. LEE - (1) Shelter, or the part or side sheltered or turned away from the wind or waves. (2) (Chiefly nautical) The quarter or region toward which the wind blows. LEEWARD - The direction toward which the wind is blowing; the direction toward which waves are travelling. LENGTH OF WAVE - The horizontal distance between similar points on two successive waves measured perpendicularly to the crest.(See Figure A-3). LEVEE - A dike or embankment for the protection of land from inundation. LIMIT OF BACKRUSH-^ LIMIT OF BACKWASH'' See BACKWASH. A-16 LITTORAL - Of or pertaining to a shore, especially of the sea, A coastal region, LITTORAL CURRENT - See CURRENT, LITTC81AL LITTORAL DEPOSITS - Deposits of littoral drift, LITTORAL DRIFT - The material moved in the littoral zone under the in¬ fluence of waves and currents, LITTORAL TRANSPORT - The movement of material along the shore in the lit¬ toral zone by waves and currents, LONGSHORE CURRENT - A current in the surf zone moving essentially parallel to the shore, usually generated by waves breaking at an angle to the shore line, LOOP - That part of a STANDING WAVE or CLAPOTIS where the vertical motion is greatest and the horizontal velocities are least, LOOPS, (some¬ times called ANTINODES) are associated with CLAPOTIS, and with SEICHE action resulting from resonant wave reflections in a harbor or bay, (See also NODE) LOWER HIGH WATER (LHW) - The lower of the two high waters of any tidal day, (See Figure A-10) LOWER LOW WATER (LLW) - The lower of the two low waters of any tidal day. The single low water occurring daily during periods when the tide is diurnal is considered to be a lower low water, (See Figure A-10), LOW TIDE (LOW WATER, LW) - The minimum height reached by each falling tide. See TIDE, (See Figure A-10) LOW WATER DATUM - An approximation to the plane of mean low water that has been adopted as a standard reference plane. See also DATUM PLANE. LOW WATER LINE - The intersection of any standard low tide datum plane with the shore. LOW WATER OF ORDINARY SPRING TIDES (LWOST) - A tidal datum appearing in some British publications, based on low water of ordinary spring tides, MANC3^0VE - A particular kind of tropical tree or shrub with thickly matted roots, confined to low-lying brackish areas, MARIGRAM - A graphic record of the rise and fall of the tide, MARSH - A tract of soft, wet or periodically inundated land, generally treeless and usually characterized by grasses and other low growth, MARSH, SALT - A marsh periodically flooded by salt water. MASS TRANSPORT - The net transfer of water by wave action in the direc¬ tion of wave travel. See under ORBIT, Moy 1961 A-17 MEAN DIAMETER, GEOMETRIC - The diameter equivalent of the arthmetic mean of the logarithmic frequency distribution. In the analysis of beach sands it is taken as that grain diameter determined graphically by the intersection of a straight line through selected boundary sizes (generally points on the distribution curve where 16 and 84 percent of the sample by weight is coarser) and a vertical line through the median diameter of the sample, MEAN HIGHER HIGH WATER (MHHW) - The average height of the higher high waters over a 19-year period. For shorter periods of observation, corrections are applied to eliminate known variations and reduce the result to the equivalent of a mean 19 year value, MEAN HIGH WATER (MHW) - The average height of the high waters over a 19- year period. For shorter periods of observations, corrections are applied to eliminate known variations and reduce the result to the equivalent of a mean 19-year value. All high water heights are included in the average where the type of tide is either semidiurnal or mixed. Only the higher high water heights are included in the average where the type of tide is diurnal. So determined, mean high water in the latter case is the same as mean higher high water, MEAN HIGH WATER SPRINGS - The average height of the high waters occurring at the time of spring tide. Frequently abbreviated to High Water Springs. MEAN LOWER LOW WATER (MLLW) - Frequently abbreviated lower low water. The average height of the lower low waters over a 19-year period. For shorter periods of observations, corrections are applied to eliminate known variations and reduce the result to the equivalent of a mean 19-year value, MEAN LOW WATER (MLW) - The average height of the low waters over a 19-year period. For shorter periods of observations, corrections are applied to eliminate known variations and reduce the result to the equivalent of a mean 19-year value. All low water heights are included in the average where the type of tide is either semidiurnal or mixed. Only the lower low water heights are included in the average where the type of tide is diurnal. So determined, mean low water in the latter case is the same as mean lower low water, MEAN LOW WATER SPRINGS - Frequently abbreviated low water springs. The average height of low waters occurring at the time of the spring tides. It is usually derived by taking a plane depressed below the half-tide level by an amount equal to one-half the spring range of tide, necessary corrections being applied to reduce the result to a mean value. This plane is used to a considerable extent for hydrographic work outside of the United States and is the plane of reference for the Pacific approaches to the Panama Canal, A-18 MEAN SEA LEVEL - The average height of the surface of the sea for all stages of the tide over a 19-year period, usually determined from hourly height readings» See also SEA LEVEL DATUM. MEAN TIDE LEVEL - Also called half-tide level. A plane midway between mean high water and mean low water, MEDIAN DIAMETER - The diameter which marks the division of a given sample into two equal parts by weight, one part containing all grains larger than that diameter and the other part containing all grains smaller. MINIMUM DURATION - The time necessaiy for steady state wave conditions to develop for a given wind velocity over a given fetch length. MIXED TIDE - A type of tide in which the presence of a diurnal wave is conspicuous by a large inequality in either the high or low water heights with two high waters and two low waters usually occurring each tidal day. In strictness all tides are mixed but the name is usually applied without definite limits to the tides intermediate to those predominantly semidiurnal and those predominantly diurnal. (See Figure A-10) MDLE - In coastal terminology, a massive solid-fill structure of earth, (generally revetted), masoniy, or large stone. It may serve as a breakwater or pier. MONOLITHIC - Like a single stone or block. Therefore in (say) breakwaters, the type of construction in which the structure's component parts are bound together to act as one. MUD - A fluid-to-plastic mixture of finely divided particles of solid material and water. NAUTICAL MILE - The length of a minute of arc, l/21,600 of an average great circle of the earth. Generally one minute of latitude is considered equal to one nautical mile. The accepted United States value is 6,080.20 feet,,,, approximately 1.15 times as long as the statute mile of 5,280 feet. Also GEOGRAPHICAL MILE. NEAP TIDE - A tide occurring near the time of quadrature of the moon. The neap tidal range is usually 10 to 30 percent less than the mean tidal range, NEARSHORE (ZONE) - In beach terminology an indefinite zone extending sea¬ ward from the shore line somewhat beyond the breaker zone. It defines the area of NEARSHORE CURRENTS. The SHOREFACB. (See Figure A-l) NEARSHORE CIRCULATION - The ocean circulation pattern composed of the NEARSHORE CURRENTS and COASTAL CURRENTS. See under CURRENT. A-19 NEARSHORE CURRENT SYSTEM - The current system caused primarily by wave action in and near the breaker zone, and which consists of four parts: the shoreward mass transport of water; long-shore currents; seaward return flow, including rip currents; and the longshore movement of the expanding heads of rip currents, NECK - The narrow band of water flowing seaward through the surf. Also RIP, NIP - The cut made by waves in a shore line of emergence, NODAL ZONE - An area at which the predominant direction of the littoral transport changes, NODE - That part of a STANDING WAVE or CLAPOTIS where the vertical motion is least and the horizontal velocities are greatest. Nodes are associated with CLAPOTIS, and with SEICHE action resulting from resonant wave reflections in a harbor or bay, (See also LOOP) NOURISHMENT - The process of replenishing a beach. It may be brought about by natural means, e,g, littoral drift, or by artificial means, e,g, by the deposition of dredged materials, OCEANOGRAPHY - That science treating of the oceans, their forms, physical features, and phenomena, OFFSHORE (n, or adj.) - (1) In beach terminology, the comparatively flat zone of variable width, extending from the breaker zone to the sea¬ ward edge of the continental shelf, (2) A direction seaward from the shore (See Figure A-1) OFFSHORE CURRENT - (1) Any current in the offshore zone, (2) Any current flowing away from shore, OFFSHORE WIND - A wind blowing seaward from the land in the coastal area, ONSHORE - A direction landward from the sea, ONSHORE WIND - A wind blowing landward from the sea in the coastal area, OPPOSING WIND - In wave forecasting, a wind blowing in the opposite direction to that in which the waves are travelling, ORBIT - In water waves, the path of a water particle affected by the wave motion. In deep water waves the orbit is nearly circular and in shallow water waves the orbit is nearly elliptical. In general, the orbits are slightly open in the direction of wave motion giving rise to MASS TOANSPORT, (See Figure A-3) ORBITAL CURRENT - The flow of water accompanying the orbital movement of the water particles in a wave. Not to be confused with wave-generated LITTORAL CURRENTS, (See Figure A-3) ORTHOGONAL - On a refraction diagram, a line drawn perpendicular to the wave crests, (See Figure A-6) May 1961 A-20 OSCILLATION - A periodic motion to and fro, or up and down, OSCILLATORY WAVE - A wave in which each individual particle oscillates about a point with little or no permanent change in positioii. The term is commonly applied to progressive oscillatory waves in which only the form advances, the individual particles moving in closed orbits. Distinguished from a WAVE of TRANSLATION, See Also ORBIT, OUTFALL - (1) The vent of a river, drain, etc, (2) A structure extending into a body of water for the purpose of discharging sewage, storm runoff, or cooling water, OVERTOPPING - The amount of water passing over the top of a structure as a result of wave run-up or surge action, OVERWASH - That portion of the upru.*’’ that carries over the crest of a berm or of a structure, PARAPET - A low wall built along the edge of a structure as on a seawall or quay, PARTICLE VELOCITY - For waves, the velocity induced by wave motion with which a specific water particle moves, PASS - In hydrographic usage a navigable channel, through a bar, reef, or shoal, or between closely adjacent islands, PEBBLES - See SOIL CLASSIFICATION. PENINSULA - An elongated portion of land nearly surrounded by water, and connected to a larger body of land, PERIODIC CURRENT - A current caused by the tide-producing forces of the moon and the sun, a part of the same general movement of the sea that is manifested in the vertical rise and fall of the tides. See also CURRENT, FLOOD and CURRENT, EBB, PERMAFROST - Permanently frozen subsoil, PERMANENT CURRENTS - A current that runs continuously independent of the tides and te(iq)orary causes. Permanent currents include the fresh water discharge of a river and the currents that form the general circulatory systems of the oceans, PERMEABLE GROIN - See under GROIN PETROGRAPHY - The description and systematic classification of rocks, PIER - A structure, extending out into the water from the shore, to serve as a landing place, a recreational facility, etc,, rather than to afford coastal protection. May 1961 A-21 PILE - A long, slender piece of wood, concrete, or metal to be driven or jetted into the earth or sea bed to serve as a support or protection. PILING - A group of piles. PILE, SHEET - A pile with a generally flat cross-section to be driven into the groiind or sea bed and meshed or interlocked with like members to form a diaphragm, wall, or bulkhead. PINNACLE - A tall, slender, pointed, rocky mass. See also REEF PINNACLE. PLAIN - An extent of level or nearly level land. PLAIN, COASTAL - A plain fronting the coast and generally representing a strip of recently emerged sea bottom. PLANEORM - The outline or shape of a body of water as determined by the still water line. PLATEAU - An elevated plain, table land, or flat-topped region of consider¬ able extent. PLUNGE POINT - (See Figure A-l) (l) For a plunging wave, the point at which the wave curls over and falls; (2) The final breaking point of the waves just before they rush up on the beach. PLUNGING BREAKER - See under BREAKER. POINT - The extreme end of a cape; or the outer end of any land area pro¬ truding into the water, usually less prominent than a cape. PORT - A place where vessels may discharge or receive cargo; may be the entire harbor including its approaches and anchorages or may be the commercial part of a harbor, where the quays, wharves, facilities for transfer of cargo, docks, repair shops, etc. are situated. POTENTIAL ENERGY OF WAVES - In a progressive oscillatory wave, the energy resulting from the elevation or depression of the water siirface from the undisturbed level. This energy advances with the wave form. PROFILE,BEACH - The intersection of the ground surface with a vertical plane; may extend from the top of the dune line to the seaward limit of sand movement. (See Figure A-l) PROGRESSION - See ADVANCE, PROGRESSIVE WAVE - A wave which is manifested by the progressive movement of the wave form. PROMONTORY - A high point of land projecting into a body of water; a head¬ land. A-22 PROPAGATION OF WAVES - The transmission of waves through water. PROTOTYPE - In laboratory usage, the original structure, concept, or phenomenon used as a basis for constructing a scale model or copy. QUAY - (pronounced KEY) - A stretch of paved bank, or a solid artificial landing place parallel to the navigable waterway, for use in loading and unloading vessels. QUICKSAND - Loose, yielding, wet sand which offers no support to heavy objects. The upward flow of the water has a velocity that eliminates contact pressures between the sand grains, and causes the sand-water mass to behave like a fluid. RECESSION (OF A BEACH) - (l) A continuing landward movement of the shore line. (2) A net landward movement of the shore line over a specified time. Also RETROGRESSION. REEF - A chain or range of rock or coral, elevated above the surroimdlng bottom of the sea, generally submerged and dangerous to surface navigation. REEF, ATOLL - A ring-shaped, coral reef, often carrying low sand islands, enclosing a body of water. REEF, BARRIER - A reef which roughly parallels land but is some distance offshore, with deeper water intervening. REEF, fringing - A reef attached to an insular or continental shore. REEF, SAND - Synonymous with BAR. REFERENCE PLANE - See DATUM PLANE. REFERENCE POINT - A specified location (in plan and/or elevation) to which measurements are referred. REFERENCE STATION - A station for which tidal constants have previously been determined and which is used as a standard for the comparison of simultaneous observations at a second station; also a station for which independent daily predictipns are given in the tide or current tables from which corresponding predictions are obtained for other stations by means of differences or factors. REFLECTED WAVE - The wave that is retiarned seaward when a wave impinges upon a very steep beach, barrier, or other reflecting siirfaces. REFRACTION OF WATER WAVES - (l) The process by which the direction of a wave moving in shallow water at an ang]e to the contours is changed. The part of the wave advancing in shallower water moves more slowly than that part still advancing in deeper water, causing the wave crest to bend toward alignment with the underwater contours. (2) The bending A-23 of wave crests by ciirrentSo (See Figure A-5) REFRACTION COEFFICIENT - The square root of the ratio of the spacing between adjacent orthogonals in deep water and in shallow water at a selected pointo When multiplied by the SHOALING FACTOR, this becomes the WAVE HEIGHT COEFFICIENT or the ratio of the refracted wave height at any point to the deep water wave heighto Also the square root of the ENERGY COEFFICIENT. REFRACTION DIAGRAM - A drawing showing positions of wave crests and/or orthogonals in a given area for a specific deep water wave period and direction. (See Figure A-6) RETARDATION - The amount of time by which corresponding tidal phases grow later day by day (averages approximately 50 minutes). RETROGRESSION OF A BEACH - (l) A continuing landward movement of the shore line; (2) A net landward movement of the shore line over a specified time. Also RECESSION. REVETMENT - A facing of stone, concrete, etc., built to protect a scarp, embankment or shore structure against erosion by the wave action or currents. RIA - A long narrow inlet, with depth gradually diminishing inward. RIDE-UP - See RUN-UP. RIDGE, BEACH - An essentially continuous mound of beach material that has been shaped up by wave or other action. Ridges may occur singly or as a series of approximately parallel deposits. (See Figure A-7) In England they are called FULLS. RILL MARKS - Tiny drainage channels in a beach caused by the flow seaward of water left in the sands of the upper part of the beach after the retreat of the tide or after the dying down of storm waves. RIP - A body of water made rough by waves meeting an opposing current, particularly a tidal ciorrent; often found where tidal currents are converging and sinking. A TIDE RIP. RIPARIAN - Pertaining to the banks of a body of water. RIPARIAN RIGHTS - The rights of a person owing land containing or bordering on a watercourse or other body of water in or to its banks, bed, or waters. RIP CURRENTS - A strong surface current of short duration flowing seaward from the shore. It usually appars as a visible band of agitated water and is the return movement of water piled up on the shore by incoming waves and wind. With the seaward movement concentrated in a limited A-24 band its velocity is somewhat accentuated, A rip consists of three parts: the FEEDER CURRENT flowing parallel to the shore inside the breakers; the NECK, where the feeder currents converge and flow through the breakers in a narrow bank or "rip"; and the HEAD, where the current widens and slackens outside the breaker line. A rip current is often miscalled a RIP TIDE, Also RIP SURF, (See Figure A-7). RIP SURF - See under RIP CURRENT. RIPPLE - (1) The ruffling of the surface of water, hence a little curling wave or undulation. (2) A wave controlled to a significant degree by both surface tension and gravity. See WAVE, CAPIILARY AND WAVE, gravity. RIPPLE MARKS - Small, fairly regular ridges in the bed of a waterway or on a land surface caused by water currents or wind. As their form is approximately normal to the direction of current or wind, they indicate both the presence and the direction of currents or winds, RIPRAP - A layer, facing, or protective mound of stones randomly placed to prevent erosion, scour, or sloughing of a structure or embankment; also the stone so used, RISE, TIDAL - The height of tide as referred to the datum of a chart. roadstead - (Nautical) A sheltered area of water near shore where vessels may anchor in relative safety. Also ROAD, ROCK - (1) (Engineering) A natural aggregate of mineral particles connected by strong and permanent cohesive forces. In igneous and metamorphic rocks, it consists of interlocking crystals; in sedimentary rocks, of closely packed mineral grains, often bound together by a natural cement. Since the terms "strong" and "permanent" are subject to dif¬ ferent interpretations, the boundary between rock and soil is neces¬ sarily an arbitrary one, (2) (Geological) The material that forms the essential part of the earth's solid crust, and includes loose inco¬ herent masses, such as a bed of sand, gravel, clay or volcanic ash, as well as the very firm, hard, and solid masses of granite, sandstone, limestone, etc. Most rocks are aggregates of one or more minerals, but some are composed entirely of glassy matter, or of a mixture of glass and minerals, ROLLER - An indefinite term, sometimes considered to be one of a series of long-crested, large waves which roll in upon a coast, as after a storm. RL'BBLE - (1) Loose angular water-worn stones along a beach. (2) Rough, irregular fragments of broken rock, RUBBLE-MOUND STRUCTURE - A mound of random-shaped and random-placed stones protected with a cover layer of selected stones or specially shaped concrete armor units. (Armor units in primary cover layer may be placed in orderly manner or dumped at random.) RUNNEL - A corrugation (trough) of the foreshore (or the bottom just off¬ shore), formed by wave and/or tidal action. Larger than the trough between ripple marks. May 1961 A-25 RUN-UP - The rush of water up a structure on the breaking of a waveo Also UPRUSH. The amount of run-up is the vertical height above still water level that the rush of water reacheSo SALTATION - That method of sand movement in a fluid in which individual particles leave the bed by bounding nearly vertically and, because the motion of the fluid is not strong or turbulent enough to retain them in suspension, return to the bed at some distance downstreamo The travel path of the particles is a series of hops and boundSo SALT MARSH - A marsh periodically flooded by salt watero SAND - See SOIL CLASSIFICATIONo SAND BAR - (l) See BAR, (2) In a river, a ridge of sand built up to or near the surface by river currents. SAND REEF - Synonymous with BAR, SCARP - A more or less continuous line of cliffs or steep slopes facing in one general direction which are caused by erosion or faulting. Also ESCARPMENT. SCARP, BEACH - An almost vertical slope along the beach caused by erosion by wave action. It may vary in height from a few inches to several feet, depending on wave action and the nature and composition of the beach, SGCUR - Erosion, especially by moving water. See also ERCSICN, SEA - (l) An ocean, or alternatively a large body of (usually) salt water less than an ocean; (2) Waves caused by wind at the place and time of observation; (3) State of the ocean or lake surface in regard to waves, SEA (state CF) - Description of the sea surface with regard to wave action. SEA BREEZE - (l) A breeze blowing from the sea toward the land; (2) A light wind blowing toward the land caused by unequal heating of land and water masses. SEA CLIFF - A cliff situated at the seaward edge of the coast, SEA LEVEL - See MEAN SEA LEVEL, SEA MOUNT - A submarine mountain rising more than 5CC fathoms above the ocean floor, SEA PUSS - A dangerous longshore current, n rip current, caused by return flow, loosely the submerged channel or inlet through a bar caused by those currents. A-26 SEASHORE - The SHORE of a sea or oceano SEAWALL - A structure separating land and water areas primarily designed to prevent erosion and other damage due to wave actiono See also BULKHEAD„ SEICHE - A periodic oscillation of a body of water whose period is determined by the resonant characteristics of the containing basin as controlled by its physical dimensionso These periods generally range from a few minutes to an hour or moreo (Originally the term was applied only to lakes but now also to harbors, bays, oceans, etc.)o SEISMIC SEA WAVE - (TSUNAMI) - A generally long period wave caused by an underwater seismic disturbance or volcanic eruption. Commonly mis¬ named "tidal wave", SEMIDIURNAL TIDES - A tide with two high and two low waters in a tidal day, with comparatively little diurnal inequality. (See Figure A-IO) SET OF CURRENT - The direction toward which a current flows. SET-UP, WIND - (l) The vertical rise in the still water level on the leeward side of a body of water caused by wind stresses on the surface of the water; (2) The difference in still water level between the wind¬ ward and the leeward sides of a body of water caused by wind stresses on the siarface of the water; (3) Synonymous with WIND TIDE. WIND TIDE is usually reserved for use on the ocean and large bodies of water. WIND SET-UP is usually reserved for use on reservoirs and small bodies of water. (See Figure A-ll) SHALLOW WATER - (l) Commonly; water of such a depth that surface waves are noticeably affected by bottom topography. It is customary to consider water of depths less than half the surface wave length as shallow water. See TRANSITIONAL WATER; (2) More strictly; in hydrodynamics with regard to progressive gravity waves, water in which the depth is less than l/25th the wave length. Also called VERY SHALLOW WATER, SHEET PILE - See \inder PILE. SHELF, CONTINENTAL - The zone bordering a continent extending from the line of permanent immersion to the depth (usually about 100 fathoms) where there is a marked or rather steep descent toward the great depths. SHELF, INSULAR - The zone surrounding an island extending from the line of permanent immersion to the depth (usually about 100 fathoms) where there is a marked or rather steep descent toward the great depths. SHINGLE - (l) Loosely and commonly; any beach material coarser than ordinary gravel, especially any having flat or flattish pebbles; (2) Strictly and accurately; beach material of smooth, well-rounded pebbles that are roughly the same size. The spaces between pebbles are not filled A-27 with finer materials» Shingle gives out a musical note when stepped on. SHOAL (noun) - A detached elevation of the sea bottom comprised of any material except rock or coral, and which may endanger surface navigation. SHOAL (verb) - (l) to become shallow gradually; (2) to cause to become shallow; (3) to proceed from a greater to a lesser depth of water. SHOALING COEIFFICIENT - The ratio of the height of a wave in water of any depth to its height in deep water with the effect of refraction eliminat¬ ed. Sometimes SHOALING FACTOR or DEPTH FACTOR. See also ENERGY COEFFICIENT and REFRACTION COEFFICIENT. SHORE - The strip of ground bordering any body of water. A shore of un¬ consolidated material is usually called a BEACH. (See Figure A-l) SHORE FACE - The narrow zone seaward from the low tide SHORE LINE per¬ manently covered by water, over which the beach sands and gravels actively oscillate with changing wave conditions. SHORE LINE - The intersection of a specified plane of water with the shore or beach, (e.g. the high water shore line would be the intersection of the plane of mean high water with the shore or beach). The line de¬ lineating the shore line on U. S. Coast and Geodetic Survey nautical charts and surveys approximates the mean high water line. SIGNIFICANT WAVE - A statistical term denoting waves with the average height and period of the one^third highest waves of a given wave group. The composition of the higher waves depends upon the extent to which the lower waves are considered. Experience so far indicates that a careful observer who attempts to establish the character of the higher waves will record values which approximately fit the definition. A wave of significant wave period and significant wave height. SIGNIFICANT WAVE HEIGHT - The average height of the one-third highest waves of a given wave group. Note that the composition of the highest waves depends upon the extent to which the lower waves are considered. In wave record analysis, the average height of the highest l/3 of a selected number of waves, this number being determined by dividing the time of record by the significant period. Also CHARACTERISTIC WAVE HEIGHT. SIGNIFICANT WAVE PERIOD - An arbitraiy period generally taken as the period of the 1/3 highest waves within a given group. Note that the composition of the highest waves depends upon the extent to which the lower waves are considered. In wave record analysis, this is determined as the average period of the most frequently recurring of the larger well- defined waves in the record \ander study. SILT - See SOIL CLASSIFICATION. A-28 SLACK TIDE (SLACK WATER) - The state of a tidal current when its velocity is near zero, especially the moment when a reversing current changes direction and its velocity is zero. Sometimes considered the intermediate period between ebb and flood currents during which the velocity of the currents is less than 0,1 knot. See TIDAL STANDo SLIP - A space between two piers, wharves, etc, for the berthing of vessels, SLOPE - The degree of inclination to the horizontal. Usually expressed as a ratio, such as 1:25 or 1 on 25 , indicating 1 iinit rise in 25 iinits of horizontal distance; or in a decimal fraction (0,04); degrees (2° 18'); or percent (4^). It is sometimes described by such adjectives as; steep, moderate, gentle, mild, or flat, SLOUGH - (pronounced - sloo): - (l) A small muddy marshland or tidal water¬ way which usually connects other tidal areas; (2) A tide-land or bottom-land creek, SOIL CLASSIFICATION (Size) - An arbitrary division of a continuous scale of sizes such that each scale unit or grade may serve as a convenient class interval for conducting the analysis or for expressing the results of an analysis. There are many classifications used; some of those most often used are presented below: 1, Wentworth's Size Classification Grade Limits (Diameters) Name Above 256 mm. Boulder Cobble Pebble Granule Very coarse sand Coarse sand .Medium sand Fine sand Veiy fine sand 256 - 64 mm, 64-4 mm. 4-2 mm, 2-1 mm, 1 - 1/2 mm. mm. 1/2 - 1/4 mm, 1/4 - 1/8 mm, 1/8 - 1/16 mm. 1/2 - 1/4 mm. Silt Clay 2, U, S, Army Corps of Engineers' Classification Grade Limits (Diameters) U,S, Standard Sieve Size Name Above 76 mm, 76 mm, - 19 mm, 19 mm, - 4 o 76 ram. Above 3" 3" to 3/4” 3 / 4 ” to No, 4 No, 4 to No, 10 No, 10 to No, 40 No, 40 to No_, 200 Below No, 200 Fine gravel Coarse sand Indium sand Fine sand Cobbles Coarse gravel 4o76 - 2,00 mm, 2,00 - 0,42 mm, 0,42 - 0,074 inni. Below 0,074 nun. Silt or clay A-29 3o U. S. Bureau of Soils Classification Grade Limits (Diameters) 2 - 1 mm, 1 - 1/2 mm, 1/2 - 1/4 mm, 1/4 - 1/10 mm, 1/10 - 1/20 mm, 1/20 - 1/200 mm. Below 1/200 mm, 4 . Atterberg *s Grade Limits (Diameters) 2,000 - 200 mm, 200 - 20 mm, 20-2 mm, 2 - 0,2 mm, 0o2 - O 0 O 2 mm, 0,02 - 0,002 mm. Below 0,002 nmio Name Fine gravel Coarse sand Medium sand Fine sand Very fine sand Silt Clay Classification Name Blocks Cobbles Pebbles Coarse sand Fine sand Silt Clay SOLITARY WAVE - A wave consisting of a single elevation (above the water surface) of height not necessarily small compared to the depth, and neither followed nor preceded by another elevation or depression of the water surfaces, SORTING COEFFICIENT - A coefficient used in describing the distribution of grain sizes in a sample of unconsolidated material. It is defined as S^ . ^Qi/Q 3 , where is the diameter which has 75^ of the cumulative size-frequency (by wt,) distribution smaller than itself and 25 ^ larger than itself, and Q 3 is that diameter having 25 % smaller and 15 % larger than itself', SOUND (noun) - (l) A wide waterway between the mainland and an island, or a wide waterway connecting two sea areas. See also STRAIT, (2) A relatively long arm of the sea or ocean forming a channel between an island and a mainland or connecting two larger bodies , as a sea and the ocean, or two parts of the same body; usually wider and more ex¬ tensive than a strait, SOUND (verb) - To measure or ascertain the depth of water as with sound¬ ing lines, SOUNDING - A measured depth of water. On hydrographic charts the soxindings are adjusted to a specific plane of reference (SOUNDING DATUM), A-30 SOUNDING DATUM - The plane to which soundings are referred» See also DATUM, CHART, SOUNDING LINE - A line, wire, or cord used in soundingo It is weighted at one end with a plummet (sounding lead). Also LEADLINEo SPILLING BREAKER - See under BREAKER. SPIT - A small point of land or submerged ridge running into a body of water from the shore. (See Figure A-9). SPRING TIDE - A tide that occurs at or near the time of new and full moon and which rises highest and falls lowest from the mean level. STAND OF TIDE - An interval at high or low water when there is no sensible change in the height of the tide. The water level is stationary at high and low water for only an instant, but the change in level near these times is so slow that it is not usually perceptible. See TIDE, SUCK. STANDING WAVE - A type of wave in which the surface of the water oscillates vertically between fixed points, called nodes, without progression. The points of maximum vertical rise and fall are called antinodes or loops. At the nodes, the underlying water particles exhibit no vertical motion but maximum horizontal motion. At the antinodes the underlying water particles have no horizontal motion and maximum vertical motion. They may be the result of two equal progressive wave trains travelling through each other in opposite directions. Sometimes called STATIONARY WAVE” STATIONARY WAVE - A wave of essentially stable form which does not move with respect to a selected reference point; a fixed swelling. Some¬ times called STANDING WAVE. STILL WATER LEVEL - The elevation of the surface of the water if all wave action were to cease. (See Figure A-3) STONE - (l) Rock or rocklike matter used as a material for building; (2) A small piece of rock or a specific piece of rock. STONE, DERRICK - Stone of a sufficient size as to require handling in individual pieces by mechanical means, generally 1 ton up. STORM TIDE - The rise of water accompanying a storm caused by wind stresses on the water surface. See also SET-UP, WIND. STRAIT - A relatively narrow waterway between two larger bodies of water. See also SOUND (noun). STREAM - (1) A course of water flowing along a bed in the earth; (2) A current in the sea formed by wind action, water density differences, etc. (Gulf Stream) See also CURRENT, STREAM. A-31 SURF- The wave activity in the area between the shore line and the outer¬ most limit of breakers. SURF BEAT - Irregular oscillations of the nearshore water level, with periods of the order of several minutes. SURF ZONE - The area between the outermost breaker and the limit of wave uprush (See Figures A-2 and A-5), SURGE - (1) The name applied to wave motion with a period intermediate between that of the ordinary wind wave and that of the tide, say from 1/2 to 60 minutes. It is of low height; usually less than 0,3 foot. See also SEICHE, (2) In fluid flow, long interval variations in ve¬ locity and pressure, not necessarily periodic, perhaps even transient in nature, SURGE, STORM - That rise above normal water level on the open coast due only to the action of wind stress on the water surface. Storm surge resulting from a hurricane also includes that rise in level due to atmospheric pressure reduction as well as that due to wind stress, (See SET-UP, WIND) SURGING BREAKER - See under Breaker. SWASH - The rush of water up onto the beach following the breaking of a wave. Also UPRUSH, RUN-UP. SWASH (CHANNEL - (1) On the open shore, a channel cut by flowing water in its return to the parent body (e.g, a rip channel); (2) A secondary channel passing through or shoreward of an inlet or river bar, (See Figure A-9) SWASH MARK - The thin wavy line of fine sand, mica scales, bits of seaweed, etc, left by the uprush when it recedes from its upward limit of move¬ ment on the beach face, SWELL - Wind-generated waves that have advanced into regions of weaker winds or calm. TERRACE - A horizontal or nearly horizontal natural or artificial topographic feature interrupting a steeper slope, sometimes occurring in a series, TIDAL CURRENT - A current caused by the tide-producing forces of the moon and the sun, a part of the same general movement of the sea that is manifested in the vertical rise and fall of the tides. Also CURRENT, PERIODIC. See also CURRENT, FLOOD and CURRENT, EBB. TIDAL DATUM - See DATUM, CHART, and DATUM PLANE. TIDAL DAY - The time of the rotation of the earth with respect to the moon, or the interval between two successive upper transits of the moon over the meridian of a place, about 24,84 solar hours (24 hours and 50 minutes) in length or 1,035 times as great as the mean solar day, (See Figure A-10) May 1961 A-32 TIDAL FLATS - Marshy or muddy land areas which are covered and uncovered by the rise and fall of the tide. TIDAL INLET - (l) A natural inlet maintained by tidal flow; (2) Loosely any inlet in which the tide ebbs and flows. Also TIDAL OUTLET. TIDAL PERIOD - The interval of time between two consecutive like phases of the tide. (See Figure A-IO) TIDAL POOL - A pool of water remaining on a beach or reef after recession of the tide. TIDAL PRISM - The total amount of water that flows into the harbor or out again with movenient of the tide, excluding any fresh water flow, TIDAL RANGE - The difference in height between consecutive high and low waters. (See Figure A-IO) TIDAL RISE - The height of tide as referred to the datum of a chart. (See Figure A-IO) TIDAL WAVE - See TSUNAMI. TIDE - The periodic rising and falling of the water that results from gravitational attraction of the moon and sun acting upon the rotating earth. Although the accompanying horizontal movement of the water resulting from the same cause is also sometimes called the tide, it is preferable to designate the latter as TIDAL CURRENT, reserving the name tide for the vertical movement. TIDE, DAILY RETARDATION OF - The amount of time by which corresponding tides grow later day by day. TIDE, DIURNAL - A tide with one high water and one low water in a tidal day. (See Figure A-IO) TIDE, EBB - That period of tide between a high water and the succeeding low water; falling tide, (See Figure A-IO) TIDE, FLOOD - That period of tide between low water and the succeeding high water; a rising tide, (See Figure A-10) * TIDE, MIXED - A type of tide in which the presence of a diurnal wave is conspicuous by a large inequality in either the high or low water heights with two high waters and two low waters usually occurring each tidal day. In strictness all tides are mixed but the name is usually applied without definite limits to the tides intermediate to those predominantly semidiurnal and those predominantly diurnal. (See Figure A-IO) A-33 TIDE, NEAP - A tide occiorring near the time of quadratvire of the moon. The neap tidal range is usually 10 to 30 percent less than the mean tidal range,- TIDE, SEMIDIURNAL - A tide with two high and two low waters in a tidal day, with comparatively little diurnal inequality, (See Figure A-IO) TIDE, SLACK - The state of a tidal ciirrent when its velocity is near zero, especially the moment when a reversing current changes direction and its velocity is zero. Sometimes considered the intermediate period between ebb and flood currents during which the velocity of the currents is less than 0,1 knot. See TIDAL STAND, Also SLACK WATER, TIDE, SPRING - A tide that occurs at or near the time of new and full moon and which rises highest and falls lowest from the mean level, TIDE, STORM - The rise af water accompanying a storm caused by wind stresses on the water surface. See also SET-UP, WIND, TOMBOLO - An area of unconsolidated material, deposited by wave action or > cuirents, that connects a rock, or Island, etc, to the mainland or to another island, (See Figure A-9) TOPOGRAPHY - The configuration of a surface including its relief, the position of its streams, roads, buildings, etc, TRAINING WALL - A wall or jetty to direct current flow, TRANSITIONAL ZONE TRANSITIONAL WATER - In regard to progressive gravity waves, water whose depth is less than l/2 but more than l/25 the wave length. Often called SHALLOW WATER, TROCHOIDAL WAVE - A progressive oscillatory wave whose form is that of a prolate cycloid or trochoid. It is approximated by waves of small amplitude. See also WAVE, CYCLOIDAL, TROUGH OF WAVE - The lowest part of a wave form between successive crests. Also that part of a wave below still water level, (See Figure A-3), TSUNAMI - A generally long period wave caused by underwater seismic dis¬ turbance or volcanic eruption. Commonly misnamed ”tidal wave", UNDERTOW - A cirrrent, below water surface, flowing seaward; also the receding water below the surface from waves breaking on a shelving beach. Actually "iindertow" is largely mythical. As the backwash of each wave flows down the beach, a current is formed which flows seaward, however, it is a periodic phenomenon. The most common phenomena expressed as "undertow" are actually the rip currents in the surf. Often uniform retiarn flows seaward or lakeward are termed "undertow" though these flows will not be as strong as rip currents. See also CURRENT SYSTEM, NEARSHORE, A-34 UNDERWATER GRADIENT - The slope of the sea bottonio See also SLOPE, UNDULATION - A continuously propagated motion to and fro, in any fluid or elastic mediiom, with no permanent translation of the particles themselves, UPCOAST - In United States usage; the coastal direction generally trending towards the north, UPDRIFT - The direction opposite that of the predominant movement of littoral materialso UPLIFT - The upward water pressure on the base of a structure or pavement, UPRUSH - The rush of water up onto the beach following the breaking of a wave. Also SWASH, RUN-UP, (See Figure A-2) VALLEY, SEA - A submarine depression of broad valley form without the steep side slopes which characterize e canyon, VALLEY, SUBMARINE - A prolongation of a land valley into or across the continental or Insular shelf, which generally gives evidence of having been formed by stream erosion, VARIABILITY OF WAVES - (l) The variation of heights and periods between individual waves within a wave train, (Wave trains are not composed of waves of equal height and period, but rather of heights and periods which vary in a statistical manner), (2) The variation in direction of propagation of waves leaving the generating area, (3) The variation in height along the crest. This is usually called "variation along the wave", VELOCITY OF WAVES - The speed with which an individual wave advances, VISCOSITY - Internal friction due to molecular cohesion in fluids. The internal properties of a fluid which offer resistance to flow, WATER LINE - A juncture of land and sea. This line migrates, changing with the tide or other fluctuation in the water level. Where waves are present on the beach, this line is also known as the limit of backrush, (Approximately the intersection of the land with the still water level), WAVE - A ridge, deformation, or undulation of the surface of a liquid, WAVE AGE - The ratio of wave velocity to wind velocity, WAVE, CAPILLARY - A wave whose velocity of propagation is controlled primarily by the surface tension of the liquid in which the wave is travelling. Water waves of a length less than one inch are considered to be capillary waves. WAVE CREST - The highest part of a waveo Also that part of the wave above still water levelo (See Figure A-3) WAVE CREST LENGTH - The length of a wave along its crest. Sometimes called CREST WIDTH. WAVE, CYCLOIDAL - A very steep, symmetrical wave whose crest forms an angle of 120°. The wave form is that of a cycloid. A trochoidal wave of maximum steepness. See also WAVE, TROCHOIDAL. WAVE DECAY - The change which waves undergo after they leave a generating area(fetch) and pass through a calm, or region of lighter or opposing winds. In the process of decay, the significant wave height decreases and the significant wave length Increases. WAVE DIRECTION - The direction from which a wave approaches. WAVE FORECASTING - The theoretical determination of future wave character¬ istics, usually from observed or predicted meteorological phenomena. WAVE GENERATION - (l) The creation of waves by natural or mechanical means. (2^ In wave forecasting, the growth of waves caused by a wind blowing over a water siorface for a certain period of time. The area involved is called the GENERATING AREA or FETCH. WAVE, GRAVITY - A wave whose velocity of propagation is controlled primarily by gravity. Water waves of a length greater than 2 inches are considered gravity waves. WAVE GROUP - A series of waves in which the wave direction, wave length, and wave height vary only slightly. See also GROUP VELOCITY. WAVE HEIGHT - The vertical distance between a crest and the preceding trough. See also SIGNIFICANT WAVE HEIGHT. WAVE HEIGHT COEFFICIENT - The ratio of the wave height at a selected point to the deep water wave height. The refraction coefficient multiplied by the shoaling factor. WAVE HINDCASTING - The calculation from historic synoptic wind charts of the wave characteristics that probably occurred at some past time. WAVE LENGTH,- The horizontal distance between similar points on two successive waves measured perpendicularly to the crest. WAVE, OSCILLATORY - A Wave in which each individual particle oscillates about a point with little or no permanent change in position. The term is commonly applied to progressive oscillatory waves in which only the form advances, the individual particles moving in closed or nearly closed orbits. Distinguished from a WAVE of TRANSLATION. See also ORBIT. A-36 WAVE PSIIOD - The time for a wave crest to traverse a distance equal to one wave length. The time for two successive wave crests to pass a fixed point. See also SIGNIFICANT WAVE PERIOD, WAVE, PROGRESSIVE - A wave which is manifested bj the progressive movement of the wave form. WAVE PROPAGATION - The transmission of waves through water. WAVE RAY - See ORTHOGONAL. WAVE, REFLECTED - The wave that is returned seaward when a wave impinges upon a very steep beach or barrier. WAVE REFRACTION - (l) The process by which the direction of a train of waves moving in shallow water at an angle to the contours is changed. The part of the wave train advancing in shallower water moves more slowly than that part still advancing in deeper water, causing the wave crests to bend toward alignment with the \mderwater contours. (See Figures A-5 and A-6). (2) The bending of wave crests by cixrrents. WAVE, SEISMIC - A TSUNAMI - A generally long period wave caused by an underwater seismic disturbance or volcanic eruption. Commonly misnamed "tidal wave". WAVE, SOLITARY - A wave consiting of a single elevation (above the water siirface) of height not necessarily small compared to the depth and neither followed nor preceded by another elevation or depression of the water surfaces. WAVE, STANDING - A type of wave in which there are nodes, or points of no vertical motion and maximum horizontal motion, between which the water oscillates vertically. The points of maximum vertical motion and least horizontal motion are called antinodes or loops. It is caused by the meeting of two similar wave groups travelling in opposing directions. WAVE, STATIOIUUIY - A wave of essentially stable form which does not move with respect to a selected reference point. WAVE STEEPNESS - The ratio of a wave’s height to its length. WAVE TRAIN - A series of waves from the same direction. WAVE OF TRANSLATION - A wave in which the water particles are permanently displaced to a significant degree in the direction of wave travel. Distinguished from an OSCILLATORY WAVE. WAVE, TTiOCHOIDAL - A progressive oscillatory wave whose form is that of a prolate cycloid or trochoid. It is approximated by waves of small amplitude. See also WAVE, CYCLOIDAL. A-37 WAVE TROUGH - The lowest part of a wave form between successive crestso Also that part of a wave below still water level» WAVE VARIABILITY - (l) The variation of heights and periods between individual waves within a wave train. (Wave trains are not composed of waves of equal height and period, but rather of heights and periods which vary in a statistical manner). (2) The variation in direction of propagation of waves leaving the generating area. (3) The variation in height along the crest. This is usually called "variation along the wave". WAVE VELOCITY - The speed with which an individual wave advances. WAVE, WIND - A wave that has been formed and built up by the wind. WAVES, INTERNAL - Waves that occur within a fluid whose density changes with depth, either abruptly at a sharp surface of discontinuity (an interface) or gradually. Their amplitude is greatest at the density discontinuity or, in the case of a gradual density change, somewhere in the interior of the fluid and not at the free upper surface where the surface waves have their maximum amplitude. WHARF - A structure built on the shore of a harbor, river, canal, etc., so that vessels may lie alongside to receive and discharge cargo, passengers, etc. WHITECAP - On the crest of a wave, the white froth caused by wind. WIND - The horizontal natural movement of air; air naturally in motion with any degree of velocity. WIND CHOP - The short-crested waves that may spring up quickly in a fairly moderate breeze, and break easily at the crest. WIND, FOLLOWING - In wave forecasting, wind blowing in the same direction that waves are travelling. WIND, OFFSHORE - A wind blowing seaward over the coastal area. WIND, ONSHORE - A wind blowing landwaid over the coastal area. Wj.ND, opposing - In wave forecasting, wind blowing in the opposite direction to that in which the waves are travelling. WIND SET-UP - (l) The vertical rise in the still water level on the leeward side of a body of water caused by wind stresses on the sxrface of the water; (2) the difference in still water levels on the windward and the leeward sides of a body of water caused by wind stresses on the surface of the water; (3) Synonymous with WIND TIDE. WIND TIDE is usually re¬ served for use on the ocean and large, bodies of water, WIND SET-UP is usually reserved for use on reservoirs and smaller bodies of water (See Figure A-ll) A-38 WIND TIDE - See WIND SET-UP WINDWARD - The direction from which the wind is blowing, WIND WAVES (l) Waves being formed and built up by the wind, (2) Loosely, any wave generated by wind. February 1957 A-39 A-40 Wave crest L = Wave Length i Direction of Wove travel t'^^ve trough Still woter level d- depth Wave Trough WAVE CHARACTERISTICS Oceon bottom ,r ;v\V.V\avv\^WA\V 5»V\vv//-WA=\\\Jl\g\ V//’A\W=\\\\\V-/\\Wa///VJ\///A^\ arWW-a/WwswvN^/^w^/gwvww = //\\v&\\\V Wave direction->- L ^ r BEACH GRASS SHOWS THE DIRECTION OF MOVEMENT OF WATER PARTICLES UNDER VARIOUS PARTS OF A SHALLOW WATER WAVE. (Wiegel,l953) FIGURE A-3 WAVE CHARACTERISTICS AND DIRECTION OF WATER PARTICLE MOVEMENT A- 41 SPILUBS BBKAKffi OF SPILLING BREAKERS PLDKiR maucs SKETCH SHOWING THE GENERAL CHARACTER OF PLUNGING BREAKERS SUHGIXG BBXAESB I SKETCH SHOWING THE GENERAL CHARACTER OF SURGING BREAKERS Both Photographs And Diagrams Of The Three Types Of Breakers Are Presented Above, The Sketches Consist Of A Series Of Profiles Of The Wave Form As It Appears Before Breaking, During The Breoking ,, And After Breaking. The Numbers Opposite The Profile Lines Indicate The Relative Times Of The Occurrences. FIGURE A-4 BREAKER TYPES A-42 Pt, Pinos. California UaTes BOTing oTor a subBarine ridga conoentrate to giTa larga ■are heighta on a point. j Halfapon Bar. California Note the increaaing width of the surf zone with increasing degree of eiposure to the south. PurieiBs Pt.. California fief Faction of waves around a headland produces low waves a narrow surf zone where banding is greatest. FIGURE A-5 REFRACTION OF WAVES (Wiegel, 1953) A-43 FIGURE A-6 REFRACTION DIAGRAM A-44 FIGURE A-7 BEACH FEATURES (Wiegel, 1953) A-45 FIGURE A-8 SHORELINE FEATURES (Wiegel , 1953 ) A-46 FIGURE A-9 bar AND BEACH FORMS A-47 SEMI - DIURNAL RGURE A-IOTYPES OF TIDES (Wiegal, 1953) A-48 Ospth, m. Wind Tide FIGURE A-11 (After Hellstrom, 1941) February 1957 A-49 p' g.' /I i >‘ 1 Cl ’•'ll i. ■ lwtg Pr ,.' ,'* k ■ ■i ^ ■ f ij. -■■>4 ;t;' ■ ' V ‘ “ V* r . I » ;-*j «i •! - J ; \'%^\v r J ; i x 1 • w^>‘"' ^ ’ \y ^ ' S'- ■ :. '':d^^T^\ . '-^•f ■''HWS' He ‘1^'-T3^ r- -f— !; -A 3«IU0t^ • ’ . ■ ' ■ . ■ ■* - Itiw< ‘ 1-^ 1 .. ...* APPENDIX B LIST OF COMMON SYMBOLS B APPENDIX B LIST OF SYMBOLS AS USED IN THIS REPORT Basic Units (in P, L, Example in Symbol Definition T System) American Units A Area l2 feet^ a Acceleration, Also: L/r^ ft/sec Amplitude L f eet ^2 Horizontal displacement of a water particle L f eet a*.as Length of semi-major axis of orbit of water particle L feet "a Subscript ”a” may refer to active earth pressures — — ‘*av Subscript ”av** refers to average — — B Breakwater gap width L feet B Rise in lake level L feet Bo Height of wind setup above mean Lake level L feet B.F. Beaufort wind force tmam — b Length of wave crest between or- thogonals (measured perpendicular to the local direction of travel) L feet t>z Vertical displacement of a water particle L feet b’.bg Length of semi-minor axis of orbit of water particle L feet “b Subscript ”b" refers to breaking wave conditions May 1961 B-l Symbol Definition Basic Units (in F, L, T System) Example in American Units C ,c Wave velocity L/T ft/sec, knots ^d»S Coefficient of drag ~ — Group velocity L/T ft/sec, knots Ch Velocity of waves of finite height L/T f t/sec Cm Coefficient of mass — — Co Deep water wave velocity L/T ft/sec (knots as noted) D Decay distance L miles or nautical miles Also: Diameter Also: A depth L L feet feet Dd Coefficient of shoaling (H/H©) — — Effective decay distance L miles or nautical miles Correction factor — — d Depth of water, measured from the still water level to the bottom L feet or fathoms ‘'b Depth of water at a breaker's position L feet or fathoms E Portion of total energy of one wave per unit length of crest transmitted forward with the wave form LFA f t-lbs/f t of crest Ef Related to wave energy l2 feet^ Ek Mean kinetic energy of one wave per unit length of crest LFA ft-lbs/ft of crest Moy 1961 B-2 p t fbi ^j> Symbol Definition Mean potential energy of one wave per unit length of crest Mean total energy of one wave per unit length of crest P Force, one of the basic units Also: a function of one or more variables, as P(x,y), Also: fetch length Fp Total horizontal drag force on a pile at a given instant F^ Maximum value of Fp for a given wave P(H) Percent of wave heights below the height H '■h ■■i Pm Horizontal component of force Total horizontal inertial force on a pile at a given instant Maximum value of F£ for a given wave Maximum value of Pp and P^ combined Minimum fetch length Py Vertical component of force Pjj(x) Prob, (X^x) * cumulative distri¬ bution function of the random variable X, f Function of one or more variables, as f(x,y) Also: Coriolis parameter (f = 2 sin 0) Also: Frequency Basic Units (in P, L, Example in T System) American Units LPA- ft-lbs/ft of crest LP/L ft-lbs/ft of crest P pounds L miles or nautical miles P pounds P pounds F pounds P F P L F pounds pounds pounds miles or nautical miles pounds lA radian/sec i/r 1/secs. May 1961 B-3 Symbol Definition Basic Units (in P, L, T System) Example in American Units f Friction factor — — f’c Compressive strength of concrete psi Horizontal drag force per unit length of vertical pile FA pounds/square f . 1 Horizontal inertial force per unit length of vertical pile PA f oot pounds/square Also: Minimum wave frequency lA foot waves/second ^max Maximum wave frequency lA waves/second fL Lower limit of wave frequency lA wave s/second fo Upper limit of wave frequency lA waves/second g Acceleration of gravity LA^ f t/sec^ Also: Function of one or more variables, as g(x,y) — — H Wave height L feet ”VlO Also: Horizontal component of force P pounds Average height of the highest one- tenth of the waves for a specified period of time L feet ”V3 Also: Hg Average height of the highest one- third of the waves (significant wave height) for a specified period of time L feet Hb Wave height on breaking L feet »D Significant wave height at end of decay distance L feet Hp Significant wave height at end of fetch L feet May 196 B-4 Symbol ^av ^ax H h Definition Average of the wave heights for a specified period of time Highest wave for a specified period of time Horizontal component (force etc.) A height Also: Elevation of wave trough above bottom Height of the mean level of the clapotis (orbit center) above SWL Distance between two underwater contours as used in the orthogonal method of wave refraction coefficient determination Sub-surface pressure response coefficient Also: Any constant Also: Refraction coefficient Damage coefficient Also: Wave force factor for drag effect on a pile at a given phase position “■Dm Maximum value of Kp for a wave of given relative height and depth Also;K, Refraction coefficient Refraction coefficient for breakers Wave force factor for inertial effect on a pile at a given phase position Basic Units (in P, T Svst< L L L L L L L/T^ L/r^ L/t2 Example in American Units feet feet feet feet feet feet feet/sec^ feet/sec2 feet May 1961 B-5 Symbo1 K* i k L La Lb Lp L o L s L* Basic Units (in P, L, Definition T System) Maximum value of for a wave of given relative height and depth L/T^ Diffraction coefficient Also: Dimensionless coefficient (Iribarren) -- Experimental coefficient for quarrystone — Beach slope, as specified L/L Also: >^-1 V m2 ♦ n^ where m * 2TrA-( long- crested waves), and n = 2ffL* lA (short-crested waves) lA Wave length (distance between two successive crests in the direction of propagation) L Also: Length, one of the basic units L Airy approximation of wave length L Wave length of breaking L Wave length at end of decay L Also: Wave length at deeper water depth (Minikin formula) L Wave length at end of fetch L Deep water wave length (Airy theory) L Shallow water wave length (Airy theory) L Wave length in the crest direction (Short-crested theory) L Example in American Units feet/sec2 Ver, rise (ft) Hor, dist, (ft) 1/feet 1/feet feet feet feet feet feet feet feet feet feet feet May 1961 B-6 Symbol Definit ion Basic Units (in P, L, T System) Example in American Units i a length L feet V crest length (the linear distance measured along the wave crest be¬ tween consecutive intersections of the crest and the still water level) L feet M Mass, a basic unit in the (M,L,T) ^ PT2 n > system slugs Also; Energy coefficient — MM Also: Moment LP foot-pounds ^Dm Externail moment on pile, about the bottom, associated with maximum drag force LP foot-pounds External moment on pile, about the bottom, associated with the maximum inertial force LF foot-pounds m 2Tr/L 1/L 1/feet “m In wave forecasting subscript ”m" refers to minimum conditions N A number — — n Number of layers of armor units — — Also: Ratio of group velocity to wave velocity MM Also: Crest interval in refraction drawings MM MM Also: 2 ttA-' lA 1/feet Also: Fraction of wave energy propogated with wave velocity LP/F ft-lbs/ft of Also; Number of standing waves or nodes — crest ”o Subscript "o” refers to deep water conditions May 1961 B-7 Basic Units (in P, L, Example in Symbol Definition T System) Anerican Units P Power transmitted by one wave per LPA f t-lbs/sec unit length of crest L per foot Also: Pressure FA^ pounds/Square foot Also: Force F pounds P Probability — — Also: Sub-surface pressure associated with wave motion FA^ pounds/square foot Also: Atmospheric pressure fA-l milibars Also: Any pressure FA^ pounds/square foot Also: Angle of earth fill surface to the horizontal MB V degrees Prob (A) or p (A) The probability of a statement^ A — ~ n Subscript "p” refers to passive K earth pressures “ — q Water particle velocity in direction of interest LA f t/sec R A resultant force F pounds Also: Distance between contours measured along an orthogonal L feet Also: Run-up L feet Reynolds number — — r Radius Also: Coefficient of energy partition Also: Distance from end of breakwater — ~ to point (x,y) in diffraction theory L feet Also: Thickness of armor unit layer L feet S Distance above the ocean bottom L feet May 1961 B-8 Symbol Basic Units (in F, L, Example in Definition T System) American Units S Sd s, 1 Sr s s T Also: Sum of the real parts of dif¬ fraction function X ® Also: Specific gravity Vertical position of action of total drag force on pile above bottom L feet Vertical position of action of total inertial force on pile above bottom L feet Specific gravity Sheltering coefficient lA-^ 1/ft^ Also: Specific gravity Also: Real part of diffraction Function f(-u) = s V iw Also: Slope as specified L/L Ver, rise (ft) Hor, dist, (ft) Subscript ”s” refers to surface terms Wave period; seiche period T seconds Also: Temperature Also: Time, a basic unit T degrees F, seconds T (or T) Average wave period T T^jTg Significant wave period T Tp Significant wave period at end of decay T Tp Significant wave period at end of fetch T T£ Maximum wave period T T^ Lower limit of significant period T Ty Upper limit of significant period T Tmax Period of wave of greatest energy in energy-frequency spectrum T seconds seconds seconds seconds seconds seconds seconds seconds May 1961 B-9 Symbol Definition Basic Units (in F, L, T System) Example in American Uni t A time T seconds Travel time of waves (from end of fetch to end of decay distance) T hours td Wind duration, interval of time wind blows at constant velocity in generating waves T hours t Modulus of wave decay due to viscosity T hours Minimum duration of wind in fetch area T hours U Velocity of surface wind L/T knots “U Velocity of mass transport LA f t/sec U* Horizontal velocity of motion left after wave motion has been destroyed (Gerstner Theory) LA f t/sec Also: i'^proximate velocity of surface wind LA knots u Water particle (horizontal component positive in the direction of wave advance) orbital velocity LA f t/sec Also: Upper limit of integral term in solutions of diffraction problem — ~ u Mass transport velocity LA feet/second du/dt Total horizontal acceleration of fluid particles at a given point LA^ feet/second^ dvi/dt Local horizontal acceleration of fluid particles at a given point LA^ p feet/second V A velocity LA f t/sec Also: A volume L^ feet^ vp Velocity of storm or fetch front LA knots May 1961 B-IO Definition Basic Units (in P, L, T System) Example in American Units Geostrophic wind velocity L/T knots, m.p.h. Vertical component (force etc.) — ~ Water particle (vertical component orbital velocity) L/T ft/sec Weight F pounds Also: Work performed by one wave per unit length of crest LF/L f t-lbs/f t of crest Unit weight Ibs/ft^ Also: Imaginary part of diffraction function f(“u) = S ♦ iw — — Unit weight of rock P/L^ Ibs/f t^ Unit weight of water F/L^ Ibs/ft^ Extraneous force in x direction F pounds Coordinate, usually horizontal L feet Also: Horizontal distance in direction of wave travel rela¬ tive to a crest. L feet Same as x but measured from a fixed point L feet Equilibrium position of particle, horizontal coordinate L feet Horizontal displacement of particles from equilibrium position L feet Extraneous force in y direction F pounds A coordinate, usually vertical L feet Equilibrium position of particle vertical coordinate L feet Symbol “y Z Zt z* a (alpha) ^3 P (beta) May 1961 Basic Units (in F, L, Examples in Definition T System) American Units Vertical displacement of particle from equilibrium position L feet Extraneous force in z direction P pounds Also: Section modulus per length of wall ft^ Time between successive weather maps T hours Also: Greenwich mean time T hours A coordinate, usually horizontal perpendicular to x and y L feet Also: Elevation measured upward from still water level (forces on piles) L feet Elevation measured upward from the bottom L feet Angle of wave crest to bottom con- tours — degrees Also; Angle of wave approach, mea¬ sured between the shore line and the line of wave advance degrees Also: Angle between gradient and surface winds degrees Also: Phase difference between axis of **f” and ”g” terms in diffraction theory radians Also: Angle of breakwater slope with horizontal — degrees Skewness coefficient ~ — Angular position of the water particle when the maximum horizontal particle velocity occurs degrees Also: Wave age, the ratio of wave velocity to the velocity of the generating wind Also: An angle — degrees Also: Phase position at which =Piai — radians B-12 Symbol Definition Basic Units (in P, L, T System) Example in American Units Po Value of p for the Airy theory radians 72 (gcuama) Resistance coefficient applicable to wind ~ — A (delta) Change — as stated 8 (delta) Wave steepness, H/L — — V (eta) Wave surface elevation L feet Also: Elevation of surface above still water at position 0 L feet \ Elevation of crest above still water level L feet 9 (theta) Angle of wall face with horizontal (for earth pressures) Mi degrees Also: Angulcir displacement Also: Temperature, a basic unit Also: Angle of variability in the direction of wave travel Also: Angle of the face of a structure with the horizontal — Radians degrees P, degrees degrees X (lamda) Wave length L feet (mu) Arithmetic mean Also: Absolute viscosity Also: Coefficient of friction PT/L^ appropriate units Ib-sec/f V (nu) Kinematic viscosity l?/T f t^/sec TT (pi) 3,1416 - Ratio of circumference of a circle to the diameter — — P Correlation coefficient — — (rho) Pa Also: Mass density, any substance Mass density of air pt^/l"^ (MA^) PT^A"^ (MA^) slugs/f t^ slugs/f t^ May 1961 B-13 Symbol Definition Basic Units (in F, L, T System) Example in American Units Mass density of water (W/g) ft^/l"^ (Salt water = 2,00 #/ft^/sec^) (MA^) slugs/ft^ cr (Fresh water* 1,96 ff/ft^/sec^) Standard deviation appropriate (sigma) Also: Ztx/T lA units radians/sec a w Surface tension of water LPA^ ft-lbs/ft^ T Drag force per unit area FA^ Ibs/ft^ (tau) 0 Velocity potential L^A ft^/sec (phi) Also: Angle of internal friction of earth degrees Also: Phase of diffracted wave ~ radians (psi) Azimuth of direction of wave travel, in the direction of travel degrees Also: Stream function L^A ft^/sec Also: Direction from which the wave comes Compass di- a Angular velocity of the earth lA rection (true) radians/sec (omega) Angular velocity (2 tt/T) 1/T radians/sec (omega) May 1961 B-14 APPENDIX C BIBLIOGRAPHY C APPENDIX C BIBLIOGRAPHY Anderson, H. 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Beach Erosion Board, Corps of Engineers, "By-passing Littoral Drift at a Harbor Entrance", The Bulletin of the Beach Erosion Board, Volume 5, No. 3, 1 July 1951. Benezit, M. "Treatise on Vertical Ocean Breakwaters", Annals des Fonts et Chausses, Paris, France, pp. 125-160, 1923. Translated by Lieutenant L. H. Hewitt, Corps of Engineers, U. S. Arny. Blue, F. L. and J. W. Johnson, "Diffraction of Water Waves Passing Through A Breakwater Gap", Transactions American Geophysical Union, Vol. 30, No. 5, October 1949. Boase, A. J., "Shore Protection by Permeable Groins", Shore and Beach, Vol. vii. No. 3, July 1939. Bowditch, N., American Practical Navigator, U. S. Hydrographic Office, 1943. 14. Bretschneider, C. L., "Revised Wave Forecasting Relationships", Proceedings of Second Conference on Coastal Engineering, 1952. 15. Brown, C. V., "Relationship of Beach Slopes to Sand Characteristics", Beach Erosion Board, 16 December 1937 (unpublished). 16. Carr, J. H., "Mobile Breakwater Studies", California Institute of Technology, Report No. N-64.2, December 1950. 17. Chaney, C. A., "Marinas - Recommendations for Design, Construction, and Maintenance", The National Association of Engine and Boat Manufacturers, Inc., 1939. 18. Chrystal, G., "On the Hydrodynamical Theory of Seiches, with a Bibliographical Sketch", Transactions of the Royal Society of Edinburgh, Vol. XLI, Part 3, No. 25,(London, England, Robert Grant and Son, 1905K 19. Chrystal, G., "Some Results in the Mathematical Theory of Seiches", Proceedings of the Royal Society of Edinburgh, Session 1903- 1904 , Vol. XXV, Part 4? (London, England, Robert Grant and Son, 1904 ), pp. 328-337. 20. Coast and Geodetic Survey, U. S. Department of Gommerce, Tide and Current Glossary, Special Publication No. 228, Revised (1949) Ed. 21. Goast and Geodetic Survey, U. S. Department of Commerce, Tide Tables, East Coast, North and South America, 1952. 22. Coast and Geodetic Survey, U. S. Department of Commerce, Tide Tables, West Coast, North and South America, 1952. 23 . Corps of Engineers, U. S. Amy, "Retaining Walls”, Engineering Manual for Civil Works, Part CXXV, Chapter 2, July 1945. 24 . Corps of Engineers, U. S. Army, "Beach Erosion Studies", Engineering Manual for Civil Works, Part CXXXIII, April 1947. 25 . Corps of Engineers, U. S. Army, "Stresses and Criteria for Structiiral Design", Engineering Manual for Civil Works, Part CXXI, Chapter 1, August 1947 . 26. Corps of Engineers, U. S. Army, "Wall Design", Engineering Manual for Civil Works, Part CXXV, Chapter 1, January 1948. 27 . Corps of Engineers, U. S. Arny, "Harbor and Shore Protection in the Vicinity of Port Hueneme, California", Los Angeles District, 1 October 1948. 28. Corps of Engineers, U. S. Amy, "Design of Miscellaneous Structures - Breakwaters and Jetties", Engineering Manual for Civil Works, Part CXXIX, Chapter 4. C-2 29. Corps of Engineers, U. S. Army, ’’Report on Beach Erosion Control of Illinois Shore Line, Lake Michigan", Chicago District, 1 June 1949 . House Doc. 28, 83rd Congress. 30 . Corps of Engineers, U. S. Army, "Beach Erosion Control Report on Cooperative Study of the Pacific Coast Line of the State of California, Point Mugu to San Pedro Breakwater", Los Angeles District, 1 September 1950. 31 . Corps of Engineers, U. S. Army, "Soil Mechanics Design", Engineering Manual for Civil Works, Part CXIX, Chapter 2, February 1952. 32 . Corps of Engineers, U. S. Army, "Hydraulic Design - Waves and Wave Pressiares", Engineering Manual for Civil Works, Part CXVI, Chapter 8, July 1952 (preliminary). 33. Corps of Engineers, U. S. Army, "Stability of Rubble-Mound Break¬ waters", Technical Memorandum No. 2-365, Waterways Experimental Station. 34 . Corps of Engineers, U. S. Army, "Report on Concrete Block Groins", New York District,letter 22 May 1952. 35 . Corps of Engineers, U. S. Am^r, "Measurement and Analysis of Sediment Loads in Streams" Reports No. 1-9, St. Paul District Sub-Office at Iowa City, Iowa (publishers) 36 . Creager, W. P., J. 0. Justin, and J. Hinds, "Engineering for Dams", Vol. II, John Wiley & Sons, 1945. 37. Defant, A., "Gezeithen problems des Meeres in Ia.ndnahe", Probleme der Kosmischen Physik, VI, Hamburg, 1925. 38. Dunham, J. W., "Refraction and Diffraction Diagrams", Proceedings of the First Conference on Coastal Engineering, 1951. 39 . Eaton, R. 0. "Littoral Processes on Sandy Coasts", Proceedings of the First Conference on Coastal Engineering, 1951. 40 . Einstein, H. A., "The Bed-Load function for Sediment Transportation in Open Channel Flows", Technical Bulletin No. 1026, U. S. Department of Agricultvire, September 1950. 41 . Einstein, H. A., "Estimating Quantities of Sediment Supplied by Streams to a Coast", Proceedings of First Conference on Coastal Engineering, 1951. 42 . Gaillard, D. D., "Wave Action Relation to Engineering Structures", Corps of Engineers, U. S. Army, Professional Paper No. 31, 1904. C-3 43. Gesler, E. E., R. 0. Eaton, and J. V. Hall, Jr., "Report on Break¬ waters", XVIII International Navigation Congress,Ocean Navigation Section, Question I, Rome 1953. 44. Hall, F. Jr., "A Review of the Theory of Breakwater Construction, with Comments on Notable Breakwater Failiires", (Thesis) Cornell University, 1938. 45. Hall, J. V., Jr., "Artificially Nourished and Constructed Beaches", Technical Memorandiun No. 29 , Beach Erosion Board, Corps of Engineers, December 1952. 46 . Handin, J. W. and J. C. Ludwick, "Accretion of Sand Behind a Detached Breakwater", Technical Memorandum No. I 6 , Beach Erosion Board, Corps of Engineers, 1950. 47 . Handin, J. W., "The Geological Aspects of Coastal Engineering", Proceedings of First Conference on Coastal Engineering, 1951. 48 . Harris, R. A., "Currents, Shallow-water Tides, Meteorological Tides, and Miscellaneous Matters", Manual of Tides, Part V, Report of the Superintendent of the Coast and Geodetic Survey, Appendix 6, Washington, D. C., 1907. 49 . Hedar, Per Anders, "Design of Rock-fill Breakwaters">Proceedings Minnesota International Hydraulic Convention, September 1953. 50 . Hellstrom, B., "Wind Effect on Lakes and Rivers" Handlenger Ingeniors Vetenkass Akademien, No. 158, 1941. yl. Henry, A. J., "Wind Velocity and Fluctuations of Water Level of Lake Erie”, Weather Bureau, U. S. Department of Agricultiire, 1902. 52 . Hidaka, K., "Seiches due to a Submarine Bank: A Theory of Shelf Seiches; and Seiches in a Channel”, Memoirs of the Imperial Marine Observatory, Kobe, Japan, 1934-1935. 53 . Hitchcock, A. S.,"Manual of Grasses for the United States", Miscellaneous Publication No. 6 OO, Department of Agriculture. 54 . Honda, K., and others, "Secondary Undulations of Oceanic Tides", Journal of the College of Science, Imperial University, Tokyo, Japan, Vol. 24, (Tolgro, A. P. Zaruya and Co. Ltd, Tori Sanchome, Nihonbashi, 1908). 55 . Hudson, R. Y., "Wave Forces on Breakwaters", Transactions of the American Society of Civil Engineers, Vol. 118, 1953, p. 653. 56 . Inman, D. L., "Aerial and Seasonal Variations in Beach and Nearshore Sediments at laJo11a,California", Technical Memorandum No. 39, Beach Erosion Board, Corps of.Engineers, 1953. C-4 57. Inman, D. L., "Sorting of Sediments in the Light of Fluid Mechanics", Journal of Sedimentary Petrology, Vol. 19, pp. 51-70, August 1949. 58. Iribarren Cavanilles, R., "A Formula for the Calculation of Rock Fill Dikes", Revista de Obras Publicas, July 1938. Trans¬ lation in The Bulletin of the Beach Erosion Board, Vol. 3, No. 1, January 1949. 59. Iribarren Cavanilles, R., and C. Nogales y Olano, "Generalization of the Formula for Calculation of Rock F^l Dikes and Verification of its Coefficients", Revista de Obras Publicas May 1950, Translation in the Bulletin of the Beach Erosion Board, Vol. 5, No. 1, January 1951. 60. Iversen, H. W., "Discussion of Results from Studies of Wave Trans¬ formation in Shoaling Water”, Technical Report Series 3, No. 331, University of California, Institute of Engineering Research. 61. Iversen, H. W., R. G. Crooke, M. J. Larocess, and R. L. Wiegel, "Beach Slope Effect on Breakers and Stirf Forecasting",Technical Report 155-38, University of California, December 1950, (Restricted) 62. Johnson, A. G., "Santa Monica Bay Shore Line Development Plans", Proceedings of First Conference on Coastal Engineering, 1951. 63. Johnson, D. W., Shore Processes and Shore Line Development, John Wiley and Sons, Inc., 1919. 64 . Johnson, J. W., and M. 0. O’Brien, and J. D. Isaacs, "Graphical Construction of Wave Refraction Diagrams", Publication No. 605, U. S. Hydrographic Office, U. S. Navy Department, January 1948. 65 . Johnson, J. W., "Engineering Aspects of Diffraction and Refraction", Proceedings of the American Society of Civil Engineers, Vol. 118, 1953. 66. Johnson, J. W. "Generalized Wave Diffraction Diagrams", Proceedings of First Confer.ence on Coastal Engineering, 1951. 67 . Kaplan, K., "Effective Height of Seawalls", The Bulletin of the Beach Erosion Board, Vol. 6, No. 2, April 1952, pp. 1-18. 68. Kaplan, K. "Analysis of Moving Fetches for Wave Forecasting", Technical Memorandum No. 35, Beach Erosion Board, Corps of Engineers, 1953. 69 . Kaplan, K. and H. E. Pape, Jr., "Design of Breakwaters", Proceedings of First Conference on Coastal Engineering, 1951. C-5 70. Kauffmann, M., "Notes on Defensive Works on the French Coast", XV International Congress of Navigation, Ocean Navigation Section, 2nd Question, Venice 1931. 71. Kerr, R. C. and J. 0. Nigra, "Analysis of Aeolian Sand Control", Arabian American Oil Company, New York, N. Y., August 1951. 72. Keulegan, G. H., "Wind Tides in Small Closed Channels", Journal Res. Nat. Bur. Stds., RP2207, V. 46, N. 5, 1951. 73. Keulegan, G. H., "The Form Factor in Wind Tide Formulas", Nat. Bur. Stds., Rpt. 1835, 1952. 74* Kieuper, E., "The Construction of the Harlingen Breakwater", Permanent International Association of Navigation Congresses, B. Bulletin No. 34, Chapter 8, 1950. 75. Krecker, F. K., "Periodic Oscillations in Lake Erie", Franz Theodore Stone laboratory, Ohio State University, Contribution No. 1, 192 8. 76. Krumbein, W. C., and F. J. Pettljohn, "Manual of Sedimentary Petrography",D. Appleton-Century Co., Inc.,. 1938. 77. Krumbein, W. C., "Shore Currents and Sand Movements on a Model Beach", Technical Memorandum No. 7, Beach Erosion Board, Corps of Engineers, 1944- 78. Krumbein, W. 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'.AJ 1 |m '• '■' f .?. '. .■• .0101 V#t '«ni>o?T;-iijJi'N<; ;,Ui , •3tia6l'9»fit ^ tWft rj- kii,> ^osJ m ., ?!•■■• *, 1 V • ‘ * . * ■ t 1 j I ^9 * . V »’ cl E"*' iT'*rtot2 v-'fcaiituH ir^iityll’n'l a .ttoslth ,i.x%,yf' \vV •,; i.’ 1 ^ . •' ««r.'l£|S ’ ;*S '*' ‘ * !rVsK 'V ;1 m “ V I t • 'i- • it . : rff.t%>i-^ *-C‘TTT?5F7r7T«.-4jrt ‘l•#4 i, .Aij ••€!'', i '.f APPENDIX D MISCELLANEOUS TABLES AND GRAPHS D APPENDIX D MISCELLANEOUS TABLES AND GRAPHS PLATES AND TABLES „ Page Table D-1, Functions of d/L for even increments of d/L© D-3 to D-15 Table D-2, Functions of d/L for even increments of d/L D-15 to D-27 Plate D-l,x Relationship between wave period, length and depth D-28 & D-la. ^ & D-29 Plate D-2, Relationship between wave period, velocity and depth D-30 Plate D-3, Illustration of various functions of d/L^ D-31 Table D-3, Conversion Table for wind forces; Beaufort scale D-32 Table D-4, Sea Scale - D-32 Table D-5, Swell Scale D-33 Table D-6, International code 69 for wave period D-33 Table D-7, International code 42 for wave height D-33 Plate D-4, Relationship between wave energy, wave length, and wave height D-34 Table D-8, Deep water wave length and velocity in terms of period D-35 Table D-9, Values of A L/Lgy and Cj/C2 D-36 Plate D-5, Graphic determination of the weight of cap armor D-42 thru units in terms of wave height and side slope to Plate D-8. D-45 Table D-10, Kp Values D-46 Table D-11, Values of cot a and slope ratio for various slopes D-47 Table D-12, Values of tan^ D-47 Plate D-9, Determination of wave height and depth of water at point of breaking D-48 Plate D-10, Change in wave direction and height due to refraction on beaches with straight, parallel depth contours D-49 Plate D-11, Tetrapods - volume, weight, thickness of layers, and dimensions, D-50 Plate D-12, Tribars - volume, weight, thickness of layers, and dimensions. D-51 Plate D-13, Quadripods - volume, weight, thickness of layers, and dimensions, D-52 Plate D-14. Modified Cubes - volume, weight, thickness of layers, and dimensions D-53 Plate D-15, Hexapods - volume, weight, thickness of layers, and dimensions D-54 Moy 1961 D-l Table D-1 - Fiinctions of d/L for Even Increments of d/L^ (139) Table D-2 - Functions of d/L^ for Even Increments of d/L (139) d/L o d/L K ratio of the depth of water at any specific location to the wave length in deep water, ratio of the depth of water at any specific location to the wave length at that same location. a pressure response factor used in connection with under¬ water pressure instruments, where K . H'/H = P/P = cosh 2u(d-z)/L o , 2trd cosh —— where P is the pressure fluctuation at a depth z below still water, Pq is the surface pressure fluctuation, d is the depth of water from still water level to the ocean bottom, L is the wave length in any particular depth of water, and H’ is the corresponding variation of head at a depth z. The values of K shown in the tables are for the instrument placed on the bottom where K - 1 ” cosh 2 TTd/L n = the fraction of wave energy that travels forward with the wave form: i.e., with the wave velocity C rather than the group velocity C^. 1 ^ * 2 n is also the ratio of group velocity C„ to wave velocity C. 1 / 4Trd/L sinh Uird/L C„/C = ratio of group velocity to deep water wave velocity v/here (j o — = “c ^ C ” ” tanh 27rd/L o o H/H* = ratio of the wave height in shallow water to what its wave height would have been in deep water if unaffected by re¬ fraction. M 1 1 1 2 • n * C/C o = an energy coefficient defined as _7r2_ 2 tanh^ (2 TT d/L) D-2 TABLE D-1 FUNCTIONS OF d/L FOR EVEN IMCRE^ENTS OF d/L^ d/L d/L 2frd/L TANKS SINK from 0.0001 COSH K ' 0 2trd/L 2 77 d/L 2 77 d/L 0 0 0 0 0 1 1 .0001000 .003990 .02507 .02506 .02507 1.0003 .9997 .0002000 . 00561*3 .0351*6 .0351*1* .0351*7 1.0006 .9991* .0003000 .006912 .01*31*3 .01*31*0 .01*31*1* 1.0009 .9991 .0001*000 . 007982 .05015 .05011 .05018 1.0013 .9987 .0005000 .008925 .05608 .05602 .05611 1.0016 .9981* .0006000 .009778 .0611*1* .06136 .0611*8 1.0019 .9981 .0007000 .01056 .06637 .06627 .0661*2 1.0022 .9978 .0008000 .01129 .07096 .07081* .07102 1.0025 .9975 .0009000 .01198 .07527 .07513 .07531* 1.0028 .9972 .001000 .01263 .07935 .07918 .0791*3 1.0032 .9969 .001100 .01325 .08323 .08301* .08333 1.0035 .9966 .001200 . 01381* .08691* .08672 .08705 1.0038 .9962 .001300 .011*1*0 .09050 .09026 .09063 1.001*1 .9959 .0011*00 . 011*95 .09393 .09365 .091*07 1.001*1* .9956 .001500 .0151*8 .09723 .09693 .09739 1.001*7 .9953 .001600 .01598 .1001* .1001 .1006 1.0051 .991*9 .001700 .0161*8 .1035 .1032 .1037 1.0051* .991*6 .001800 .01696 .1066 .1062 .1068 1.0057 .991*3 .001900 .0171*3 .1095 .1091 .1097 1.0060 .991*0 .002000 .01788 .1123 .m9 .1125 1.0063 .9937 .002100 .01832 .1151 .111*6 .1151* 1.0066 .9931* .002200 .01876 .1178 .1173 .1181 1.0069 .9931 .002300 .01918 .1205 .1199 .1208 1.0073 .9928 .0021*00 .01959 .1231 .1225 .1231* 1.0076 .9925 .002500 .02000 .1257 .1250 .1260 1.0079 .9922 .002600 .0201*0 .1282 .1275 .1285 1.0082 .9919 .002700 .02079 .1306 .1299 .1310 1.0085 .9916 .002800 .02117 .1330 .1323 .1331* 1.0089 .9912 .002900 .02155 .1351* .131*6 .1358 1.0092 .9909 .003000 .02192 .1377 .1369 .1382 1.0095 .9906 .003100 .02228 .11*00 .1391 .11*05 1.0098 .9903 .003200 .02261* .1^23 .11*13 .11*27 1.0101 .9900 .003300 .02300 .11*1*5 .11*35 .11*1*9 i.oioi* .9897 .0031*00 .02335 .11*67 .ia56 .11*72 1.0108 .9893 .003500 .02369 .11*88 .11*77 .11*91* 1.0111 .9890 .003600 .021*03 .1510 .11*98 .1515 1.0111* .9887 .003700 .021*36 .1531 .1519 .1537 1.0117 .9881* .003800 .021*69 .1551 .1539 .1558 1.0121 .9881 .003900 .02502 .1572 .1559 .1579 1.0121* .9878 .001*000 .02531* .1592 .1579 .1599 1.0127 .9875 .001*100 .02566 .1612 .1598 .1619 1.0130 .9872 .001*200 .02597 .1632 .1617 .1639 1.0133 .9869 .001*300 .02628 .1651 .1636 .1659 1.0137 .9865 .001*1*00 .02659 .1671 .1655 .1678 1.011*0 .9862 .001*500 .02689 .1690 .1671* .1698 1.011*3 .9859 .001*600 .02719 .1708 .1692 .1717 1.011*6 . 9656 .001*700 .0271*9 .1727 .1710 .1736 1.011*9 .9853 .001*800 .02778 .171*5 .1728 .1751* 1.0153 .981*9 .001*900 .02807 .1761* .171*6 .1773 1.0156 .981*6 .005000 .02836 .1782 .1761* .1791 1.0159 .981*3 .005100 .02861* .1800 .1781 .1809 1.0162 .981*0 .005200 .02893 .1818 .1798 .1827 1.0166 .9837 .005300 .02921 .1835 .1815 .181*5 1.0169 .9831* .0051*00 .0291*8 .1852 .1832 .1863 1.0172 .9831 .005500 .02976 .1870 .181*8 .1880 1.0175 .9828 .005600 .03003 .1887 .1865 .1898 1.0178 .9825 .005700 .03030 .1901* .1881 .1915 1.0182 .9822 .005800 .03057 .1921 .1897 .1932 1.0185 .9818 .005900 .03083 .1937 .1913 .191*9 1.0188 .9815 to 1.000 UTTd/L SIKH COSH ” Cq/C h/h; u l*7rd/L 1+rd/L \r 0 0 0 0 1 1 0 oC oc .05011* .05016 1.001 .9998 .02506 a. 167 7,855 .07091 .07097 1.003 .9996 .035a3 3.757 3,928 .08686 .08697 l.OOl* .999a .oa336 3.395 2,620 .1003 .1005 1.005 .9992 .05007 3.160 1,965 .1122 .1121* 1.006 .9990 .05596 2.989 1,572 .1229 .1232 1.006 .9988 .06128 2.856 1,311 .1327 .1331 1.009 .9985 .06617 2.11x9 1,12a .1149 .11*21* 1.010 .9983 .07072 2.659 933.5 .1505 .1511 1.011 .9981 .07a99 2.582 87a.3 .1587 .1591* 1.013 .9979 .07902 2.515 787.0 .1665 .1672 1.011* .9977 .08285 2.a56 715.6 .1739 .171*8 1.015 .9975 .08651 2.aoa 656.1 .1810 .1820 1.016 .9973 .09001 2;357 605.8 .1879 .1890 1.018 .9971 .09330 2.31a 562.6 .191*5 .1957 1.019 .9969 .09663 2.275 525 .2009 .2022 1.020 .9967 .09977 2.239 193 .2071 .2086 1.022 .9965 .1028 2.205 163 .2131 .211*7 1.023 .9962 .1058 2.17a 138 .2190 .2207 1.021* .9960 .1087 2.ia5 ai5 .221*7 .2266 1.025 .9958 .Ilia 2.119 39a .2303 .2323 1.027 .9956 .liai 2.09a 376 .2357 .2379 1.028 .995a .1161 2.070 359 .21*10 .21*33 1.029 .9952 .1193 2.01 x 1 313 .21*62 .2^87 1.031 .9950 .1219 2.025 329 .2513 .251*0 1.032 .99a8 .I2a3 2.005 316 .2563 .2592 1.033 .99a6 .1268 1.986 301 .2612 .261*2 1.031* .99aa .1292 1.967 292 .2661 .2692 1.036 .99a2 .1315 1.950 282 .2708 .271*1 1.037 .9939 .1338 1.933 212 .2755 .2790 1.038 .9937 .1360 1.917 263 .2800 .2837 1.01*0 .9935 .1382 1.902 255 .281*5 .2881* 1.01*1 .9933 .laoa 1.887 217 .2890 .2930 1.01*2 .9931 .ia25 1.873 210 .2931* .2976 1.01*3 .9929 .iaa6 1.860 233 .2977 .3021 1.01*5 .9927 .ia66 i.sa? 226 .3020 .3065 1.01*6 .9925 .ia87 1.83a 220 .3061 .3109 1.01*7 .9923 .1507 1.822 21a .3103 .3153 1.01*9 .9921 .1527 1.810 208 .311*1* .3196 1.050 .9919 .i5a6 1.799 203 .3181* .3238 1.051 .9917 .1565 1.788 198 .3221* .3280 1.052 .9915 .158a 1.777 193 .3263 .3322 1.05a .9912 .1602 1.767 189 .3302 .3362 1.055 .9910 .1621 1.756 181 .331*1 .31*03 1.056 .9908 .i6ao 1.7a6 180 .3380 .31*1*1* 1.058 .9906 .1658 1.737 176 .3^7 .31*83 1.059 .990a .1676 1.727 172 .31*51* .3523 1.060 .9902 .1693 1.718 169 .31*91 .3562 1.062 .9900 .1711 1.709 165 .3527 .3601 1.063 .9898 .1728 1.701 162 .3561* .361*0 1.06a .9696 .17a6 1.692 159 .3599 .3678 1.066 .989a .1762 1.68a 156 .3635 .3715 1.067 .9892 .1779 1.676 153 .3670 .3753 1.068 .9889 .1795 1.669 150 .3705 .3790 1.069 .9887 .1811 1.662 117 .3739 .3827 1.071 .9885 .1827 1.65a 115 .3771* .3861* 1.072 .9883 .18a3 1.6a7 112 .3808 .3900 1.073 .9881 .1859 i.6ao ilo .381*1 .3937 1.075 .9879 .187a 1.633 137 .3875 .3972 1.076 .9877 .1890 1.626 135 ♦Also; bs/as, C/Co> L/Lg D-3 Table D-1 Cont'd d/L„ d/L 2^^ d/L TANH 27rd/L SINK 2^ d/L COSH K a-^dA 2 Trd/L SINK a^^dA COSH n C A hTTd/L Q ° h/h; M .006000 .03110 .195a .1929 .1967 1.0192 .9812 .3W8 .aoo8 1.077 .9875 .1905 1.620 133 130 128 .006100 .03136 .1970 .19a5 .1983 1.0195 .9809 .39ai .aoaa 1.079 .9873 .1920 1.6ia .006200 .03162 .1987 .1961 .2000 1.0198 .9806 .3973 .a079 1.080 .9871 .1935 1.607 .006300 .03188 .2003 .1976 .2016 1.0201 .9803 .ao06 .ana 1.081 .9869 .1950 1.601 126 .006li00 .03213 .2019 .1992 .2033 1.0205 .9799 .ao38 .aia® 1.083 .9867 .1965 1.595 12a .006500 .03238 .2035 .2007 .20a9 1.0208 .9796 .a070 .ai83 1.08a .9865 .1980 1.589 1.583 123 121 .006600 .0326a .2051 .2022 .2065 1.0211 .9793 .aioi .a2i7 1.085 .9863 .199a .006700 .03289 .2066 .2037 .2081 1.021a .9790 .ai33 .a25i 1.087 .9860 .2009 1.578 119 .006800 .03313 .2082 .2052 .2097 1.0217 .9787 .ai6a .a285 1.088 .9858 .2023 1.572 117 .006900 .03338 .2097 .2067 .2113 1.0221 .978a .ai95 .a3i9 1.089 .9856 .2037 1.567 116 .007000 .03362 .2113 .2082 .2128 1.022a .9781 .a225 .a352 1.091 .985a .2051 1.561 na ,007100 .03387 .2128 .2096 .2iaa 1.0227 .9778 .a256 .a386 1.092 .9852 .2065 1.556 112 ,007200 .03ail .2ia3 .2111 .2160 1.0231 .mix .a286 .aai9 1.093 .9850 .2079 1.551 111 ,007300 .03a35 .2158 .2125 .2175 1.023a .9771 .a316 .hh 52 1.095 .98a8 .2093 1.5a6 109 ,007li00 .03a59 .2173 .2139 .2190 1.0237 .9768 .a3a6 .aasa 1.096 .98a6 .2106 i.5ai 108 ,007500 .03a82 .2168 .215a .2205 1.02ao .9765 .a376 .a5i7 1.097 .98aa .2120 1.536 106 ,007600 .03506 .2203 .2168 .2221 i.02aa .9762 .aao6 .a5a9 1.099 .98a2 .213a 1.531 105 007700 .03529 .2218 .2182 .2236 1.02a7 .9759 .aa35 .a582 1.100 .98ao .2ia7 1.526 10a 007800 .03552 .2232 .2196 .2251 1.0250 .9756 .aa6a .a6ia 1.101 .9838 .2160 1.521 102 007900 .035/6 . 221 x 1 .2209 .2265 1.0253 .9753 .aa93 .a6a6 1.103 .9836 .2173 1.517 101 .008000 .03598 .2261 .2223 .2280 1.0257 .9750 . 1 x 522 .a678 i.loa .983a .2186 1.512 100 .008100 .03621 .2275 .2237 .2295 1.0260 .97a7 .a55l .a709 1.105 .9832 .2199 1.508 98.6 .008200 .036aa .2290 .2250 .2310 1.0263 .97aa .a579 .a7ai 1.107 .9830 .2212 1.503 97.5 ,008300 .03666 .230a .226a .232a 1.0266 .97ai . 1 x 601 . hn 2 1.108 .9827 .2225 i.a99 96.3 ,0081400 .03689 .2318 .2277 .2338 1.0270 .9131 . 1 x 636 .a8o3 1.109 .9825 .2237 i.a95 95.2 ,008500 .03711 .2332 .2290 .2353 1.0273 .973a .a66a .a83a 1.111 .9823 .2250 i.an 9a.1 ,006600 .03733 .23a6 .2303 .2367 1.0276 .9731 .a691 .a865 1.112 .9821 .2262 i.a87 93.0 ,008700 .03755 .2360 .2317 .2381 1.0280 .9728 .a719 .a896 1.113 .9819 .2275 i.a82 91.9 008800 .03777 .231 y .2330 .2396 1.0283 .9725 .a7a7 .a927 1.115 .9817 .2287 i.a78 90.9 008900 .03799 .2387 .23a3 .2aio 1.0286 .9722 .huh .a957 1.116 .9815 .2300 i.a7a 89.9 .009000 .03821 .2aoi .2356 . 2 lx 2 lx 1.0290 .9718 .aaol .a988 1.118 .9813 .2312 i.a7i 88.9 .009100 .038a2 .2aia .2368 .2a38 1.0293 .9815 .a828 .5018 1.119 .98U .232a i.a67 88.0 .009200 .0386a . 2 h 2 d .2381 .2a52 1.0296 .9712 .a855 .5oa9 1.120. .9809 .2336 i.a63 87.1 .009300 .03885 .2aai .239a .2a65 1.0299 .9109 .h662 .5079 1.122 .9807 .23a8 i.a59 86.1 .009ii00 .03906 .2a55 .2ao7 .2a79 1.0303 .9706 .h909 .5109 1.123 .9805 .2360 i.a56 85.2 .009500 .03928 . 21 x 66 .2ai9 .2a93 1.0306 .9703 .a936 .5138 1.12a .9803 .2371 i.a52 8a.3 .009600 .039a9 .2a8i .2a31 .2507 1.0309 .9700 .a962 .5168 1.126 .9801 .2383 i.aa8 83.5 .009700 .03970 .2a9a . 2 Ux 3 .2520 1.0313 .9697 .a988 .5198 1.127 .9799 .239a i.aa5 82.7 .009800 .03990 .2507 . 21 x 56 .253a 1.0316 .969a .501a .5227 1.128 .9797 .2a06 i.aaa 81.8 .009900 .oaoii .2520 . 21 x 66 .25a7 1.0319 .9691 .5oao .5257 1.130 .979a .2ai7 i.a38 81.0 .01000 .oao32 .2533 .2a80 .2560 1.0322 .9688 .5066 .5286 1.131 .9792 .2a29 i.a35 80.2 .01100 .oa233 .2660 .2598 .2691 1.0356 .9656 .5319 .557a l.ia5 .9772 .2539 i.ao3 73.1 .01200 .oaa26 .2781 .2711 .2817 1.0389 .9625 .5562 .5853 1.159 .9751 .26a3 1.375 67.1 .01300 .oa6i2 .2898 .2820 .2938 l.oa23 .959a .5195 .6125 1.173 .9731 .27a3 1.350 62.1 .oUoo .oa791 .3010 .292a .3056 l.oa56 .956a .6020 .6391 1.187 .9710 .2838 1.327 57.8 .01500 .oa96a .3119 .3022 .3170 l.oa90 .9533 .6238 .6651 1.201 .9690 .2928 1.307 5a.0 .01600 .05132 .3225 .3117 .3281 1.052a .9502 .6a50 .6906 1.215 .9670 .301a 1.288 50.8 .01700 .05296 .3328 .3209 .3389 1.0559 .9a71 .6655 .7158 1.230 . 96 h 9 .3096 1.271 hl .9 .01800 .05a55 .3a28 .3298 .3a95 1.0593 .9hhO .6856 .7ao5 1.2a]i. .9629 .3176 1.255 h 5.3 .01900 .05611 .3525 .3386 .3599 1.0628 .9a09 .7051 .7650 1.259 .9609 .3253 i.2ao a3.o ,02000 .05763 .3621 .3a70 .3701 1.0663 .9378 .72a2 .7891 1.27a .9588 .3327 1.226 ai.o ,02100 .05912 .37ia .3552 .3800 1.0698 .93a8 .lh29 .8131 1.289 .9568 .3399 1.213 39.1 ,02200 .06057 .3806 .3632 .3898 1.0733 .9317 .7612 .8368 1.30a .95a8 .3168 1.201 37.a ,02300 .06200 .3896 .3710 .3995 1.0768 .9287 .7791 .8603 1.319 .9528 .3535 1.189 35.9 ,02U00 .063ao .398a .3786 .ao9o 1.080a .9256 .7967 .8837 1.335 .9508 .3600 1.178 3a.a .02500 .06a78 .a070 .3860 .ai8a i.osao .9225 .oiao .9069 1.350 .9a88 .3662 1.168 33.1 ,02600 .06613 .ai55 .3932 .a276 1.0876 .9195 .8310 .9310 1.366 . 9 h 66 .3722 1.159 31.9 ,02700 .067a7 .a239 .aoo2 . 1 x 361 1.0912 .916a .8a78 .9530 1.381 .9aa8 .3781 1.150 30,8 ,02800 .06878 .a322 .ao7i . 1 x 1 x 51 1.09a9 .9133 .86a3 .9760 1.397 .9a28 .3838 i.iai 29.8 ,02900 .07007 . 1 x 1 x 03 .a38 .U 5 lx 6 1.0985 .9103 .8805 .9988 l.ai3 .9ao8 .3893 1.133 28.8 D-4 Table D-1 Corit*d dAn d/h 21T d/L TAUH SINH COSH K hTTd/L SINH COSH n Cp/C H/H' M 0 20- dA 2Vd/h 2 IT d/L hrrd/L U^d/L ° 0 .03000 .07135 .(4(483 .(4205 .(463(4 1.1021 .9073 .8966 1.022 1.(430 .9388 .39(47 1.125 27.9 .03100 .07260 .(4562 .(4269 .(4721 1.1059 .90(42 .912(* l.Ohh 1.(4(46 .9369 .(4OOO 1.118 27.1 .03200 .07385 .(46(40 .(4333 .(*808 1.1096 .9012 .9280 1.067 1.(462 .93(49 .(4O51 1.111 26.3 .03300 .07507 .(4717 .(4395 .(489(4 1.1133 .8982 .9(43(4 1.090 1.(479 .9329 .(4IOO 1.10(4 25.6 .O3U0O .07630 .(479(4 .(4(457 .(4980 1.1171 .8952 .9588 1.113 1.(496 .9309 .(41(49 1.098 2(*.8 .03500 .077(48 .U868 .(4517 .506(4 1.1209 .8921 .9737 1.135 1.513 .9289 .a96 1.092 2(4.19 .03600 .07867 .(49(43 .(4577 .51(47 1.12U7 .8891 .9886 1.158 1.530 .9270 .(42(*2 1.086 23.56 .03700 .0798(4 .5017 .(4635 .5230 1.1285 .8861 1.0033 1.180 1.5(*7 .9250 .U287 1.080 22.97 .03800 .08100 .5090 .(4691 .5312 1.132(4 .8831 1.018 1.203 1.56(4 .9230 .(*330 1.075 22. (*2 .03900 .08215 .5162 .(47(47 .539(4 1.1362 .8801 1.032 1.226 1.582 .92U .(4372 1.069 21.90 .oUooo .08329 .5233 .(4802 .5(475 l.UOl .8771 1.0(47 1.2(48 1.600 .9192 .(4(41(4 1.06(4 21.(40 .oUoo .08(4(42 .530(4 .(4857 .5556 1.1(4(40 .87(41 1.061 1.271 1.617 .9172 .(*(*55 1.059 20.92 .0U200 .08553 .537(4 .(4911 .5637 I.K479 .8711 1.075 1.29(4 1.636 .9153 .(4(495 1.055 20.(46 .CU300 .0866(4 .5Uhh .(496(4 .5717 1.1518 .8688 1.089 1.317 1.65(» .9133 .(453(4 1.050 20.03 .0)4ii00 .0877(4 .5513 .5015 .5796 1.1558 .8652 1.103 1.3(40 1.672 .911(4 .(*571 1.0(46 19.62 .0(4500 .08883 .5581 .5066 .5876 1.1599 .8621 1.116 1.363 1.691 .9095 .(4607 1.0(42 19.23 .0(4600 .08991 .56(49 .5116 .595(4 1.1639 .8592 1.130 1.306 1.709 .9076 .(*6(43 1.038 18.85 .0(4700 .09098 .5717 .5166 .6033 1.1679 .8562 1.1(43 1.(409 1.728 .9057 .(*679 1.03(4 10.(*9 .0(4800 .09205 .578(4 .5215 .6111 1.1720 .8532 1.157 1.(433 1.7(47 .9037 .(4713 1.030 18.15 .0(4900 .09311 .5850 .5263 .6189 1.1760 .0503 1.170 1.(456 1.766 .9018 .(47(46 1.026 17.82 .05000 .09(416 .5916 .5310 .6267 1.1802 .8(473 1.183 1.(479 1.786 .8999 .(4779 1.023 17.50 .05100 .09520 .5981 .5357 .63(4(4 1.18(43 .8(4(4(4 1.196 1.503 1.805 .8980 .(*811 1.019 17.19 .05200 .09623 .60(46 .5(403 .6(421 1.188(4 .0(415 1.209 1.526 1.825 .8961 .(48(42 1.016 16.90 .05300 .09726 .6111 .5(4(49 .6(499 1.1926 .8385 1.222 1.550 1.8(45 .89(43 .(4873 1.013 16.62 .05(400 .09829 .6176 .5(49(4 .6575 1.1968 .8356 1.235 1.57(4 1.865 .892(4 .(4903 1.010 16.35 .05500 .09930 .6239 .5538 .6652 1.2011 .8326 1.2(*8 1.598 1.085 .8905 .(4932 1.007 16.09 .05600 .1003 .6303 .5582 .6729 1.2053 .8297 1.261 1.622 1.906 .8886 .(*960 1.00(4 15.8(4 .05700 .1013 .6366 .5626 .6805 1.2096 .8267 1.273 1.6(46 1.926 .8867 .(4988 1.001 15.60 .05800 .1023 .6(428 .5668 .6880 1.2138 .8239 1.286 1.670 1.9(47 .88(49 .5015 .9985 15.36 .05900 .1033 .6(491 .5711 .6956 1.2181 .8209 1.298 1.695 1.968 .8830 .50(42 .9958 15.13 .06000 .10(43 .6553 .5753 .7033 1.2225 .8180 1.311 1.719 1.989 .8811 .5068 .9932 1(4.91 .06100 .1053 .6616 .579(4 .7110 1.2270 .8150 1.3231 1.7(4(4 2.011 .8792 .509(4 .9907 1(4.70 .06200 .1063 .6678 .583(4 .7187 1.2315 .8121 1.336 1.770 2.033 .8773 .5119 .9803 1(4.50 .06300 .1073 .6739 .587(4 .7256 1.2355 .8093 1.3(48 1.795 2.055 .3755 .51(43 .9860 1(4.30 .06(400 .1082 .6799 .591(4 .7335 1.2(402 .8063 1.360 1.819 2.076 .8737 .5167 .9837 1(4.11 .06500 .1092 .6860 .595(4 .7(ai 1.2(4(47 .0035 1.372 1.8(45 2.098 .8719 .5191 .9015 13.92 .06600 .1101 .6920 .5993 .7(486 1.2(492 .8005 1.38(4 1.870 2.121 .8700 .521(4 .9793 13.7(4 .06700 .1111 .6981 .6031 .7561 1.2537 .7977 1.396 1.896 2.1(4(4 .8682 .5236 .9772 13.57 .06800 .1120 .7037 .6069 .7633 1.2580 .79(48 1.(408 1.921 2.166 .866(4 .5250 .9752 13.(40 .06900 .1130 .7099 .6106 .7711 1.2628 .7919 1.(420 1.9(48 2.109 .86(46 .5279 .9732 13.2(4 .07000 .1139 .7157 .61(4(4 .7783 1.2672 .7890 1.(432 1.97(4 2.213 .8627 .5300 .9713 13.08 .07100 .11(49 .7219 .6181 .7863 1.2721 .7861 1. (4(4(4 2.000 2.236 .8609 .5321 .969(4 12.92 .07200 .1158 .7277 .6217 .7937 1.2767 .7033 1.(455 2.026 2.260 .8591 .53(4! .9676 12.77 .07300 .1168 .7336 .6252 .8011 1.2813 .780(4 1.(467 2.053 2.28(4 .3572 .5360 .9658 12.62 .07(400 .1177 .7395 .6289 .8086 1.2861 .7775 1.(479 2.080 2.308 .855(4 .5380 .96(41 12.(48 .07500 .1186 .7(453 .632(4 .8162 1.2908 .77(47 1.(490 2.107 2.332 .8537 .5399 .962(4 12.3(4 .07600 .1195 .7511 .6359 .8237 1.2956 .7719 1.502 2.135 2.357 .8519 .5(417 .9607 12.21 .07700 .1205 .7569 .6392 .8312 1.300(4 .7690 1.51(4 2.162 2.382 .8501 .5(435 .9591 12.00 .07800 .121(4 .7625 .6(427 .8386 1.3051 .7662 1.525 2.189 2.(407 .8(483 .5(452 .9576 11.95 .07900 .1223 .7603 .6(460 .8(462 1.3100 .763(4 1.537 2.217 2.(432 .8(465 .5(469 .9562 11.83 .08000 .1232 .77(41 .6(493 .8538 1.31(49 .7605 1.5(48 2.2(45 2.(458 .8(4(48 .5(485 .95(48 11.71 .08100 .12(41 .7799 .6526 .861(4 1.3198 .7577 1.560 2.27(4 2.(48(4 .8(430 .5501 .953(4 11.59 .08200 .1251 .785(4 .6558 .8687 1.32(46 .75(49 1.571 2.303 2.511 .5(413 .5517 .9520 11.(47 .08300 .1259 .7911 .6590 .8762 1.3295 .7522 1.583 2.331 2.537 .8395 .5533 .9506 11.36 .08(400 .1268 .7967 .6622 .8837 1.33(45 .7(49(4 1.59(4 2.360 2.563 .8378 .55(48 .9(493 11.25 .08500 .1277 .8026 .6655 .8915 1.3397 .7(46(4 1.605 2.389 2.590 .8360 .5563 .9(481 11.1(4 .08600 .1286 .8080 .6685 .8989 1.3(4(46 .7(437 1.616 2.(4lB 2.617 .83(42 .5577 .9(469 11.0(4 .08700 .1295 .8137 .6716 .906(4 1.3(497 .7(409 1.626 2.(4148 2.6(4!* .0325 .5591 .9(457 10.9!* .08800 .130(4 .6193 .67(47 .91(4! 1.35(48 .7381 1.639 2.(478 2.672 .8308 .5605 .9(4(45 10.8!* .08900 .1313 .8250 .6778 .9218 1.3600 .7353 1.650 2.508 2.700 .8290 .5619 .9(433 10.7(4 D-5 Table D-1 Cont'd d/Lo d/L 27rd/L TAlIH SIUH • COSH K a 77 d/L SINK COSH n C_/C H/H' 2rd/L 27rd/L 2trd/L arr d/L arr d/L ° ° 0 .09000 .1322 .8306 .6808 .9295 1.3653 .7321+ 1.661 2.538 2.728 .8273 .5632 .9a22 .09100 .1331 .8363 .6838 .9372 1.3706 .7296 1.672 2.568 2.756 .8255 .56a5 .9aii .09200 .131+0 .81+20 .6868 .91+50 1.3759 .7268 1.68a 2.599 2.785 .8238 .5658 .9aoi .09300 .131*9 .81+71+ .6897 .9525 1.3810 .721+1 1.695 2.630 2.8ia .8221 .5670 .9391 .091+00 .1357 .8528 .6925 .9600 1.3862 .7211* 1.706 2.662 2.8a3 .820a .5682 .9381 .09500 .1366 .8583 .6953 .9677 1.3917 .7186 1.717 2.693 2.873 .8187 .5693 .9371 .09600 .1375 .8639 .6982 .9755 1.3970 .7158 1.728 2.726 2.903 .8170 .57oa .9362 .09700 .1381+ .8691+ .7011 .9832 1.1*023 .7131 1.739 2.757 2.933 .8153 .5716 .9353 .09800 .1392 .871+9 .7039 .9908 l.J+077 .7101+ 1.750 2.790 2.963 .8136 .5727 .93aa .09900 .11+01 .8803 .7066 .9985 1.1+131 .7076 1.761 2.822 2.99a .8120 .5737 .9335 .1000 .11+10 .8858 .7093 1.006 1.1+187 .701+9 1.772 2.855 3.025 .8103 . 57 k 7 .9327 .1010 .11+19 .8913 .7120 1.011+ 1.1+21+2 .7022 1.783 2.888 3.057 .8086 .5757 .9319 .1020 .11*27 .8967 .711+7 1.022 1.1+297 .6991* 1.793 2.922 3.088 .8069 .5766 .9311 .1030 .11+36 .9023 .7173 1.030 1.1+351+ .6967 1.805 2.956 3.121 .8052 .5776 .930a .101+0 .11+1+5 .9076 .7200 1.037 i.i+i+io .691+0 1.815 2.990 3.153 .8036 .5785 .9297 .1050 .11+53 .9130 .7226 1.01+5 1.1*1*65 .6913 1.826 3.02a 3.185 .8019 .579a .9290 .1060 .11+62 .9181+ .7252 1.053 1.1+523 .6886 1.837 3.059 3.218 .8003 .5803 .9282 .1070 .11+70 .9239 .7277 1.061 1.1+580 .6859 i.aas 3.09a 3.251 .7986 .5812 .9276 .1080 .11+79 .9293 .7303 I.O69 1.1*638 .6833 1.858 3.128 3.28a .7970 .5820 .9269 .1090 .11+88 .931+3 .7327 1.076 1.1+692 .6806 1.869 3.16a 3.319 .795a .5828 .9263 .1100 .11+96 .91+00 .7352 1.085 1.1+752 .6779 1.880 3.201 3.353 .7937 .5836 .9257 .1110 .1505 .91+56 .7377 1.093 1.1+811+ .6752 1.891 3.237 3.388 .7920 .58a3 .9251 .1120 .1513 .9508 .71+02 1.101 1.1+871 .6725 1.902 3.27a 3.a23 .790a .5850 .92a5 .1130 .1522 .9563 .71+26 1.109 1.1+932 .6697 1.913 3.312 3.a59 .7886 .5857 .9239 .111+0 .1530 .9616 .7150 1.117 1.1+990 .6671 1.923 3.3a8 3.a9a .7872 .586a .923a .1150 .1539 .9670 .71+71+ 1.125 1.5051 .661+5 1.93a 3.385 3.530 .7856 .5871 .9228 .1160 .151+7 .9720 .71+97 1.133 1.5108 .6619 i.9aa 3.a23 3.566 .78ao .5878 .9223 .1170 .1556 .9775 .7520 1.11+1 1.5171 .6592 1.955 3.a62 3.603 .782a .588a .9218 .1180 .1561+ .9827 .751+3 1.11+9 1.5230 .6566 1.966 3.501 3.6ai .7808 .5890 .921a .1190 .1573 .9882 .7566 1.157 1.5293 .6539 1.977 3.5ao 3.678 .7792 .5396 .9209 .1200 .1581 .9936 .7589 1.165 1.5356 .6512 1.987 3.579 3.716 .7776 .5902 .92oa .1210 .1590 .9989 .7612 1.171+ 1.51+18 .61+86 1.998 3.620 3.755 .7760 .5907 .9200 .1220 .1598 1.001+ .7631* 1.182 1.51+79 .61+60 2.008 3.659 3.793 .77a5 .5913 .9196 .1230 .1607 1.010 .7656 1.190 1.551+6 .61*33 2.019 3.699 3.832 .7729 .5918 .9192 .121+0 .1615 1.015 .7678 1.198 1.5605 .61+07 2.030 3.7ao 3.871 .7713 .5922 .9189 .1250 .1621+ 1.020 .7700 1.207 1.5671* .6381 2.oai 3.782 3.912 .7698 .5926 .9186 .1260 .1632 1.025 .7721 1.215 1.5731* .6356 2.051 3.82a 3.952 .7682 .5931 .9182 .1270 .161+0 1.030 .771+2 1.223 1.5795 .6331 2.061 3.865 3.992 .7667 .5936 .9178 .1280 .161+9 1.036 .7763 1.231 1.5862 .6305 2.072 3.907 a.033 .7652 .5940 .9175 .1290 .1657 i.oia .7783 1.21+0 1.5927 .6279 2.082 3.950 a.o7a .7637 .59aa .9172 .1300 .1665 1.01+6 .7801+ 1.21+8 1.5990 .6251+ 2.093 3.992 a.U5 .7621 .59a8 .9169 .1310 .1671+ 1.052 .7821+ 1.257 1.6060 .6228 2.10a a.036 a.l58 .7606 .5951 .9166 .1320 .1682 1.057 .781+1* 1.265 1.6121+ .6202 2.11a a. 080 a.201 .7591 .595a .916a .1330 .1691 1.062 .7865 1.273 I.6I9I .6176 2.125 a.125 a.2a5 .7575 .5958 .9161 .131*0 .1699 1.068 .7885 1.282 1.6260 .6150 2.135 a.l69 a.288 .7560 .5961 .9158 .1350 .1708 1.073 .7905 1.291 1.633 .6123 2.ia6 a.217 a.33a .7sa5 .596a .9156 .1360 .1716 1.078 .7925 1.300 1.61+0 .6098 2.156 a. 262 a.378 .7530 .5967 .915a .1370 .1721+ 1.081+ .791+5 1.308 1.61+7 .6073 2.167 a. 309 a.a23 .7515 .5969 .9152 .1380 .1733 1.089 .7961+ 1.317 1.651+ .601+7 2.177 a.355 a.a68 .7500 .5972 .9150 .1390 .171+1 I.09I+ .7983 1.326 1.660 .6022 2.188 a.ao2 a.5ia .7afl5 .5975 .9ia8 -.11+00 .171+9 1.099 .8002 1.331+ 1.667 .5998 2.198 a.aso a.561 .7a71 .5978 .9ia6 -.11+10 .1758 1.105 .8021 1.31+3 1.675 .5972 2.209 a.a98 a.607 .7a56 .5980 .9iaa .11+20 .1766 1.110 .8039 1.352 1.681 .591+7 2.219 a.5a6 a.65a .7aai .5982 .9ia2 .11+30 .1771+ 1.115 .8057 1.360 1.688 .5923 2.230 a. 595 a.663 .7a26 .598a .9iai .11+1+0 .1783 1.120 ,8076 1.369 1.696 .5898 2.2ao a.6aa a.75i .7ai2 .5986 .9iao .11+50 .1791 1.125 .8091+ 1.378 1.703 .5873 2.251 a.695 a.800 .7397 .5987 .9139 .11+60 .1800 1.131 .8112 1.388 1.710 .581+7 2.261 a.7a6 a.850 .7382 .5989 .9137 .11+70 .1808 1.136 .8131 1.397 1.718 .5822 2.272 a.798 a.901 .7368 .5990 .9136 .11+80 .1816 1.11+1 .811+9 1.1+05 1.72s .5798 2.282 a.8a7 a.951 .735a .5992 .9135 .11+90 .1825 1.11+6 .8166 1.1+15 1.732 .5773 2.293 a. 901 5.001 .7339 .5993 .913a D-6 K 10.65 10.55 10.1+6 10.37 10.29 10.21 10.12 10 . 01 + 9.962 9.681+ 9.808 9.731+ 9.661 9.590 9.519 9.1+51 9.381+ 9.318 9.251+ 9.191 9.129 9.068 9.009 8.950 8.891 8.835 8.780 8.726 8.673 8.621 8.569 8.518 6.1+68 8.1+19 8.371 8.321+ 8.278 8.233 8.189 8.11+6 8.103 8.061 8.020 7.978 7.937 7.897 7.857 7.819 7.761 7.71+1+ 7.707 7.671 7.636 7.602 7.567 7.533 7.1+99 7.1+65 7.1+32 7.1+00 Tgble D-1 Cont'd d/L 2rr d/L TANK SINH COSH K aiTd/L SINH COSH n c„/c h/H' M ' 0 2ff d/L 2ir d/L 2^ d/L hrrd/L ai d/L 0 0 0 .1500 .1833 1.152 .8183 1.1*21* i.7ao .57a8 2.303 a. 95a 5.05a .7325 .5996 .9133 7.369 .1510 .181*1 1.157 .8200 1.1*33 1.7a7 .5723 2.31a 5.007 5.106 .7311 .5996 .9133 7.339 .1520 .1850 1.162 .8217 1.1*1*2 1.755 .5699 2.32a 5.061 5.159 .7296 .5995 .9132 7.309 .1530 .1858 1.167 .8231* 1.1*51 1.762 .5675 2.335 5.115 5.212 .7282 .5996 .9132 7.279 .15I40 .1866 1.173 .8250 1.1*60 1.770 .5651 2.3a5 5.169 5.265 .7268 .5996 .9132 7.250 .1550 .1875 1.178 .8267 1.1*69 1.777 .5627 2.356 5.225 5.320 .7256 .5997 .9131 7.221 .1560 .1883 1.183 .8281* 1.1*79 1.785 .5602 2.366 5.283 5.376 .7260 .5998 .9130 7.191 .1570 .1891 1.188 .8301 1.1*88 1.793 .5577 2.377 5.339 5.a32 .7226 .5999 .9129 7.162 .1580 .1900 I.I9I* .8317 1.1*98 1.801 .5552 2.387 5.398 5.a9o .7212 .5998 .9130 7.136 .1590 .1908 1.199 .8333 1.507 1.809 .5528 2.398 5.a5a 5.5aa .7198 .5998 .9130 7.107 .1600 .1917 I.20I* .831*9 1.517 1.817 .550a 2.ao8 5.513 5.603 .7186 .5998 .9130 7.079 .1610 .192? 1.209 .8365 1.527 1.825 .5a80 2.ai9 5.571 5.660 .7171 .5998 .9130 7.052 .1620 .1933 1.215 .8381 1.536 1.833 .5a56 2.a29 5:630 5.718 .7157 .5998 .9130 7.026 .1630 .191*1 1.220 .8396 1.51*6 1.8ai .5a32 2.aao 5.690 5.777 .7166 .5998 .9130 7.000 .161*0 .1950 1.225 .81*11 1.555 1.8a9 .5ao9 2.a5o 5.751 5.837 .7130 .5998 .9130 6.975 .1650 .1958 1.230 .81*27 1.565 1.857 .5385 2.a6i 5.813 5.898 .7117 .5997 .9131 6.969 .1660 .1966 1.235 .81*1*2 1.571* 1.865 .5362 2.a7i 5.87a 5.959 .7103 .5996 .9132 6.926 .1670 .1975 1.21*0 .81*57 1.581* 1.873 .5339 2.a82 5.938 6.021 .7090 .5996 .9132 6.900 .1680 .1983 1.21*6 .81*72 1.591* 1.882 .5315 2.a92 6.003 6.085 .7076 .5995 .9133 6.876 .1690 .1992 1.251 .81*86 1.601* 1.890 .5291 2.503 6.066 6.ia8 .7063 .5996 .9133 6.853 .1700 .2000 1.257 .8501 1.61a 1.899 .5267 2.513 6.130 6.212 .7050 .5993 .9136 6.830 .1710 .2008 1.262 .8515 1.62a 1.907 .52a3 2.523 6.197 6.275 .7036 .5992 .9135 6.807 .1720 .2017 1.267 .8529 1.63a 1.915 .5220 2.53a 6.262 6.3a2 .7023 .5991 .9136 6.786 .1730 .2025 1.272 .851*1* i.6aa 1.92a .5197 2.5aa 6.329 6.a07 .7010 .5989 .9137 6.761 .171*0 .2033 1.277 .8558 1.65a 1.933 .517a 2.555 6.395 6.a73 .6997 .5988 .9138 6.738 .1750 .201*2 1.282 .8572 1.66a i.9ai .5151 2.565 6.a65 6.5ai .6986 .5987 .9139 6.716 .1760 .2050 1.288 .8586 1.675 1.951 .5127 2.576 6.53a 6.610 .6971 .5985 .9160 6.696 .1770 .2058 1.293 .8600 1.685 1.959 .5ioa 2.586 6.603 6.679 .6958 .5986 .9161 6.672 .1780 .2066 1.298 .8611* 1.695 1.968 .5081 2.597 6.672 6.767 .6966 .5982 .9162 6.651 .1790 .2075 1.30i* .8627 1.706 1.977 .5058 2.607 6.7aa 6.818 .6933 .5980 .916J* 6.631 .1800 .2083 1.309 .861*0 1.716 1.986 .5036 2.618 6.818 6.891 .6920 .5979 .9165 6.611 .1810 .2092 1.311* .8653 1.727 1.995 .5013 2.629 6.890 6.963 .6907 .5977 .9166 6.591 .1820 .2100 1.320 ,86^ 1.737 2.00a .a990 2.639 6.963 7.035 .6895 .5975 .9168 6.571 .1830 .2108 1.325 .8680 i.7a8 2.013 .a967 2.650 7.038 7.109 .6882 .5976 .9169 6.550 .leao .2117 1.330 .8693 1.758 2.022 .a9a5 2.660 7.113 7.183 .6870 .5972 .9150 6.530 .1850 .2125 1.335 .8706 1.769 2.032 .a922 2.671 7.191 7.260 .6857 .5969 .9152 6.511 .i860 .2131* 1.31*1 .8718 1.780 2.oai .a899 2.681 7.267 7.336 .6865 .5967 • 7 .9156 6.692 .1870 .211*2 1.31*6 .8731 1.791 2.051 .a876 2.692 7.3a5 7.ai2 .6832 .5965 • 7^y^ 9155 w « *47^ 6 li7li .1880 .2150 1.351 .871*3 1.801 2.060 .a85a 2.702 7.a2i 7.688 .6820 • 7 ^yy 9157 .1890 .2159 1.356 .8755 1.812 2.070 .a832 2.712 7.500 7.566 .6808 • ✓ 7^J .5961 .9159 6.638 .1900 .2167 1.362 .8767 1.823 2.079 .a809 2.723 7.581 7.667 .6796 .5958 .9161 6.621 .1910 .2176 1.367 .6779 1.83a 2.089 .a787 2.73a 7.663 7.728 .6786 .5955 .9163 6.603 .1920 .2181* 1.372 .8791 i.8a5 2.099 .a765 2.7aa 7.7a6 7.810 .6772 .5952 .9165 6.385 .1930 .2192 1.377 .8803 1.856 2.108 .a7a3 2.755 7.827 7.891 .6760 .5950 .9167 6.368 .191*0 .2201 1.383 .8815 1.867 2.118 .a72i 2.765 7.911 7.976 .6768 .5968 .9169 6.351 .1950 .2209 1.388 .8827 1.879 2.128 .U 699 2.776 7.996 8.059 .6736 .5966 .9170 6.336 .I960 .2218 1.393 .8839 1.890 2.138 .a677 2.787 8.083 8.165 .6726 .5966 .9172 6.317 .1970 .2226 1.399 .8850 1.901 2.ia8 .a655 2.797 8.167 8.228 .6712 .5961 .9176 6.300 .1980 .2231* 1.1*01* .8862 1.913 2.158 .a633 2.808 8.256 8.316 .6700 .5938 .9176 6.286 .1990 .221*3 1.1*09 .8873 1.92a 2.169 .a6u 2.819 8.3a6 8.606 .6689 .5935 .9179 6.268 .7000 .2251 1.1*11* .8881* 1.935 2.178 .a59o 2.829 8.a36 8.695 .6677 .5932 .9181 6.253 .2010 .2260 1.1*20 .8895 1.9a7 2.189 .1*569 2.8ao 8.52a 8.583 .6666 .5929 .9183 6.237 .2020 .2268 1.1*25 .8906 1.959 2.199 .a5a7 2.850 8.616 8.676 .6656 .5926 .9186 6.222 .2030 .2277 1.1*30 .8917 1.970 2.210 .a526 2.861 8.708 8.766 .6662 .5923 .9188 6.206 .201*0 .2285 1.1*36 .8928 1.962 2.220 .a5oa 2.872 8.803 8.860 .6631 .5920 .9190 6.191 .2050 .2293 1.1*1*1 .8939 1.99a 2.231 .aa83 2.882 8.897 6.953 .6620 .5917 .9193 6.176 .2060 .2302 1.1*1*6 .8950 2.006 2.2a2 .aa62 2.893 8.99a 9.050 .6608 .5916 .9195 6.161 .2070 .2310 l.a5l .8960 2.017 2.252 .aaai 2.903 9.090 9.iaa .6597 .5911 .9197 6.167 .2080 .2319 1.1*57 .8971 2.030 2.263 .aai9 2.91a 9.187 9.260 .6586 .5908 .9200 6.133 .2090 .2328 1.1*62 .8981 2.oa2 2.27a .a398 2.925 9.288 9.362 .6576 .5905 .9202 6.119 D-7 Table D-1 Cont'd d/L. dA 2 frd/L TANK SINH COSH K 1*77 d/L SINH COSH n Cp/C h/h> U o ZrTd/L 2 ^d/L 2/7 d/L l*’ 2 d/L a^d/L G 0 0 .2100 .2336 1.U68 .8991 2.055 2.285 .1*377 2.936 9.389 9.aa2 .6563 .5901 .9205 6.105 .2110 .231*1* 1.1*73 .9001 2.066 2.295 .1*357 2.91*6 9.1*90 9 . 5 k 2 .6552 .5898 .9207 6.091 .2120 .2353 1.1*79 .9011 2.079 2.307 .1*336 2.957 9.590 9.6a2 .65ai .589a .9210 6.077 .2130 .2361 1.1*81* .9021 2.091 2.318 .1*315 2.967 9.693 9.7aa .6531 .5891 .9213 6.06a .211»0 .2370 1.1*89 .9031 2.103 2.329 .1*291* 2.978 9.796 9.8a7 .6520 .5888 .9215 6.051 .2150 .2378 1.1*91* .90la 2.115 2.31*0 .1*271* 2.989 9.902 9.952 .6509 .588a .9218 6.037 .2160 .2387 1.500 .9051 2.128 2.351 .1*253 2.999 10.01 10.06 .6a98 .5881 .9221 6.02a .2170 .2395 1.506 .9061 2.11*2 2.361* .1*232 3.010 10.12 10.17 .6a88 .5878 .9223 6.011 .2180 .21*01, 1.511 .9070 2.151* 2.375 .1*211 3.021 10.23 10.28 .6a77 .587a .9226 5.999 .2190 .21*12 1.516 .9079 2.166 2.386 .1*191 3.031 10.31* 10.38 . 6 U 67 .5871 .9228 5.987 .2200 .21*21 1.521 .9088 2.178 2.397 .1*171 3.01*2 10.1*5 10.50 . 6 U 56 .5868 .9231 5.975 .2210 .21*29 1.526 .9097 2.192 2.1*09 .1*151 3.052 10.56 10.61 ,6lj*6 .586a .923a 5.963 .2220 .21*38 1.532 .9107 2.201* 2.1*21 .1*131 3.063 10.68 10.72 .6a36 .5861 .9236 5.951 .2230- .21*1*6 1.537 .9116 2.218 2.1*33 .1*111 3.071* 10.79 10.8a .6a25 .585^7 .9239 5.939 . 221*0 • 21*55 1 . 9*2 .9125 2.230 2.1*1*1* .1*091 3.085 10.91 10.95 .6aia .58?a .92a2 5.927 .2250 .21*63 1.51*8 .9131* 2.21*1* 2.1*57 .1*071 3.095 11.02 11.07 .6aoa .5850 .92a5 5.915 .2260 .21*72 1.553 .911*3 2.257 2.1*69 .1*051 3.106 11.15 11.19 .639a .58a6 .92a8 5.903 .2270 .21*81 1.559 .9152 2.271 2.1*81 .1*031 3.117 11.27 11.31 .6383 . 58 U 2 .9251 5.891 .2280 .21*89 1.561* .9161 2.281* 2.1*93 .1*011 3.128 11.39 ii.aa .6373 .5838 .925a 5.860 .2290 .21*98 1.569 .9170 2.297 2.506 .3991 3.138 11.51 11.56 .6363 .583a .9258 5.869 .2300 .2506 1.575 .9178 2.311 2.518 .3971 3.11*9 11.61* 11.68 .6353 .5830 .9261 5.858 .2310 .2515 1.580 .9186 2.325 2.531 .3952 3.160 11.77 11.81 .63a3 .5826 .926a 5.8a8 .2320 .2523 1.585 .9191* 2.338 2.51*3 .3932 3.171 11.90 11.93 .6333 .5823 .9267 5.838 .2330 .2532 1.591 .9203 2.352 2.556 .3912 3.182 12.03 12.07 .6323 .5819 .9270 5.827 .231*0 .251*0 1.596 .5211 2.366 2.569 .3893 3.192 12.15 12.19 .6313 .5815 .9273 5.816 .2350 .251*9 1.602 .9219 2.380 2.581 .387U 3.203 12.29 12.33 .630a .5811 .9276 5.806 .2360 .2558 1.607 .9227 2.393 2.591* .3855 3.211+ 12.1*3 12.a7 .629a .5807 .9279 5.796 .2370 .2566 1.612 .9235 2.1*08 2.607 .3836 3.225 12.55 12.59 .628a .58oa .9282 5.786 .2380 .2575 1.618 .921*3 2.1*22 2.620 .3816 3.236 12.69 12.73 .6275 .5800 .9285 5.776 .2390 .2581* 1.623 .9251 2.1*36 2.631* .3797 3.21*7 12.83 12.87 .6265 .5796 .9288 5.766 .21*00 .2592 1.629 .9259 2.1*50 2.61*7 .3779 3.257 12.97 13.01 .6256 .5792 .9291 5.756 .21*10 .2601 1.631* .9267 2.1*61* 2.660 .3760 3.268 13.11 13.15 , 62 h 6 .5788 .929a 5.786 .21*20 .2610 1.61*0 .9275 2.1*80 2.671* .371*1 3.279 13.26 13.30 .6237 .578a .9298 5.736 .21*30 .2618 1.61*5 .9282 2.1*91* 2.687 .3722 3.290 13.1*0 13.aa .6228 .5780 .9301 5.727 .21*1*0 .2627 1.650 .9289 2.508 2.700 .3701* 3.301 13.55 13.59 .6218 .5776 .930a 5.718 .21*50 .2635 1.656 .9296 2.523 2.711* .3685 3.312 13.70 13.73 .6209 .5272 .9307 5.710 .21*60 .261*1* 1.661 .9301* 2.538 2.728 .3666 3.323 13.85 13.88 .6200 .5768 .9310 5.701 .21*70 .2653 1.667 .9311 2.553 2.71*2 .361*8 3.331* 11*. 00 la.oa .6191 .576a .931a 5.692 .21*80 .2661 1.672 .9318 2.568 2.755 .3629 3.31*1* 11*. 15 La.19 .6182 .5760 .9317 5.68a .21*90 .2670 1.678 .9325 2.583 2.770 .3610 3.355 It*. 31 ia.35 .6173 .5756 .9320 5.675 .2500 .2679 1.683 .9332 2.599 2.781* .3592 3.367 11*. 1*7 ia.5i .616a .5752 .9323 5.667 .2510 .2667 1.689 .9339 2.611* 2.798 .3571* 3.377 11*. 62 ia.66 .6155 .57a8 .9327 5.658 .2520 .2696 1.691* .931*6 2.629 2.813 .3556 3.388 ll*.79 ia.82 .6ia6 .57aa .9330 5.650 .2530 .2705 1.700 .9353 2.61*5 2.828 .3537 3.399 11*. 95 ia.99 .6137 .57ao .9333 5.6ai .251*0 .2711* 1.705 .9360 2.660 2.81*2 .3519 3.1*10 15.12 15.15 .6128 .5736 .9336 5.633 .2550 .2722 1.711 .9367 2.676 2.856 .3501 3.1*21 15.29 15.32 .6120 .5732 .93ao 5.62a .2560 .2731 1.716 .9371* 2.691 2.871 .3U83 3.1*32 15.1*5 15. a9 .6111 .5728 .93a3 5.616 .2570 .271*0 1.722 .9381 2.707 2.886 .31*65 3.1*1*3- 15.63 15.66 .6102 .572a .93a6 5.608 .2580 .271*9 1.727 .9388 2.723 2.901 .31*1*7 3.1*51* 15.80 15.83 .6093 .5720 .93a9 5.600 .2590 .2757 1.732 .9391* 2.739 2.916 .31*30 3.I165 15.97 16.00 .6085 .5716 .9353 5.592 .2600 .2766 1.736 .91*00 2.755 2.931 .31*12 3.1*76 16.15 16.18 .6076 .5712 .9356 5.565 .2610 .2775 1.71*1* .91*06 2.772 2.91*6 .3391* 3.1*87 16.33 16.36 .6068 .5707 .9360 5.578 .2620 .2781* 1.71*9 .91*12 2.788 2.962 .3376 3.1*98 16.51 16.5a .6060 .5703 .9363 5.571 .2630 .2792 1.755 .91*18 2.801* 2.977 .3359 3.509 16.69 16.73 .6052 .5699 .9367 5.563 .261*0 .2801 1.760 .91*25 2.820 2.992 .331*2 3.520 16.88 16.91 .6oa3 .5695 .9370 5.556 .2650 .2810 1.766 .91*31 2.837 3.008 .3325 3.531 17.07 17.10 .6035 .5691 .9373 5.5a8 .2660 .2819 1.771 .91*37 2.853 3.023 .3308 3.51*2 17.26 17.28 .6027 .5687 .9377 5.5ai .2670 .2827 1.776 .91*1*3 2.870 3.039 .3291 3.553 17.1*5 17.a5 .6018 .5683 .9380 5.53a .2680 .2836 1.782 .91*1*9 2.886 3.055 .3271* 3.561* 17.61* 17.67 .6010 .5679 .9383 5.527 .2690 .281*5 1.788 .91*55 2.901* 3.071 .3256 3.575 17.8a 17.87 .6002 .5675 .9386 5.520 D-8 Table D-1 Coni'd d/L d/L 2”’ d/L TANH SIKH COSH K l*»rd/L SINK COSH n H/H’ M ' 0 2^ d/L 2 'Td/L 2^d/L l*^d/L l*rd/L Vj 0 0 .2700 .2851* 1.793 .91*61 2.921 3.088 .3239 3.587 18.01* 18.07 .5991* .5671 .9390 5.513 .2710 .2863 1.799 .91*67 2.938 3.101* .3222 3.598 18.21, 18.27 .5986 .5667 .9393 5.506 .2720 .2872 1.80l* .91*73 2.956 3.120 .3205 3.610 18.1*6 18.1*9 .5978 .5663 .9396 5.499 .2730 .2880 1.810 • 91*78 2.973 3.136 .3189 3.620 18.65 18.67 .5971 .5659 .91*00 5.493 .27liO .2889 1.815 .91*81* 2.990 3.153 .3172 3.631 18.86 18.89 .5963 .5655 .91*03 5.486 .2750 .2898 1.821 .91*90 3.008 3.170 .3155 3.6U2 19.07 19.10 .5955 .5651 .91*06 5.480 .2760 .2907 1.826 .91*95 3.025 3.186 .3139 3.653 19.28 19.30 .591*7 .561*7 .91*10 5.474 .2770 .2916 1.832 .9500 3.01*3 3.203 .3122 3.661* 19.1*9 19.51 .591*0 .561*3 .9413 5.468 .2780 .2921* 1.837 .9505 3.061 3.220 .3106 3.675 19.71 19.71* .5932 .5639 .9416 5.462 .2790 .2933 1.81*3 .9511 3.079 3.237 .3089 3.686 19.93 19.96 .5925 .5635 .9420 5.456 .2800 .291*2 1.81*9 .9516 3.097 3.251* .3073 3.697 20.16 20.18 .5917 .5631 .9423 5.450 .2810 .2951 1.85U .9521 3.115 3.272 .3057 3.709 20.39 20.1*1 .5910 .5627 .9426 5.444 .2820 .2960 1.860 .9526 3.133 3.289 .301*0 3.720 20.62 20.61* .5902 .5623 .9430 5.438 .2830 .2969 1.866 .9532 3.152 3.307 .3021* 3.731 20.85 20.87 .5895 .5619 .9433 5.432 .281*0 .2978 1.871 .9537 3.171 3.325 .3008 3.71,2 21.09 21.11 .5887 .5615 .9436 5.426 .2850 .2987 1.877 .951*2 3.190 3.31*3 .2992 3.751* 21.33 21.35 .5880 .5611 .9440 5.420 .2860 .2996 1.662 .951*7 3.209 3.361 .2976 3.765 21.57 21.59 .5873 .5607 .9443 5.414 .2870 .3005 1.688 .9552 3.228 3.379 .2959 3.776 21.82 21.61* .5866 .5603 .9446 5.409 .2880 .3011* 1.893 .9557 3.21*6 3.396 .291*1* 3.787 22.05 22.07 .5859 .5600 .9449 5.403 .2890 .3022 1.899 .9562 3.261* 3.1al4 .2929 3.798 22.30 22.32 .5852 .5596 .9452 5.397 .2900 .3031 1.905 .9567 3.281* 3.1*33 .2913 3.609 22.51* 22.57 .581*5 .5592 .9456 5.392 .2910 .301*0 1.910 .9572 3.303 3.1*51 .2898 3.821 22.81 22.83 .5838 .5588 .9459 5.386 .2920 .301*9 1.916 .9577 3.323 3.1*71 .2882 3.832 23.07 23.09 .5831 .5581* .9463 5.380 .2930 .3058 1.922 .9581 3.31*3 3.1*90 .2866 3.81,3 23.33 23.35 .5821* .5580 .9466 5.375 .291*0 .3067 1.927 .9585 3.362 3.506 .2851 3.855 23.60 23.62 .5817 .5576 .9469 5.371 .2950 .3076 1.933 .9590 3.382 3.527 .2835 3.866 23.86 23.88 .5810 .5572 .9473 5.366 .2960 ' .3085 1^938 .9591* 3.1*02 3.51*6 .2820 3.877 21*.12 21*. 15 .5801* .5568 .9476 5.361 .2970 .3091* 1.9U* .9599 3.1*22 3.565 .2805 3.888 21*.1*0 21*.1*2 .5797 .5561, .9480 5.356 .2980 .3103 1.950 .9603 3.1*1*2 3.585 .2790 3.900 21*.68 21*. 70 .5790 .5560 .9483 5.351 .2990 .3112 1.955 .9607 3.1*62 3.601* .2775 3.911 21*. 96 21*.98 .5781* .5556 .9486 5.347 .3000 .3121 1.961 .9611 3.1*83 3.621* .2760 3.922 25.21* 25.26 .5777 .5552 .9490 5.342 .3010 .3130 1.967 .9616 3.503 3.61*3 .271*5 3.933 25.53 25.55 .5771 .551*9 .9493 5.337 .3020 .3139 1.972 .9620 3.521* 3.663 .2730 3.91*5 25.82 25.83 .5761* .551*5 .9496 5.332 .3030 .311*8 1.978 .9621* 3.51*5 3.683 .2715 3.956 26.12 26.11* .5758 .551*1 .9499 5.328 .301*0 .3157 1.981* .9629 3.566 3.703 .2700 3.968 26.1*2 26.1*1* .5751 .5538 .9502 5.323 .3050 .3166 1.989 .9633 3.587 3.721* .2685 3.979 26.72 26.71* .571,5 .5531. .9505 5.318 .3060 .3175 1.995 .9637 3.609 3.71*5 .2670 3.990 27.02 27.01* .5739 .5530 .9509 5.314 .3070 .3181* 2.001 .961*1 3.630 3.765 .2656 1*.002 27.33 27.35 .5732 .5527 .9512 5.309 .3080 .3193 2.007 .961*5 3.651 3.786 .261*1 1*.013 27.65 27.66 .5726 .5523 .9515 5.305 .3090 .3202 2.012 .^1*9 3.673 3.806 .2627 U.021* 27.96 27.98 .5720 .5519 .9518 5.300 .3100 .3211 2.018 .9653 3.691* 3.827 .2613 1*.036 28.28 28.30 .5711* .5515 .9522 5.296 .3110 .3220 2.023 .9656 3.716 3.81,8 .2599 l,.0l*7 28.60 28.62 .5708 .5511 .9525 5.292 .3120 .3230 2.029 .9660 3.738 3.870 .2581* 1*.058 28.93 26.95 .5701 .5507 .9528 5.288 .3130 .3239 2.035 .9661* 3.760 3.891 .2570 1*.070 29.27 29.28 .5695 .5501* .9531 5.284 .311*0 .321*8 2.01*1 .9668 3.782 3.912 .2556 l*.08l 29.60 29.62 .5689 .5500 .9535 5.280 .3150 .3257 2.01*6 .9672 3.805 3.931* .251*2 l*.093 29.91* 29.96 .5683 .51*97 .9538 5.276 .3160 .3266 2.052 .9676 3.828 3.956 .2528 l*.10l* 30.29 30.31 .5678 .51,91* .9541 5.272 .3170 .3275 2.058 .9679 3.851 3.978 .2511* 1*.116 30.61* 30.65 .5672 • 51*90 .9544 5.268 .3180 .3281* 2.063 .9682 3.873 li.ooo .2500 1*.127 30.99 31.00 .5666 .51*86 .9547 5.264 .3190 .3291* 2.069 .9686 3.896 1*.022 .21*86 l*.139 31.35 31.37 .5660 .51*83 .9550 5.260 .3200 .3302 2.075 .9690 3.919 l*.ol*5 .21*72 1*.150 31.71 31.72 .5655 .51*79 .9553 5.256 .3210 .3311 2.081 .9693 3.91*3 l*.o68 .21*59 1*.161 32.07 32.08 .561*9 .51*76 .9556 5.252 .3220 .3321 2.086 .9696 3.966 1*.090 .21*1,5 1,.173 32.1*1* 32.1*6 .561*3 .51*72 .9559 5.249 .3230 .3330 2.092 .9700 3.990 l*.lll* .21*31 1*.185 32.83 32.81* .5637 .51,68 .9562 5.245 .321*0 .3339 2.098 .9703 1*.011* 1*.136 .21*18 1*.196 33.20 33.22 .5632 .51*65 .9565 5.241 .3250 .331*9 2.IOI* .9707 1*.038 1*.160 .21*01* 1*.208 33.60 33.61 .5627 .51*62 .9568 5.237 .3260 .3357 2.110 .9710 l*.06l 1*.183 .2391 1*.219 33.97 33.99 .5621 .51*58 .9571 5.234 .3270 .3367 2.115 .9713 1*.085 1*.206 .2378 1*.231 31*.37 31*. 38 .5616 .51*55 .9574 5.231 .3280 .3376 2.121 .9717 1*.110 1*.230 .2361* 1*.21*2 31*.77 31.79 .5610 .51*51 .9577 5.227 .3290 .3385 2.127 .9720 1*.135 1*.251* .2351 1*.251* 35.18 35.19 .5605 .51*1*8 .9580 5.223 D-9 Table D-1 Cont'd d/L d/L 2ir d/L TANK SINH COSH K l*7rd/L SINH COSH n H/H' M o 2frd/L 27r d/*. 27rd/L i*7rd/L UTTd/l O' 0 0 .3300 .3391* 2.133 .9723 l*.l59 1*.277 .2338 1*.265 35.58 35.59 .5599 .51*1*1* .9583 5.220 .3310 .31*03 2.138 .9726 l*.l81* 1*.301 .2325 1*.277 35.99 36,00 .559U .51*1*1 .9586 5.217 .3320 .31*13 2.11*1* .9729 1*.209 1*.326 .2312 lj.288 36.1*2 36.1*3 .5589 .51*38 .9589 5.211* .3330 .31*22 2.150 .9732 U.23I* 1*.350 .2299 1*.300 36.8U 36.85 .5581* .51*31* .9592 5.210 .33l»0 .31*31 2.156 .9735 U.259 1*.375 .2286 l*.3U 37.25 37.27 .5578 .51*31 .9595 5.207 .3350 .31*1*0 2.161 .9738 l*.26l* 1*.399 .2273 1*.323 37.70 37.72 .5573 .51*27 .9598 5.201* .3360 .3l*U9 2.167 .971*1 1*.310 l*.l*2i* .2260 1*.335 38. lU 38.15 .5568 .51*21* .9601 5.201 .3370 .31*59 2.173 .971*1* 1*.336 1*.Ij5o .221*7 l*.3l*6 38.59 38.60 .5563 .51*21 .9601* 5.198 .3380 .31*68 2.179 .971*7 l*.36l 1*.1*71* .2235 l*.358 39.02 39.01* .5558 .51*17 .9607 5.191* .3390 .31*77 2.185 .9750 1*.388 I*. 500 .2222 1*.369 39.1*8 39.1*9 .5553 .51*11* .9610 5.191 .31*00 .31*68 2.190 .9753 1*.1*13 1*.525 .2210 l*.38l 39.95, 39.96 .551*6 .51*11 .9613 5.188 .31*10 .31*95 2.196 .9756 1*.1*39 l*.55o .2198 l*.392 1*0.1*0 1*0.1*1 .551*1* .51*06 .9615 5.185 .31*20 .3501* 2.202 .9758 1*.U66 U.576 .2185 1*.1*01* 1*0.87 1*0.89 .5539 .51*05 .9618 5.182 .31*30 .3511* 2.208 .9761 l*.l*92 1*.602 .2173 1*.U16 ia.36 1*1.37 .5531* .51*02 .9621 5.179 .31*1*0 .3523 2.211* .9761* 1*.521 1*.630 .2160 1*.1*27 1*1.85 1*1.81* .5529 .5399 .9623 5.176 .31*50 .3532 2.220 .9767 1*.51*7 U.656 .211*8 l*.i*39 1*2.33 1*2.31* .552U .5396 ,9626 5.173 .31*60 .351*2 2.225 .9769 1*.575 U.682 .2136 l*.l*5i 1*2.83 1*2.81* .5519 .5392 .9629 5.171 .31*70 .3551 2.231 .9772 1*.602 1*.709 .2121* 1*.1*62 1*3.31* 1*3.35 .5515 .5389 .9632 5.168 .31*80 .3560 2.237 .9775 1*.629 1*.736 .2111 l*.l*7l* 1*3.85 1*3.86 .5510 .5386 .9635 5.165 .31*90 .3570 2.21*3 .9777 1*.657 1*.763 .2099 1*.1*86 1*1*.37 1*1*. 1*0 .5505 .5383 .9638 5.162 .3500 .3579 2.21*9 .9780 1*.685 i*.791 .2087 1*.1*98 1*1*.89 1*1*.80 .5501 .5380 .961*0 5.159 .3510 .3588 2.255 .9782 1*.713 1*.818 .2076 1*.509 1*5.1*2 1*5.1*3 .51*96 .5377 .961*8 5.157 .3520 .3598 2.260 .9785 l*.7l*l 1*.81*5 ,2061* U.521 1*5.95 1*5.96 .51*92 .5371* .961*6 5.151* .3530 .3607 2.266 .9787 1*.770 1*.873 .2052 1*.533 1*6.50 1*6.51 .51*87 .5371 .961*8 5.152 .351*0 .3616 2.272 .9790 1*.798 1*.901 .201*0 1*.51*1* 1*7.03 1*7. Ol* .51*83 .5368 .9651 5.ia9 .3550 .3625 2.278 .9792 1*.827 1*.929 .2029 1*.556 1*7.59 1*7.60 .5U79 ,5365 .9651* 5.11*7 .3560 .3635 2.281* .9795 1*.856 1*.957 .2017 1*.568 1*8.15 1*8.16 .51*71* .5362 .9657 5.11*1* .3570 .361*1* 2.290 .9797 1*.885 I*.987 .2005 1*.579 1*8.72 1*8.73 .51*70 .5359 .9659 5.11*1 .3580 .3653 2.296 .9799 lj.911* 5.015 .1991* 1*.591 1*9.29 1*9.30 .51*66 .5356 .9662 5.139 .3590 .3663 2.301 .9801 1*.9U1* 5.0UU .1983 1*.603 1*9.88 1*9.89 .51*61 .5353 .9665 5.137 .3600 .3672 2.307 .9801* l*.97l* 5.072 .1972 1*.615 50.1*7 50.1*8 .51*57 .5350 .9667 5.131* .3610 .3682 2.313 .9606 5.00I* 5.103 .1960 U.627 51.08 51.09 .51*53 .531*7 .9670 5.132 .3620 .3691 2.319 .9808 5.031* 5.132 .191*9 1*.638 51.67 51.67 .51*1*9 .531*1* .9673 5.130 .3630 .3700 2.325 .9811 5.063 5.161 .1938 l*.65o 52.27 52.26 ' .51*1*5 .531*2 .9675 5.127 .361*0 .3709 2.331 .9813 5.09I4 5.191 .1926 U.661 52.89 52.90 .51*1*1 .5339 .9677 5.125 .3650 .3719 2.337 .9815 5.121* 5.221 .1915 1*.673 53.52 53.53 .51*37 .5336 .9680 5.123 .3660 .3728 2.31*2 .9817 5.155 5.251 .1901* 1*.685 51*.15 51*. 16 .51*33 .5333 .9683 5.121 .3670 .3737 2.31*8 .9819 5.186 5.281 .1891* 1*.697 5U.78 51*.79 .51*29 .5330 .9686 5.118 .3680 .371*7 2.351* .9821 5.217 5.312 .1883 i*.7oe 55.1*2 55.1*3 .51*25 .5327 .9668 5.116 .3690 .3756 2.360 .9823 5.21*8 5.31*3 ,1872 1*.720 56.09 56.10 .51*21 .5325 .9690 5.111* .3700 .3766 2.366 .9825 5.280 5.371* .1861 1*.732 56.76 56.77 .51*17 .5322 .9693 5.112 .3710 .3775 2.372 .9827 5.312 5.1*06 .1850 l*.7l*l* 57.1*3 57.1*1* .51*13 .5319 .9696 5.110 .3720 .3785 2.378 .9830 5.31*5 5.1*38 .1839 1*.756 58.13 58.11* .51*09 .5317 .9698 51107 .3730 .3791* 2.381* .9832 5.377 5.1*69 .1828 1*.768 56.82 58.63 .51*05 .5311* .9700 5.105 .371*0 .3801* 2.390 .9831* 5.1*10 5.502 .1618 1*.780 59.52 59.53 .51*02 .5312 .9702 5.103 .3750 .3813 2.396 .9835 5.1+1*3 5.531* .1807 1*.792 60.21* 60.25 .5398 .5309 .9705 5.101 .3760 .3822 2.1*02 .9837 5.1*75 5.566 .1797 1*.803 60,95 60.95 .5391* .5306 .9707 5.099 .3770 .3832 2.1*08 .9839 5.508 5.598 .1786 l*.8l5 61.68 61,68 .5390 .5301* .9709 5.097 .3780 .381*1 2.1*13 .981*1 5.5U 5.631 .1776 1*.827 62.1*1 62.1*2 .5387 .5301 .9712 5.095 .3790 .3850 2.1*19 .98U3 5.572 5.661 .1766 1*.838 63.13 63.11* .5383 .5299 .9711* 5.093 .3800 .3860 2.1*25 .981*5 5.609 5.697 .1756 l*.85l 63.91 63.91 .5380 .5296 .9717 5.091 .3810 .3869 2.1*31 .981*7 5.61*3 5.731 .171*5 l*.862 61*. 67 61*.67 .5376 .5291* .9719 5.090 .3820 .3679 2.1*37 .981*8 5.677 5.765 .1735 1*.875 65.1*5 65.1*6 .5372 .5291 .9721 5.088 .3830 ,3888 2.1*1*3 .9850 5.712 5.798 .1725 1*.885 66.16 66.17 .5369 .5288 .9721* 5.086 .381*0 .3898 2.1*1*9 .9852 5.71*6 5.633 .1715 1*.898 67.02 67.03 .5365 .5286 .9726 5.081* .3850 .3907 2.1*55 .9851* 5.780 5.866 .1705 1*.910 67.80 67.81 .5362 .5281* .9728 5.082 .3860 .3917 2.1*61 .9855 5.811. 5.900 .1695 1*.922 68.61 68.62 .5359 .5281 .9730 5.081 .3870 .3926 2.1*67 .9857 5.850 5.935 .1685 l*.93l* 69.1*5 69.1*6 .5355 .5279 .9732 5.079 .3880 .3936 2.1*73 .9859 5.686 5.970 .1675 l*.9l*6 70.28 70.29 .5352 .5276 .9735 5.077 .3890 .391*5 2.1*79 .9860 5.921 6.005 .1665 1*.958 71.12 71.13 .531*9 .5271* .9737 5.076 D-IO Table D-1 Coni'd dA &/}. 2f^d/L TANH SINK COSH K l* 77 'd/L SINH COSH n H/H' M "'"o 2 r 7 -d/L 2 7 rd/L Zn dtL l* 7 ?'d/L a^d/L u 0 0 .3900 .3955 2 . 1*85 .9662 5.957 6.01*0 .1656 l *.970 71.97 71.98 . 53 a 5 .5271 .9739 5.07a .3910 . 3961 * 2.1*91 . 9861 * 5.993 6.076 . 161*6 1*.982 72.85 72.86 . 53 a 2 .5269 . 97 ai 5.072 .3920 .3971* 2 . 1*97 .9865 6.029 6.112 .1636 1*.993 73.72 73.72 .5339 .5267 5.071 .3930 .3983 2.503 .9867 6.066 6 . 11*8 .1627 5.005 71*.58 7a. 59 .5336 .5265 . 97 a 5 5.069 .39I4O .3993 2.509 .9869 6.103 6.185 .1617 5.017 75 . 1*8 75 . a 9 .5332 .5262 . 97 h 6 5.067 .3950 . 1*002 2.515 .9870 6 . 11*0 6.221 .1608 5.029 76 . 1*0 76 .ao .5329 .5260 .9750 5.066 .3960 . 1*012 2.521 .9872 6.177 6.258 .1598 5 . 01*1 77.31 77.32 .5326 .5258 .9752 5 . 06 a .3970 . 1*021 2.527 .9873 6.215 6.295 .1589 5.053 78 . 21 * 78.2a .5323 .5255 .975a 5.063 .3980 .1*031 2.532 .9871* 6.252 6.332 .1579 5,065 79.19 79.19 .5320 .5253 .9756 5.062 .3990 . 1 * 01*0 2.538 .9876 6.290 6.369 .1570 5.077 80.13 80.13 .5317 .5251 .9750 5.060 .Uooo .1*050 2.51*1* .9877 6.329 6 . 1*07 .1561 5.089 81.12 aL .12 .531a . 52 a 8 .9761 5.058 .I4OIO . 1*059 2.550 .9879 6.367 6 . 1 * 1*5 .1552 5.101 82.07 82.08 .5311 . 52 a 6 .9763 5.056 .U020 .1*069 2.556 .9880 6 . 1*06 6 . 1*83 . 151*2 5.113 83.06 83.06 .5308 . 52 aa .9765 5.055 . 1*030 . 1*078 2.562 .9882 6 . 1 * 1 *!* 6.521 .1533 5.125 8 i *.07 8 a .07 .5305 . 52 a 2 .9766 5.053 . 1 * 01*0 .1*088 2.568 .9883 6 .U 81 * 6.561 .1521* 5.137 85.11 85.12 .5302 . 52 ao .9768 5.052 . 1*050 .1*098 2.575 .9885 6.525 6.601 .1515 5 . 11*9 86 . 11 * 86.1a .5299 .5238 .9770 5.050 .1*060 .1*107 2.581 .9886 6 . 561 * 6 . 61*0 .1506 5.161 87.17 87.17 .5296 .5236 .9772 5 .oa 9 . 1*070 .iai 6 2.586 .9887 6.603 6.679 . 11*97 5.173 88.19 88.20 .5293 .523a .977a 5 .oa 8 .1*080 . 1*126 2.592 .9889 6 . 61 * 1 * 6.718 . 11*88 5.185 89.28 89.28 .5290 .5232 .9776 5 .oa 6 .1*090 .1*136 2.598 .9890 6 . 681 * 6.758 .11*80 5.197 90.38 90.39 .5287 .5229 .9778 5 .oa 5 . 1*100 . 1 * 11*5 2.60I* .9891 6.725 6.799 . 11*71 5.209 91.1*1* 9 i.aa .5285 .5227 .9780 5 .oaa .laio .1*155 2.610 .9892 6.766 6,839 . 11*62 5.221 92.51* 92.55 .5282 .5225 .9782 5 .oa 3 . 1*120 . 1 * 161 * 2.616 .9891* 6.806 6.879 . 11 * 51 * 5.233 93.67 93.67 .5279 .5223 .978a s.oai .1*130 .1*171* 2.623 .9895 6 . 8 i *9 6.921 . 11 * 1*5 5 . 21*5 91*. 83 9a. 83 .5277 .5221 .9786 5 .oao . 1 * 11*0 . 1*183 2.629 .9896 6.890 6.963 . 11*36 5.257 95.95 95.96 .527a .5219 .9788 5.039 .1*150 .la 93 2.635 .9898 6.932 7.00I* . 11*28 5.269 97.13 97.13 .5271 .5217 .9790 5.037 ,1*160 .1*203 2 . 61*1 .9899 6.971* 7 . 01*6 . 11*19 5.281 96.29 98.30 .5269 .5215 .9792 5.036 .1*170 . 1*212 2 . 61*7 .9900 7.018 7.088 . 11*11 5.291* 99.52 99.52 .5266 .5213 .979a 5.035 .1*180 . 1*222 2.653 .9901 7.060 7.130 . 11*03 5.305 100.7 100.7 .5263 .5211 .9795 5.03a .ia 90 .1*231 2.659 .9902 7.102 7.173 .1391* 5.317 101.9 101.9 .5261 .5209 .9797 5.033 .1*200 . 1 * 21*1 2.665 .9901* 7 . 11*6 7.215 ,1386 5.329 103.1 103.1 .5258 .5208 .9798 5.031 .1*210 .U 251 2.671 .9905 7.190 7.259 .1378 5.31*1 10 l*.l* loa.a .5256 .5206 .9800 5.030 .1*220 . 1*260 2.677 .9906 7.231* 7.303 .1369 5.353 105.7 105.7 .5253 .520a .9802 5.029 .1*230 .1*270 2.683 .9907 7.279 7 . 31*9 .1361 5.366 107.0 107.0 .5251 .5202 .980a 5.028 .1*21*0 . 1*280 2.689 .9908 7.325 7.392 .1353 5.378 108.3 108.3 . 52 a 8 .5200 .9806 5.027 .1*250 . 1*289 2.695 .9909 7.371 7 . 1*38 .131*5 5.390 109.' 109.7 . 52 a 6 .5198 .9808 5.026 .1*260 .1*298 2,701 .9910 7 . 1*12 7 . 1*79 .1337 5 . 1*02 110.9 110.9 . 52 aa .5196 .9610 5.025 .1*270 .1*308 2.707 .9911 7 . 1*57 7.521* .1329 5 . 1 * 11 * 112.2 112.2 . 52 ai .5195 .9811 5.02a . 1*280 . 1*318 2.713 .9912 7.503 7.570 .1321 5 . 1*26 113.6 113.6 .5239 .5193 .9812 5.023 .1*290 .1*328 2.719 . 991 > 7.550 7.616 .1313 5 . 1*38 115.0 115.0 .5237 .5191 .981a 5.022 .1*300 . 1*337 2.725 .9911* 7.595 7.661 .1305 5.1*50 116 . 1 * 116. a .523a .5189 .9816 5.021 .1*310 .1*31*7 2.731 .9915 7 . 61*2 7.707 .1298 5.1*62 117.8 117,8 .5232 .5187 .9818 5.020 .1*320 . 1*356 2.737 .9916 7.688 7.753 .1290 5 . 1 * 71 * 119.2 119.3 .5230 .5186 .9819 5.019 .1*330 . 1*366 2 . 71*3 .9917 7.735 7.800 .1282 5 .a 86 120.7 120.7 .5227 . 5 l 8 a .9821 5.018 .1*31*0 . 1*376 2.71*9 .9918 7.783 7 . 81*7 .1271* 5 . 1*99 122.2 122.2 .5225 .5182 .9823 5.017 . 1*350 . 1*385 2.755 .9919 7.831 7.895 .1267 5.511 123.7 123.7 .5223 .5181 .982a 5.016 .1*360 . 1*395 2.762 .9920 7.880 7.91*3 .1259 5.523 125.2 125.2 .5221 .5179 .9826 5.015 .U 370 .1*1*05 2.768 .9921 7.922 7.991 .1251 5.535 126.7 126.7 .5218 .5177 .9828 5.01a . 1*380 .Ulill* 2 . 771 * .9922 7.975 8.035 . 121 * 1 * 5.51*7 128.3 128.3 .5216 .5176 .9829 5.013 .1*390 . 1 * 1 * 21 * 2.780 .9923 8.026 8.088 .1236 5.560 129.9 129.9 .521a .517a .9830 5.012 . 1 * 1*00 .Ul* 3 l* 2.786 . 992 U 8.075 8.136 .1229 5.572 131. a 131. a .5212 .5172 .9832 5.011 . 1 * 1*10 JM 3 2.792 .9925 8 . 121 * 8.185 .1222 5 . 581 * 133.0 133.0 .5210 .5171 .9833 5.010 . 1 * 1*20 . 1 * 1*53 2.798 .9926 8.175 8.236 . 1211 * 5.596 13a. 7 13 a .7 .5208 .5169 .9835 5.009 . 1 * 1*30 .1*1*63 2 . 801 * .9927 8.228 8.285 .1207 5.608 136.3 136.3 .5206 .5168 .9836 5.008 . 1 * 1 * 1*0 . 1 * 1*72 2.810 .9928 8 . 271 * 8.331* .1200 5.620 137.9 137.9 .520a .5166 .9838 5.007 . 1 * 1*50 . 1 * 1*82 2.816 .9929 8.326 8.387 .1192 5.632 139.6 139.7 .5202 .5165 .9839 5.006 . 1 * 1*60 .1*1*92 2.822 .9930 8.379 8 . 1*38 .1185 5 . 61 * 1 * lai.a lai.a .5200 .5163 . 98 ai 5.005 . 1 * 1*70 .1*501 2.828 .9930 8 . 1*27 8 .U 86 .1178 5.657 ia 3 .i ia 3 .i .5198 • 5161 . 98 a 3 5.005 . 1 * 1*80 .1*511 2.831* .9931 8 . 1*81 8 . 51*0 .1171 5.669 iaa .8 iaa .8 .5196 .5160 . 98 aa 5.00a .1*1*90 .1*521 2 . 81*0 .9932 8.532 8.590 . 1161 * 5.681 ia 6.6 ia 6.6 .519a .5158 . 981*6 5.003 D-l I Table D-1 Cont'd d/L d/L 2 '^ d/L UNH SIKH COSH K a(7 d/L SINH COSH n h/h: M 0 in d/L 217 d/L 2 d/L U^d/L a^Td/L [j 0 0 .i*5oo .1*531 2.8^7 .9933 8.585 8.6a3 .1157 5.693 II48.U ia8.a .5192 .5157 . 9 Bk 7 5.002 .^510 .1*51*0 2.853 .993a 8.638 8.695 .1150 5.705 150.2 150.2 .5190 .5156 .96a8 5.001 .u$20 .1*550 2.859 .9935 8.693 8.750 .iia3 5.717 152.1 152.1 .5188 .515a .98a9 5.000 .1*530 .1*560 2.865 .9935 8.7a7 8.80a .1136 5.730 i5a.o 15a.0 .5186 .5152 .9851 5.000 .i*5Uo .1*569 2.871 .9936 8.797 8.85a .1129 5.7a2 155.9 155.9 .518a .5151 .9852 a.999 .1*550 .1*579 2.877 .9937 8.853 8.910 .1122 5.75a 157.7 157.7 .5182 .5150 .9853 a.998 .1*560 .1*589 2.883 .9938 8.910 8.965 .1115 5.766 159.7 159.7 .5181 .5ia8 .9855 a.997 .1*570 .1*599 2.890 .9938 8.965 9.021 .1109 5.779 161.7 161.7 .5179 .5ia6 .9857 a, 997 .1*580 .1*608 2.896 .9939 9.016 9.072 .1102 5.791 163.6 163.6 .5177 .5ia5 .9658 a.996 .1*590 .1*618 2.902 .99ao 9.07a 9.129 .1095 5.803 165.6 165.6 .5175 .5iaa .9859 a.995 .1*600 .1*628 2.908 .99ai 9.132 9.186 .1089 5.815 167.7 167.7 .5173 .5ia3 .9860 a.99a ,1*610 .1*637 2.911* .99ai 9.183 9.238 .1083 5.827 169.7 169.7 .5172 .5iai .9862 a. 99a .1*620 .1*61*7 2.920 . 99 U 2 9.2a2 9.296 .1076 5.8ao 171.6 171.6 .5170 .5iao .9863 a.993 .1*630 .1*657 2.926 .99a3 9.301 9.35a .1069 5.852 173.9 173.9 .5168 .5139 .986a a.992 .1*61*0 .1*666 2.932 .99aa 9.353 9.ao6 .1063 5.86a 176.0 176.0 .5167 .5138 .9865 a,99i .1*650 .1*676 2.938 .99aa 9.ai3 9.a66 .1056 5.876 178.2 178.2 .5165 .5136 .9867 a.991 .1*660 .1*686 2.91*1* .99a5 9.a72 9.525 .1050 5.888 180. a I80.a .5163 .5135 .9868 a.990 ,1*670 .1*695 2.951 .99U6 9.533 9.585 .ioa3 5.900 182.6 182.6 .5162 .513a .9869 a.989 .1*680 .1*705 2.957 .99146 9.586 9.638 .1037 5.912 isa.e I8a.8 .5160 .5132 .9871 a.989 .1*690 .1*715 2.963 . 99 k 7 9.6a7 9.699 .1031 5.925 187.2 187.2 .5158 .5131 .9872 a.988 .1*700 .1*725 2.969 .99I47 9.709 9.760 .1(525 5.937 189.5 189.5 .5157 .5129 .9873 a.988 .1*710 .1*735 2.975 .99a8 9.770 9.821 .1018 5.9a9 191.8 191.8 .5155 .5128 .987a a. 987 .1*720 .1*71*1* 2.981 .99a9 9.826 9.877 .1012 5.962 19a. 2 19a. 2 .515a .5127 .9875 a.986 .1*730 .1*751* 2.987 .99a9 9.888 9.938 .1006 5.97a 196.5 196.5 .5152 .5126 .9876 a.986 .1*71*0 .1*761* 2.993 .9950 9.951 10.00 .1000 5.986 199.0 199.0 .5150 .5125 .9877 a.985 .1*750 .1*77L 2.999 .9951 10.01 10.07 .099a2 5.999 201.a 201.a .5ia9 .512a .9878 a.96a .1*760 .1*763 3.005 .9951 10.07 10.12 .09882 6.011 203.9 203.9 .5ia7 .5122 .9880 a. 98a .1*770 -1*793 3.012 .9952 10.13 10.18 .09820 ' 6.023 206.5 206.5 .5ia6 .5121 .9881 a.983 .1*780 .a803 3.018 .9952 10.20 10.25 .09759 6.036 209.0 209.0 .5iaa .5120 .9882 a.983 .1*790 .1*813 3.021* .9953 10.26 10.31 .09698 6.01*8 211.7 211.7 .5ia3 .5119 .9883 a.962 .1*800 .1*822 3.030 .9953 10.32 10.37 .096ai 6.060 21a. 2 21a. 2 .5ia2 .5117 .9885 a.982 .1*810 .1*832 3.036 .995a 10.39 10.a3 .09583 6.072 216.8 216.8 .5iao .5116 .9886 a. 981 .a820 .a8U2 3.01*2 .9955 io.a5 10.50 .09523 6.085 219.5 219.5 .5139 .5115 .9887 a. 980 .1*830 .1*852 3.01*9 .9955 10.52 10.57 .09a6a 6.097 222.2 222.2 .5137 .511a .9888 a.980 .1*81*0 .1*862 3.055 .9956 10.59 10.63 .o9ao5 6.109 225.0 225.0 .5136 .5113 .9889 a.979 .1*850 .1*871 3.061 .9956 10.65 10.69 .09352 6.121 228.3 228.3 .513a .5112 .9890 a.979 .1*860 .1*881 3.067 .9957 10.71 10.76 .0929a 6.13a 230.6 230.6 .5133 .5111 .9891 a.978 .1*870 .1*891 3.073 .9957 10.78 10.83 .09236 6.ia6 233.5 233.5 .5132 .5110 .9892 a.978 .1*880 .1*901 3.079 .9958 10.85 10.90 .09178 6.159 236. a 236.a .5130 .5109 .9893 a.977 ,1*890 .1*911 3.086 .9958 10.92 10.96 .09121 6.171 239.6 239.6 .5129 .5107 .9895 a.977 .1*900 .1*920 3.092 .9959 10.99 11.03 .0906a 6.183 2a2.3 2a2.3 .5128 .5106 -9896 a. 976 .1*910 .1*930 3.098 .9959 11.05 11.09 .09010 6.195 2a5.2 2a5.2 .5126 .5105 .9897 a.976 .1*920 .1*91*0 3.I0I* .9960 11.12 11.16 .08956 6.208 2a8.3 2a8.3 .5125 .510a .9898 a.975 .1*930 .1*950 3.110 .9960 11.19 11.2a .08901 6.220 251.3 251.3 .512a .5103 .9899 a.975 .1*91*0 .1*960 3.117 .9961 11.26 11.31 .088a5 6.232 25a.5 25a. 5 .5122 .5102 .9899 a. 97a .1*950 .1*969 3.122 .9961 11.32 11.37 .08793 6.2a5 257.6 257.6 .5121 .5101 .9900 a.97a .1*960 .1*979 3.128 .9962 11. ao 11. aa .087ai 6.257 260,8 260.8 .5120 .5100 .9901 a.973 .1*970 .1*989 3.135 .9962 ii.a7 11.51 .08691 6.269 26a. 0 26a. 0 .5119 .5099 .9902 a.973 .1*980 .U999 3.11*1 .9963 11.5a 11.59 .08637 6.282 267.3 267.3 .5118 .5098 .9903 a-972 .1*990 .5009 3.11*7 .9963 11.61 11.65 .0858a 6.29a 270.6 270.6 .5116 .5097 .990a a.972 .5000 .5018 3.153 .996a 11.68 11.72 .08530 6.306 27a.0 27a.0 .5115 .5096 .9905 a.971 .5010 .5028 3.159 .996a 11.75 11.80 .06a77 6.319 277.5 277.5 .511a .5095 .9906 a.971 .5020 .5038 3.166 .996a 11.83 11.87 .o8a2a 6.331 280.8 280.8 .5113 .509a .9907 a.971 .5030 .501*8 3.172 .9965 11.91 11.95 .08371 6.3a3 26a.3 28a .*3 .5112 .5093 .9908 a. 970 .sol*o .5058 3.178 .9965 11.98 12.02 .08320 6.356 287.9 287.9 .5110 .5092 .9909 a. 970 .5050 .5067 3.18a .9966 12.05 12.09 .08270 6.368 291.a 291.a .5109 .5092 .9909 a. 969 .5060 .5077 3.190 .9966 12.12 12.16 .08220 6.380 295.0 295.0 .5108 .5091 .9910 a. 969 .5070 .5087 3.196 .9967 12.20 12.2a .08169 6.393 298.7 298.7 .5107 .5090 .9911 a. 968 .5080 .5097 3.203 .9967 12.28 12.32 .08119 6.ao5 302. a 302.a .5106 .5089 .9912 a. 968 .5090 .5107 3.209 .9968 12.35 12.39 .08068 6.ai7 306.2 306.2 .5105 .5088 .9913 a. 967 D-12 Table D-1 Coni'd d/L 0 d/L 2 n d/L TANH 277 d/L SINK 2f7’d/L COSH 277'd/L K 4^ d/L SINH 4^ d/L COSH n C„/C hrrd/L ^ ° H/H' 0 M .5100 .5117 3.215 .9968 12.1,3 12.1,7 .08022 6.430 310.0 310.0 .5104 .5087 .9914 4.967 .5110 .5126 3.221 .9968 12.50 12.51* .07972 6.442 313.8 313.8 .5103 .5086 .9915 4.967 .5120 .5136 3.227 .9969 12,58 12.62 .07922 6.454 317.7 317.7 .5102 .5086 .9915 4.966 .5130 .511*6 3.233 .9969 12,66 12.70 .07873 6.467 321.7 321.7 .5101 .5085 .9916 4.966 .51UO .5156 3.21*0 .9970 12.71* 12.78 .07624 6.479 325.7 325.7 .5100 .5084 .9917 4.965 .5150 .5166 3.21*6 .9970 12.82 12.86 .07776 6.491 329.7 329.7 .5098 .5083 .9918 4.965 .5160 .5176 3.252 .9970 12.90 12.91* .07729 6.504 333.8 333.8 .5097 .5082 .9919 4.965 .5170 .5185 3.258 .9971 12,98 13.02 .07682 6.516 337.9 337.9 .5096 .5082 .9919 4.964 .5180 .5195 3.261* .9971 13.06 13.10 .07634 6.529 342.2 342.2 .5095 .5081 .9920 4.964 .5190 .5205 3.270 .9971 13.11* 13.18 .07587 6.541 346.4 346.4 .5094 .5080 .9921 4.964 .5200 .5215 3.277 .9972 13.22 13.26 .07540 6.553 350.7 350.7 .5093 .5079 .9922 4.963 .5210 .5225 3.283 .9972 13.31 13.35 .07494 6.566 355.1 355.1 .5092 .5078 .9923 4.963 .5220 .5235 3.289 .9972 13.39 13.1*3 .07449 6.578 359.6 359.6 .5092 i5077 .9924 4.963 .5230 .521*1* 3.295 .9973 13.1*7 13.51 .07404 6.590 364.0 364.0 .5091 .5077 .9924 4.962 .521*0 .5251* 3.301 .9973 13.55 13.59 .07358 6.603 368.5 368.5 .5090 .5076 .9925 4.962 .5250 .5261* 3.308 .9973 13.61* 13.68 .07312 6.615 373.1 373.1 .5089 .5075 .9926 4.962 .5260 .5271* 3.311* .9971* 13.73 13.76 .07266 6.628 377.8 377.8 .5088 .5074 .9927 4.961 .5270 .5281* 3.320 .9971* 13.81 13.85 .07221 6.640 382.5 382.5 .5087 .5074 .9927 4.961 .5280 .5291* 3.326 .9971* 13.90 13.9I* .07177 6.652 387.3 387.3 .5086 .5073 .9926 4.961 .5290 .5301* 3.333 .9975 13.99 11*. 02 .07134 6.665 392.2 392.2 .5085 .5072 .9929 4.960 .5300 .5311* 3.339 .9975 11*. 07 11*. 10 .07091 6.677 397.0 397.0 .5084 .5071 .9930 4.960 .5310 .5323 3.31*5 .9975 111, 16 11*. 19 .07047 6.690 402.0 402.0 .5083 .5070 .9931 4.960 .5320 .5333 3.351 .9976 11*.25 11*. 28 .07003 6.702 406.9 406.9 .5082 .5070 .9931 4.959 .5330 .531*3 3.357 .9976 11*. 31* 11*. 37 .06959 6.714 412.0 412.0 .5082 .5069 .9932 4.959 .531*0 .5353 3.363 .9976 ll*.l*3 11*. 1,6 .06915 6.727 417.2 417.2 .5081 .5068 .9933 4.959 .5350 .5363 3.370 .9976 11*. 52 il*.55 .06872 6.739 422.4 422.4 .5080 .5068 .9933 4.959 .5360 .5373 3.376 .9977 11*. 61 ll*.61* .06829 6.752 427.7 427.7 .5079 .5067 .9934 4.958 .5370 .5383 3.382 .9977 11*. 70 ll*.73 .06787 6.764 433.1 433.1 .5078 .5066 .9935 4.958 .5360 .5393 3.388 .9977 11*. 79 ll*.82 .06746 6.776 438.5 436.5 .5077 .5066 .9935 4.958 .5390 .51*02 3.391* .9977 11*. 88 14.91 .06705 6.789 444.0 444.0 .5077 .5065 .9936 4.958 .51*00 .51*12 3.1*01 .9978 11*.97 15.01 .06664 6.801 449.5 449.5 .5076 .5065 .9936 4.957 ,51*10 .51*22 3.1*07 .9978 15.07 15.10 .06623 6.814 455.1 455.1 .5075 .5064 .9937 4.957 .51*20 .51*32 3.1*13 .9978 15..16 15.19 .06582 6.826 460,7 460.7 .5074 .5063 .9938 4.957 ,51*30 .51*1*2 3.1a9 .9979 15.25 15.29 .06542 6.838 466.4 466.4 .5073 .5063 .9938 4.956 .51*1*0 .5U52 3.1*26 .9979 15.35 15.38 .06501 6.851 472.2 472.2 .5073 .5062 .9939 4.956 .51*50 .51*61 3.U32 .9979 15.1*5 15.48 .06461 6.863 478.1 478.1 .5072 .5061 .9940 4.956 .51*60 .51*71 3.1*38 .9979 15.51* 15.58 .06420 6.876 484.3 484.3 .5071 .5060 .9941 4.956 .51*70 .5U81 3.1*1*1* .9980 15.6U 15.67 .06380 6.888 490.3 490.3 .5070 .5060 .9941 4.955 .51*80 .51*91 3.1*50 .9980 15.71* 15.77 .06341 6.901 496.4 496.4 .5070 .5059 .9942 4.955 .51*90 .5501 3.1*56 .9980 15.81* 15.87 .06302 6.913 502.5 502.5 .5069 .5059 .9942 4.955 .5500 .5511 3.1*63 .9980 15.91* 15.97 .06263 6.925 508.7 508.7 .5068 .5058 .9942 4.955 .5510 .5521 3.1*69 .9981 16.01* 16.07 .06224 6.937 515.0 515.0 .5067 .5058 .9942 4.954 .5520 .5531 3.1*75 .9981 16.11* 16.17 .06186 6.950 521.6 521.6 .5067 .5057 .9943 4.954 .5530 .55U1 3.1*81 .9981 16.21* 16.27 .06148 6.962 528.1 528.1 .5066 .5056 .9944 4.954 .55ao .5551 3.1*88 .9981 16.31* 16.37 .06110 6.975 534.8 534.8 .5065 .5056 .9944 4.954 .5550 .5560 3.1*91* .9982 16.U* 16.47 .06073 6.987 541.4 541.4 .5065 .5056 .9945 4.953 .5560 .5570 3.500 .9982 16.51* 16.57 .06035 7.000 548.1 548.1 .5064 .5055 .9945 4.953 .5570 .5580 3.506 .9982 16.65 16.68 .05997 7.012 554.9 554.9 .5063 .5054 .9946 4.953 .5580 .5590 3.512 .9982 16.75 16.78 .05960 7.025 562.0 562.0 .5063 .5053 .9947 4.953 .5590 .5600 3.519 .9982 16.85 16.88 .05923 7.037 569.1 569.1 .5062 .5053 .9947 4.953 .5600 .5610 3.525 .9983 16.96 16.99 .05887 7.050 576.1 576.1 .5061 .5053 .9947 4.952 .5610 .5620 3.531 .9983 17.06 17.09 .05850 7.062 583.3 583.3 .5061 .5052 .9948 4.952 .5620 .5630 3.537 .9983 17.17 17.20 .05814 7.074 590,7 590.7 .5060 .5051 .9949 4.952 .5630 .561*0 3.51*3 .9983 17.28 17.31 .05778 7.087 598.0 598.0 .5059 .5051 .9949 4.952 .561*0 .561*9 3.550 .9981* 17.38 17.41 .05743 7.099 605.0 605.0 .5059 .5050 .9950 4.951 .5650 .5659 3.556 .9981* 17.1*9 17.52 .0570? 7.112 613.2 613.2 .5058 .5050 .9950 4.951 .5660 .5669 3.562 .9981* 17.60 17.63 .05672 7.124 620.8 620.8 .5057 .5049 .9951 4.951 .5670 .5679 3.568 .9981* 17.71 17.74 .05637 7.136 628.5 628.5 .5057 .5049 .9951 4.951 .5680 .5689 3.575 .9981* 17.82 17.85 .05602 7.149 636.4 636.4 .5056 .5048 .9952 4.951 .5690 .5699 3.581 .9985 17.91* 17.97 .05567 7.161 644.3 644.3 .5056 .5048 .9952 4.950 D- 13 Table D-1 Cont'd d/L o d/L 2 IT d/L TANH 2Trd/L SINH 2Trd/L COSH 2^ d/L K Un-d/L SINH UTTd/L COSH n U^d/L C„/C 0 0 H/H' 0 M .5700 .5709 3.587 .9985 18.05 18.08 .05532 7.17I, 652.1, 652.1, .5055 .501,7 .9953 l,.95o .5710 .5719 3.593 .9985 18,16 18.19 .051,97 7.186 660.5 660.5 .5051, .501,7 .9953 1,.950 .5720 .5729 3.600 .9985 18.28 18.31 .051,63 7.199 668.8 668.8 .5051, .501,6 .9951, 1,.950 .5730 .5738 3.606 .9985 18.39 16 .1,2 .051,30 7.211 677.2 677.2 .5053 .501*6 .9951, 1j.95o .57U0 .571,8 3.612 .9985 18.5o 18.53 .05396 7.221, 685.6 685.6 .5053 .501,5 .9955 1,.950 .5750 .5758 3.618 .9986 18.62 18 .61, .05363 7.236 691,. 3 691,.3 .5052 .501,5 .9955 l,.9l,9 .5760 .5768 3.621, .9986 18.73 18.76 .05330 7.21,9 703.2 703.2 .5052 .501,1, .9956 4.9I49 .5770 .5778 3.630 .9986 18.85 18.86 .05297 7.261 7U.9 711.9 .5051 .50W, .9956 4.91,9 .5780 .5788 3.637 .9986 18.97 19.00 .05261, 7.271, 720.8 720.8 .5051 .501,3 .9957 4.949 .5790 .5798 3.61,3 .9986 19.09 19.12 .05231 7.286 729.9 729.9 .5050 .501,3 .9957 4.949 .5800 .5808 3.61,9 .9987 19.21 19.21, .05198 7.298 739.0 739.0 .501,9 .501,3 .9957 4.948 .5810 .5818 3.656 .9987 19.33 19.36 .05166 7.311 718.1 71,8.1 .501,9 .501,2 .9958 4.948 .5820 .5828 3.662 .9987 19.1,5 19.1,8 .05131, 7.323 757.5 757.5 . 501,8 .501,2 .9958 4.948 .5830 .5838 3.668 .9987 19.58 19.60 .05102 7.336 767.0 767.0 .501,8 .501,1 .9959 4.948 .58UO .581,8 3.671, .9987 19.70 19.73 .05070 7.31,8 776.7 776.7 .501*7 .501*1 .9959 4.948 .5850 .5858 3.680 .9987 19.81 19.81, .0501,0 7.361 786.5 786.5 .501,7 .501,0 .9960 4.948 .5860 .5867 3.686 .9987 19.91, 19.96 .05009 7.373 796.1, 796.1, .501,6 .501,0 .9960 4.948 .5870 .5877 3.693 .9988 20.06 20.09 .01,978 7.386 806.5 806.5 .501,6 .501*0 .9960 4.947 .5880 .5887 3.699 .9988 20.19 20.21 .01,91,7 7.398 816.5 816.5 .501,5 .5039 .9961 4.947 .5890 .5697 3.705 .9988 20.32 20.31, .01,916 7.1,11 826.7 826.7 .501,5 .5039 .9961 4.947 .5900 .5907 3.712 .9988 20.1,5 20.1,7 .01,885 7.1,23 837.1 837.1 .50UI, .5038 • 9962 4.947 .5910 .5917 3.718 .9988 20.57 20.60 .01,855 7.1,36 81,7.6 81,7.6 .SOhk .5038 .9962 4.947 .5920 .5927 3.721, .9988 20.70 20.73 .01,821, 7.1,1,8 858.2 858.2 .501,3 .5037 .9963 4.947 .5930 .5937 3.730 .9989 20.83 20.86 .01,791, 7.1,60 868.9 868.9 .501,3 .5037 .9963 4.946 .591^0 .591,7 3.737 .9989 20.97 20.99 .01,761, 7.1,73 879.8 879.8 .501,3 .5037 .9963 4.946 .5950 .5957 3.71,3 .9989 21.10 21.12 .01,735 7.1,85 890.8 890.8 .501,2 .5036 .9961, 4.946 .5960 .5967 3.71,9 .9989 21.23 21.25 .01,706 7.1,98 901.9 901.9 .501,2 .5036 .9961, 4.946 .5970 .5977 3.755 .9989 21.35 21.37 .01,677 7.510 913.1, 913.1, .5014 .5036 .9961, 4.946 .5980 .5987 3.761 .9989 21.1,9 21.51 .01,61,8 7.523 925.0 925.0 .5oai .5035 .9965 4.946 .5990 .5996 3.767 .9989 21.62 21.61, .01,619 7.535 936.5 936.5 .501,0 .5035 .9965 4.946 .6000 .6006 3.77U .9990 21.76 21.78 .01,591 7.51,8 91,8.1 916.1 .501,0 .5035 .9965 4.945 .6100 .6106 3.836 .9991 23.17 23.19 .01,313 7.673 1,071, 1,071* .5036 .5031 .9969 4.^44 .6200 .6205 3.899 .9992 21,.66 21 ,. 68 .01,052 7.798 1,217 1,217 .5032 .5028 .9972 4.943 .6300 .6305 3.961 .9993 26.25 26.27 .03806 7.923 1,379 1,527 1,379 .5029 1,527 .5026 .5025 .9975 4.942 .61*00 .61,01, l,.02l, .9991, 27.95 27.97 .03576 8.01,8 .5023 .9911 4.941 .6500 .6501, 1,.086 .999h 29.75 29.77 .03359 8.173 1,771 1,771 .5023 .5020 .9980 4.940 .6600 .6603 1,.11,9 .9995 31.68 31.69 .03155 8.298 2,008 2,275 2,008 .5021 .5018 .9982 4.940 .6700 .6703 1,.212 .9996 33.73 33.71, .02961, 8.1,23 2,275 .5019 .5017 .9983 4.939 .6800 .6803 l,.27l. .9996 35.90 35.92 .02781, 8.51,8 2,579 2,579 .5017 .5015 .9965 4.939 .6900 .6902 1,.337 .9997 38.23 38.21, .02615 8.671, 2,923 2,923 .5015 .5013 .9987 4.938 .7000 .7002 1,.1,00 .9997 1,0.71 1,0.72 .021,56 8.799 3,311, 3,311, .5013 .5012 .9988 4.938 .7100 .7102 1,.1,62 .9997 U3.31, 1,3.35 .02307 8.925 3,757 3,757 .5012 .5011 .9989 4.937 .7200 .7202 1,.525 .9998 1 ,6.11, 1,6.15 .02167 9.050 h , 2 S 8 1,,258 .5011 .5010 .9990 4.937 .7300 .7302 1,.588 .9998 1,9.13 1,9.11, .02035 9.175 1,,828 1,.828 .5010 .5009 .9991 4.937 .7100 .71,01 U.650 .9998 52.31 52.32 .01911 9.301 5,1,73 .5,1,73 .5009 .5008 .9992 4.937 ,7500 .7501 1,.713 .9998 55.70 55.71 .01795 9.1,26 6 ,201, 6,201, .5008 .5007 .9993 4.936 .7600 .7601 1.776 .9999 59.31 59.31 .01686 9.552 7,031, 7,031, .5007 .5006 .9991, 4.936 .7700 .7701 1,.839 .9999 63.15 63.16 .01583 9.677 7,976 7,976 .5006 .5005 .9995 4.936 .7800 .7801 1,.902 .9999 67.2i, 67.25 .011,87 9.803 9 ,01,2 9,01,2 .5005 .5001, .9996 4.936 .7900 .7901 h.96h .9999 71,60 71.60 .01397 9.929 10,250 10,250 .5005 .5001, .9996 4.936 .8000 .8001 5.027 .9999 76.21, 76.21, .01312 10.05 11,620 11,620 .5001, 13,180 .5001, .5001, .9996 4.936 .8100 .8101 5.090 .9999 81.16 81.19 .01232 10.18 13,180 .5001, .9996 4.936 .8200 .8201 5.153 .9999 86 .1,1, 86 .1,1, .01157 10.31 llj,9l,0 11 ,, 91,0 .5003 .5003 .9997 4.935 .8300 .8301 5.215 .9999 92.01, 92,05 .01086 10.1,3 17,31,0 17,31,0 .5003 .5003 .9997 4.935 .81,00 .81,00 5.278 1.000 98.00 98.01 .01020 10.56 19,210 19,210 .5003 .5003 .9997 4.935 .8500 .8500 5.31,1 1.000 lOl,.!, lOlt.l, .009582 10.68 21,780 21,780 .5002 .5002 .9998 4.935 .8600 .8600 5.1,01, 1.000 111.1 111.1 .009000 10.81 21,, 690 21,, 690 .5002 .5002 .9998 4.935 .8700 .8700 5.167 1.000 118.3 118.3 .0081,51 10.93 28,000 31,750 28,000 .5002 31,750 .5002 .5002 .9998 4.935 .8800 .8800 5.529 1.000 126.0 126.0 .007931, 11.06 .5002 .9998 4.935 .8900 .8900 5.592 1.000 131,.2 131.2 .0071,51, 11.18 36,000 36,000 .5002 .5002 .9998 4.935 D-14 Table D-1 Cont'd dA 0 d/L 2t7 d/L TAWH 2ir'dA SINK 2Trd/L COSH 2TTd/L K UfFd/L SINH UfTa/L COSH n l.^d/L u 0 H/H' 0 H .9000 .9000 5i655 1.000 11.2.9 11.2.9 .007000 11,31 1.0,810 1.0,810 .5001 .5001 .9999 1..935 .9100 .9100 5.718 1.000 152.1 152.1 ,006571. 11.1.1. 1.6,280 1.6,280 .5001 .5001 .9999 1..935 .9200 .9200 5.781 1.000 162.0 162.0 .006173 11.56 52,a70 52,1.70 .5001 .5001 .9999 1..935 .9300 .9300 5.81.14 1.000 172.5 172.5 .005797 11.69 59,500 59,500 .5001 .5001 .9999 1..935 .9l<00 .9l»00 5.906 1.000 183.7 183.7 .005U.5 11.81 67,1.70 67,1.70 .5001 .5001 .9999 I..935 .9500 .9500 5.969 1.000 195.6 195.6 .005113 11.91. 76,1.90 76,1.90 .5001 .5001 .9999 1..935 .9600 .9600 6.032 1.000 203.5 203.5 .001.911. 12.06 86,71.0 86,71.0 .5001 .5001 .9999 I..935 .9700 .9700 6.095 1.000 222.8 222.8 .001.1.89 12.19 98,350 98,350 .5001 .5001 .9999 1..935 .9800 .9800 6.158 1.000 236.1 236.1 .001.235 12.32 111,500 111,500 .5001 .5001 .9999 1.935 .9900 .9900 6.220 1.000 251.1. 251. U .003977 i2.i.a 126,500 126,500 ,5000 .5000 1.000 I..935 1.000 1.000 6.283 1,000 267.7 267.7 .003735 12.57 11.3,1.00 11.3,1.00 .5000 .5000 1.000 8.935 TABLE D-2 FUNCTIONS OF d/L FOR EVEN INCREMENTS OF dA from 0.0001 to 0.2890 (This covers the region where interpolation of d/L in Table I is inconvenient. Values of d/L of 0.2890 to 1,0000 can be obtained from Table I by interpolation) dA d/Lo 2 frdA TANK 2 ff dA SINH 2rrd/L COSH 2 TrdA K UtTdA SINH l.Jrd/L COSH UTTdA n Cq/Co h/h; M 0 0 .0000000 0 0 0 1.0000 1.000 0 0 1.000 1.000 0 00 00 .0001000 6283 .000000 .0006283 .0006283 .0006283 1.0000 1.000 .001257 .001257 1.000 1.000 .0006283 28.21 12,500,000 .0002000 2511. .000000 .001257 .001257 .001257 1.0000 1.000 .002513 .002513 1.000 1.000 .001257 19.95 3,125,000 .0003000 5655 .00000 .001885 .001885 .001885 1,0000 1.000 .003770 .003770 1.000 1.000 .001885 16.29 1 ,389.000 .0001.000 1005 .00000 .002513 .002513 .002513 1.0000 1.000 .005027 .005027 1.000 1.000 ,002513 lU.io 781,300 .0005000 1571 .00000 .00311.2 ,00311.2 .00311.2 1.0000 1.000 .006283 .006283 1.000 1.000 .00311*2 12.62 500,000 .0006000 2262 .00000 .003770 .003770 .003770 1.0000 1.000 .00751.0 .00751.0 1.000 1.000 .003770 11.52 31.7,200 .0007000 3079 .00000 .001.398 .001.398 .001.398 1.0000 1.000 .008796 .008797 1.000 1.000 .001.398 10.66 255,100 .0008000 1.022 .00000 .005027 .005027 .005027 1.0000 1.000 .01005 .01005 1.000 1.000 .005026 9.971. 195,X)0 .0009000 5090 ,00000 .005655 .005655 .005655 1.0000 1.000 .01131 .01131 1.000 1.000 ,005655 9.1.03 151., 300 .001000 6283 .00000 .006283 .006283 .006283 1.0000 1.000 .01257 .01257 1.000 1.000 .006283 8.921 125,000 .001100 7603 .00000 .006912 .006911 .006912 1.0000 1.000 .01382 .01382 1,000 1.000 .006911 8.506 103,300 .001200 901.8 .00751.0 .00751.0 .00751.0 1.0000 1.000 .01508 .01508 1.000 1.000 .00751.0 8.1m. 86,810 .001300 .00001062 .008168 .008168 .008168 1.0000 1.000 .01631. .01631. 1.000 1.000 .008168 7 .821. 73,970 .ooiLoo .00001231 .008796 .008796 .008797 1.0000 1.000 .01759 .01759 1.000 1.000 .008796 7.539 63,780 .001500 .000011.11. .0091.25 .0091.25 .0091.25 1.0000 1.000 .01885 .01885 1,000 1.000 .0091.21. 7.281. 55,560 .001600 .00001608 .01005 .01005 ,01005 1.0001 .9999 .02011 .02011 1.000 1.000 .01005 7.052 1.8,830 .001700 .00001816 .01068 ,01068 ,01068 1.0001 .9999 .02136 .02136 1.000 1.000 .01068 6.81.2 1 .3,260 .001800 .00002036 .01131 ,01131 ,01131 1.0001 .9999 ,02262 .02262 1.000 1.000 .01131 6.61.9 38,580 .001900 .00002269 .01191. .01191. .01191. 1.0001 .9999 .02388 .02388 1.000 1.000 .01191. 6.1.72 31., 630 D-15 Table D-? Cont'd d/L d/Lo 2Trd/L TANH 2rTd/L SINK 2rTdA COSH K 2 77 d/L 1)77 d/L SINK 8T7d/L COSH 8n'd/L n H/H- u .002000 .00002511) .01257 .01257 .01257 l.OOQl .9999 0)2513 .02518 1.000 .9999 .01257 6.308 31,250 .002100 .00002772 .01319 .01319 .01320 1.0001 .9999 .02639 .02639 1.000 .9999 .01319 6.156 28,350 .002200 .0000301)0 .01382 .01382 .01382 1.0001 .9999 .02765 .02765 1.000 .9999 .01382 6.015 25,830 .002300 .00003321) .011)1)5 .011)1)5 .011)1)5 1.0001 .9999 0)2690 .02891 1,000 .9999 .01885 5.882 23,630 .0O2UOO .00003619 .01508 .01508 .01500 l.OOQl .9999 .03016 .03016 1.000 .9999 .01508 5.759 21,700 .002500 .00003928 .01571 .01571 .01571 1.0001 .9999 .0311)2 .03182 1.000 .9999 .01571 5.682 20,000 .002600 .00001)21)8 .01631) .01633 .01631) 1.0001 .9999 .03267 .03268 1.001 .9999 .01633 5.533 18,890 .002700 .00001)579 .01696 .01696 .01697 1.0001 .9999 .03393 .03398 1.001 .9999 .01696 5.829 17,150 .002800 .00001)925 .01759 .01759 .01759 1.0002 .9998 .03S19 .03519 1.001 .9999 .01759 5.332 15,950 .002900 .00005281) .01022 .01822 .01822 1.0002 .9998 .0361)1) .03685 1.001 .9999 .01822 5.239 18,870 .003000 .00005652 .01885 .01885 .01885 1.0002 .9998 .03770 .03771 1.001 .9999 .01885 5.151 13,890 .003100 .00006039 .0191)8 .0191)8 .0191)8 1.0002 .9998 .03896 .03897 1.001 ,9999 .01987 5.067 13,010 .003200 .000061)35 . 02011 .02010 .02011 1.0002 .9998 .01)021 .08022 1.001 .9999 .02010 8,987 12,210 11,880 .003300 .0000681)1 .02073 .02073 .02073 1.0002 .9998 .01)11)7 .08188 1.001 .9999 .02073 8.911 .OO3I4OO .00007262 .02136 .02136 .02136 1.0002 .9998 .01)273 .08278 1.001 .9998 .02136 8.838 10,820 .003500 .00007697 .02199 .02199 .02199 1.0002 .9998 .01)398 .08399 1.001 .9998 .02199 8.769 10,210 .003600 .0000811)0 . 02262 .02262 .02262 1.0003 .9997 .01)521) .08525 1.001 .9998 .02261 8.702 9,688 .003700 .00008599 .02325 .02321) .02325 1.0003 .9997 .01)650 .08652 l.OOl .9998 .02328 8.638 9,138 .003800 .00009071 .02388 .02387 .02388 1.0003 .9997 .01)775 .08777 1.001 .9998 .02387 8.577 8,660 .033900 .00009551 .021)50 .021)50 .021)51 1.0003 .9997 ,01)901 .08903 1.001 .9998 .02889 8.518 8,221 .0014000 .0001005 .02513 .02513 .02513 1.0003 .9997 .05027 .05029 1.001 .9998 .02511 8.862 7,815 .ooUioo .0001056 .02 576 .02576 .02576 1.0003 .9997 .05152 .05158 1.001 .9998 .02578 8.807 7,839 .00^200 .0001106 .02639 .02638 .02639 1.0003 .9991 .05278 .05280 1.001 .9998 .02637 8.358 7,090 .0014300 .0001161 .02702 .02701 .02702 i.oool) .9996 .051)01) .05806 1.001 .9998 ,02700 8.303 6,768 .OOljilOO .0001216 .02765 .02761) .02765 i.oool) .9996 .05529 .05531 1.002 .9997 .027 63 8.258 6,860 .OOIjSOO .0001272 .02827 .02827 .02828 I.oool) .9996 .05655 .05656 1.002 .9997 .02 825 8.207 6,176 .00U600 .0001329 .02890 .02889 .02890 I.oool) .9996 .05781 ,05788 1.002 .9997 .02 888 8.161 5,911 .001j700 .0001387 .02953 .02952 .02953 I.oool) .9996 .05906 .05909 1.002 .9997 .02951 8.116 5,662 .OOljSOO .00011)1)7 .03016 .03015 .03016 1.0005 .9995 .06032 .06035 1.002 .9997 .03018 8.073 5,829 .001)900 .0001508 .03079 .03076 .03079 1.0005 .9995 .06158 .06161 1.002 .9997 .03076 8.032 5,209 .OOSOOO .0001570 .0311)2 .0311a .0311)3 1.0005 .9995 .06283 .06287 1.002 .9997 .03139 3.991 5,003 •ooSioo .0001631) .O320I) .03203 .03205 1.0005 .9995 .061)09 .06813 1.002 .9997 .03202 3.951 8,809 .005200 .0001698 .03267 .03266 .03266 1.0005 .9995 .06535 ,06539 1.002 .9996 .03265 3.913 8,626 .005300 .0001761) .03330 .03329 .03331 1.0005 .9995 .06660 .06665 1.002 .9996 .03328 3.876 8,853 .005U00 .0001832 .03393 .03392 .03391) 1.0006 .999I) .06786 .06791 1.002 .9996 .03391 3.880 8,290 .005500 .OOQ1900 .031)56 .031)55 .031)57 1.0006 .999I) .069U .06916 1.002 .9996 .03858 3.805 8,135 .005600 .0001970 .03519 .03517 .03520 1.0006 .999I) .07037 .07082 1.002 .9996 .03517 3.771 3,989 .005700 .000201)1 .03581 .03580 .03582 1.0006 .999I) .07163 .07169 1.003 .9996 .03579 3.738 3,851 .005800 .0002112 .0361)1) .0361)2 .0361)5 1.0007 .9993 .07288 .07298 1.003 .9996 .03681 3.706 3,719 .005900 .0002186 .03707 .03705 .03708 1.0007 .9993 .07811) .07820 1.003 .9995 .03703 3.675 3,598 .006000 .0002261 .03770 .03768 .03771 1.0007 .9993 .07580 .07587 1.003 .9995 .03766 3.688 3,875 .006100 .0002337 .03833 .03831 .03831) 1.0007 .9993 .07665 .07672 1.003 .9995 .03829 3.618 3,363 .006200 .00021)11) .03896 .03891) .03097 1.0008 .9992 .07791 .07798 1.003 .9995 .03892 3.588 3,255 .006300 .00021)92 .03958 .03956 .03959 1.0006 .9992 .07917 .07925 1.003 .9995 .03958 3.556 3,153 .0061400 .0002570 .01)021 .01)019 .01)022 1.0008 .9992 .08082 .08050 1.003 .9995 .08017 3.528 3,055 .006500 .0002653 .01)081) .01)062 .01)085 1.0006 .9992 .06168 .08177 1.003 .9998 .08080 3.501 2,962 .006600 .0002735 .0Ull)7 .01)11)1| .01)11)8 1.0009 .9991 .00298 .08303 1.003 .9998 .08182 3.875 2,873 .006700 .0002819 .01)210 .01)207 .Ol)2U 1.0009 .9991 .OB8I9 .08828 1.008 .9998 0)8208 3.889 2,788 .006800 .OOO29OI) .01)273 .01)270 .01)271) 1.0009 .9991 .08585 .00555 1.008 .9998 .08267 3.823 2,707 .006900 .0002990 .01)335 .01)333 .01)336 1.0009 .9991 .08671 .08681 1.008 .9998 .08330 3.390 2,629 .007000 .0003077 .01)398 .01)395 .01)399 1.0010 .9990 .08796 .00807 1.008 .9998 .08392 3.378 2,558 .007100 .0003165 .01)1)61 .01)1)58 .01)1)62 1.0010 .9990 .00922 .08933 1.008 .9993 .08855 3.350 2,883 .007200 .OOO325I) .01)521) .01)521 .01)525 1.0010 .9989 .09088 ,09060 1.008 .9993 .08516 3.327 2,815 .007300 .000331)6 .01)587 .01)581) .01)589 l.OOU .9989 .09173 .09105 1.008 .9993 .08581 3.308 2,389 .0071)00 .00031)39 .01)650 .01)61)6 .01)652 1.0011 .9989 .09299 .09312 1.008 .9993 ,08688 3.201 2,286 .007500 .0003532 .01)712 .01)709 .01)711) l.OOU .9989 .09825 .09838 1.008 .9993 .08706 3.260 2,226 .007600 .0003627 .OU775 .01)772 .01)777 l.OOU .9989 .09550 .09565 1.005 ,9992 .08768 3.238 2,167 .007700 .0003722 .01)838 .01)831) .01)81)0 1.0012 .9986 .09676 .09681 1.005 .9992 .08830 3.217 2,112 .007800 .0003820 .01)901 .01)897 .01)903 1.0012 .9988 .09802 .09817 1.005 .9992 .08893 3.197 2,058 .007900 .0003918 .01)961) .01)960 .01)966 1.0012 .9988 .09927 .09983 1.005 .9992 .08956 3.176 2,006 D-16 Table D-2 Coni'd d/L d/Lo 2 IT d/L TANH 2TTd/L SINK 2 TTd/L COSH 2 TTd/L K 1* TTd/L SINH 1* TTd/L COSH 1* TTd/L n Cq/Co H/H'o M .008000 .000li0l8 .05027 .05022 .05029 1.0013 .9987 .1005 .1007 1.005 .9992 .05018 3.157 1,956 .006100 .oooliiia .05089 .05085 .05091 1.0013 .9987 .1018 .1020 1.005 .9991 .05080 3.137 1,909 .008200 .000li221 .05152 -05ll*7 .05151* 1.0013 .9987 .1030 .1032 1.005 .9991 .0511*2 3.118 1,862 .008300 .000lt32l« .05215 .05210 .05217 1.0011* .9986 .101,3 .101*5 1.005 .9991 .05205 3.099 l,8l8 •OOSljOO .OOOUIj29 .05278 .05273 .05280 l.OOU* .9986 .1056 .1058 1.006 .9991 .05268 3.081 1,775 .008500 •OOOI4536 .0531*1 .05336 .0531*3 l.ooU* .9986 .1068 .1070 1.006 .9991 .05331 3.062 1,733 .008600 .OOOU6I4I4 .051*01* .05398 .051*06 1.0015 .9985 .1081 .1083 1.006 .9990 .05391* 3.01*1* 1,693 .008700 .OOOU751 .051*66 .051*61 .051*69 1.0015 .9985 .1093 .1095 1.006 .9990 .051*56 3.027 1,655 .008800 .OOOI486O .05529 .05521* .05533 1.0015 .9985 .1106 .1108 1.006 .9990 .05518 3.010 1,617 .008900 .0001*972 .05592 .05586 .05595 1.0016 .9984 .1U8 .1121 1.006 .9990 .05580 2.993 1,581 .009000 .0005081* .05655 .0561*9 .05658 1.0016 .9981* .1131 .1133 1.006 .9989 .0561*3 2.977 1,5U6 .009100 .0005198 .05718 .05712 .05721 1.0016 .9981* .111*1* .111*6 1.006 .9989 .05706 2.960 1,513 .009200 .0005312 .05781 .0577U .05781* 1.0017 .9983 .1156 .1158 1.007 .9989 .05768 2.91*1* 1,1*80 .009300 .00051*27 .0581*3 .05836 .0581*6 1.0017 .9983 .1169 .1171 1.007 .9989 .05830 2.929 1,1*1*9 .009U00 .000551*5 .05906 «c05899 .05909 1.0017 .9983 .1181 .1181* 1.007 .9988 .05892 2.913 1,1*18 .009500 .0005661* .05969 .05962 .05973 1.0018 .9982 .1191* .1196 1.007 .9988 .05955 2.898 1,388 .009600 .0005781* .06032 .06025 .06036 1.0018 .9982 .1206 .1209 1.007 .9988 .06018 2.882 1,360 .009700 .0005905 .06095 .06087 .06099 1.0019 .9981 .1219 .1222 1.007 .9988 .06080 2.867 1,332 .009800 .0006027 .06158 .06150 .06162 1.0019 .9981 .1232 .1235 1.008 .9987 .0611*2 2.853 1,305 .009900 .0006150 .06220 .06212 .06221* 1.0019 .9981 .121*1* .121,7 1.008 .9987 .06201* 2.839 1,279 .01000 .0006275 .06283 .06275 .06287 1.0020 .9980 .1257 .1260 1.0079 .9987 .06267 2.825 1,253 .ouoo .0007591 .06912 .06901 .06917 1.0021* .9976 .1382 .1387 1.0096 .998U .06890 2.691* 1,036 .01200 .0009031 .0751*0 .07526 .0751*7 1.0028 .9972 .1508 .1513 1.0111* .9981 .07511 2.580 871.0 .01300 .001060 .08168 .08150 .08177 1.0033 .9967 .1631* .161*1 1.0131* .9978 .08131 2.1*80 71*2.9 .OlliOO .001228 .06795 .08771* .08808 1.0039 .9961 .1759 .1768 1.0155 .9971* .08751 2.389 61a.l .01500 .0011*10 .091*25 .09397 .091*39 1.00i*l* .9956 .1885 .1896 1.0178 .9970 .09369 2.310 558.9 .01600 .001603 .1005 .1002 .1007 1.0051 .991*9 .20U .2021* 1.0203 .9966 .09986 2.*238 1*91.6 .01700 .001809 .1068 .1061* .1070 1.0057 .99143 .2136 .2153 1.0229 .9962 .1060 2.172 1*35.8 .01800 .002027 .1131 .1126 .1133 I.OO6I* .9936 .2262 .2281 1.0257 .9958 .1121 2.112 389.1 .01900 .002258 .1191* .1188 .1197 1.0071 .9929 .2388 .21*10 1.0286 .9953 .1183 2.056 31*9.5 .02000 .002500 .1257 .1250 .1260 1.008 .9922 -2513 .25U0 1.032 .991*7 .121*1* 2.005 315.8 .02100 .002755 .1320 .1312 .1323 1.009 .9911* .2639 .2669 1.035 .991*2 .1305 1.958 286.8 .02200 .003022 .1382 .1371* .1387 1.010 .9905 .2765 .2800 1.038 .9937 .1365 1.915 261.5 .02300 .003301 .11*1*5 .11*35 .11*50 1.011 .9896 .2890 .2931 1.01*2 .9931 .11*25 1.873 239.6 .021*00 .003592 .1508 .11*97 .1511* l.OU .9887 .3016 .3062 1.01*6 .9925 .11*85 1.831* 220.3 .02500 .003895 .1571 .1558 .1577 1.012 .9878 .311*2 .319U 1.050 .9919 .151*5 1.799 203.3 .02600 .001*2-10 .1631* .1619 .I6la 1.013 .9868 .3267 .3326 1.051* .9912 .1605 1.765 188.2 .02700 .001*537 .1697 .1680 .1705 1.011* .9858 .3393 .31*58 1.058 .9905 .1665 1.733 171*.8 .02800 .001*876 .1759 .171*1 .1768 1.016 .981*7 .3519 .3592 1.063 .9898 .1721* 1.703 162.7 .02900 .005226 .1822 .1802 .1832 1.017 .9836 .361*1* .3725 1.067 .9891 .1783 1.675 151.9 .03000 .005589 .1885 .1863 .1896 I.OI8 .9825 .3770 .3860 1.072 .9881* .181*1 1.61*8 1U2.2 .03100 .005963 .19^8 .1921* .I960 1.019 .9813 .3896 .3995 1.077 .9876 .1900 1.622 133.1* .03200 .00631*7 .2011 .1981* .2021* 1.020 .9801 .1*021 .1*131 1.082 .9868 .1958 1.598 125.1* .03300 .00671*6 .2073 .201*1* .2088 1.022 .9789 .1*11*7 .1*267 1.067 .9860 .2016 1.575 118.1 .031*00 .007155 .2136 .2101* .2153 1.023 .9776 .1*273 .1*1*01* 1.093 .9851 .2073 1.553 111.1* .03500 .007575 .2199 .a6a .2217 1.021* .9763 .1*398 .1*51*1 1.098 .981*3 .2130 1.532 105.3 .03600 .008007 .2262 .2221* .2281 1.026 .97U9 .1*521, .1*680 l.lOl* .9831* .2187 1.512 99.75 .03700 .0061*50 .2325 .2281* .231*6 1.027 .9736 .1*650 .1*819 1.110 .9821* .221*1* 1.1*93 9l*.6l .03800 •008905 .2388 .231*3 .21*10 1.029 .9722 .1*775 .1*959 1.116 .9815 .2300 1.1*75 89.88 .03900 .009370 .21*50 .21*03 .2527 1.030 .9708 .1*901 .5099 1.123 .9805 .2356 1.1*57 85.50 .01*000 .00981*7 ,2513 .21*62 .251*0 1.032 .9693 .5027 .52la 1.129 .9795 . 21*11 1.1*1*0 81.1*3 .olaoo .01033 .2576 .2521 .2605 1.033 .9677 .5152 .5383 1.136 .9785 .21*67 1.1*21, 77.67 .01*200 .01063 .2639 .2579 .2670 1.035 .9662 .5278 .5526 1.11*3 .^775 .2521 1.1*08 71*. 17 .01*300 .01131* .2702 .2638 .2735 1.037 .961*6 .51*01* .5670 1.150 .9765 .2576 1.393 70.91 .01*1*00 .01186 .2765 ,2696 .2600 1.039 .9630 .5529 .5815 1.157 .9751* .2630 1.379 67.88 .01*500 .01239 .2827 .2751, .2865 1.01*0 .9613 .5655 .5961 1.161* .971*3 .2681* 1.365 65.05 .01,600 .01291* .2890 .2812 .2931 1.01*2 .9596 .5781 .6108 1.172 .9732 .2737 1.352 62.39 .01*700 .0131*9 .2953 .2870 .2996 1.01*1* .9579 .5906 .6256 1.180 .9721 .2790 1.339 59.91 .01*800 .011*05 .3016 .2928 .3062 1.01*6 .9562 .6032 .61*01* 1.188 .9709 .281*3 1.326 57’. 57 .01*900 .011*63 .3079 .2985 .3128 1.01*8 .951*1* .6158 .6551* 1.196 .9697 .2895 1.311* 55.38 D-17 Table D-2 Cont'd d/L d/Lo 2-n-dA TANH 2irdA SINK 2Trd/L COSH 2Trd/L K 4TrdA SINK 4/7 d/L COSH 4/7 d/L n H/Hi u .05000 .01521 .3142 .3042 .3194 1.050 .9526 .6283 .6705 1.204 .9685 .2947 1.303 53.32 .05100 .01580 .3204 .3099 .3260 1.052 .9508 .6409 .6857 1.213 .9673 .2998 1.291 51.38 .05200 .016U. .3267 .3156 .3326 1.054 .9489 .6535 .7010 1.221 .9661 .3049 1.281 49.55 .05300 .01702 .3330 .3212 .3392 1.056 .9470 .66^ .7164 1.230 .9649 ,3099 1.270 47.82 .051i00 .01765 .3393 .3269 .3458 1.058 .9451 .6786 .7319 1.239 .9636 .31U9 1.260 46.19 .05500 .01829 .3456 .3325 .3525 1.060 .9431 .6912 .7475 1.249 .9623 .3199 1.250 44.65 .05600 .01893 .3519 .3380 .3592 1.063 .9411 .7037 .7633 1.258 .9610 .3248 1.241 43.19 .05700 .01958 .3581 .3436 .3658 1.065 .9391 .7163 .7791 1.268 .9597 .3297 1.231 41.80 .05800 .02025 .3644 .3491 .3726 1.067 .9371 .7289 .7951 1.278 .9583 .3346 1.222 40.49 .05900 .02092 .3707 .3546 .3793 1.070 .9350 .7414 .8112 1.288 .9570 .3394 1.214 39.24 .06000 .02161 .3770 .3601 .3860 1.072 .9329 .7540 .8275 1.298 .9556 .3441 1.205 38.06 .06100 .02230 .3833 .3656 .3927 1.074 .9308 .7666 .8439 1.308 .9542 .3488 1.197 36.93 .06200 .02300 .3896 .3710 .3995 1.077 .9286 .7791 .8604 1.319 ,9528 .3534 1.189 35.86 .06300 .02371 .3958 .3764 .4062 1.079 .9265 .7917 .8770 1.330 .9514 .3581 1.182 34.83 .06UOO .02i(Uj .4021 .3816 .4130 1.082 .9243 .8043 .8938 1.341 .9499 .3626 1.174 33.86 .06500 .02516 .4o84 .3671 .4199 1.085 .9220 .8168 .9107 1.353 .9484 .3672 1.167 32.93 .06600 .02590 .4147 .3925 .4267 1.087 .9198 .8294 .9278 1.364 .9470 .3716 1.160 32.04 .06700 .02665 .4210 .3978 .4335 1.090 .9175 .8419 .9450 1.376 .9455 .3761 1.153 31.19 .06800 .02739 .4273 .4030 .4404 1.093 .9152 .8545 .9624 1.388 .9440 .3804 1.147 30.38 .06900 .02817 .4335 .4083 .4473 1.095 .9128 .8671 .9799 1.400 .9424 .3848 1.140 29.61 .07000 .02895 .4398 .4135 .4541 1.098 .9105 .8796 .9976 1.412 .9409 .3891 1.134 28.86 .07100 .02973 .4461 .4187 .4611 1.101 .9081 .8922 1.015 1.425 .9393 .3933 1.128 28.15 .07200 .03052 .4524 .4239 .4680 1.104 .9057 .9058 1.033 1.438 .9378 .3975 1.122 27.47 .07300 .03132 .4587 .4290 .4749 1.107 .9033 .9173 1.052 1.451 .9362 .4016 1.116 26.81 .07U00 .03213 .4650 .4341 .4819 1.110 .9008 .9299 1.070 1.464 .9346 .4057 1.110 26.18 .07500 .0329U .4712 .4392 .4889 1.113 .8984 .9425 1.088 1.478 .9330 .4098 1.105 25.58 .07600 .03377 .4775 .41*43 .4958 1.116 .8959 .9551 1.107 1.492 .9314 .4138 1.099 25.00 .07700 ,03^60 .4838 .4493 .5029 1.119 .8934 .9676 1.126 1.506 .9298 .4177 1.094 24.45 .07800 .035143 .4901 .4542 .5100 1.123 .8909 .9802 1.145 1.520 .9281 .4216 1.089 23.92 .07900 .03628 .4964 .4593 .5170 1.126 .8883 .9927 1.164 1.534 .9264 .4255 1.084 23.40 .06000 <0371U .5027 .4642 .5241 1.129 .8857 1.005 1.183 1.549 .9248 .4293 1.079 22.90 .08100 .03799 .5089 .4691 .5312 1.132 .8831 1.018 1.203 1.564 .9231 .4330 1.075 22.i*2 .08200 .03887 .5152 .4740 .5383 1.136 .8805 1.030 L,223 1.580 .9214 .4367 1.070 21.96 .08300 .03975 .5215 .4789 .5455 1.139 .8779 1.043 1.243 1.595 .9197 .4404 1.066 21.52 .oBUoo .04063 .5278 .4837 .5526 1.143 .8752 1.056 1.263 1.611 .9179 .4440 1-061 21.09 .08500 .04152 .5341 .4885 .5598 1.146 .8726 1.068 1.283 1.627 .91o2 .4476 1.057 20.68 .08600 .04242 .5404 .4933 .5670 1.150 .8699 1.081 1.304 1.643 .9145 .4511 1.053 20.28 .08700 .04333 .5466 .4980 .5743 1.153 .8672 1.093 1.324 1.660 .9127 .4545 1.049 19.90 .08800 .04424 .5529 .5027 .5815 1.157 .8645 1.106 1.346 1.676 .9109 .4579 1.045 19.53 .08900 .04516 .5592 .5074 .5888 1.160 .8617 1.118 1.367 1.693 .9092 .4613 1.041 19.17 .09000 .04608 .5655 .5120 .5961 1.164 .8590 1.131 1.388 1.711 .9074 .4646 1.037 18.82 .09100 .04702 .5718 .5167 .6034 1.168 .8562 1.144 1.410 1.728 ,9056 .4679 1.034 18-49 .09200 .04796 .5781 .5213 .6108 1.172 .8534 1.156 1.431 1.746 .9038 .4711 1.030 18.16 .09300 .04890 .5843 .5258 .6182 1.176 .8506 1.169 1.453 1.764 .9020 .4743 1.027 17.85 .091400 .04985 .5906 .5303 .6256 1.180 .8478 1.181 1.476 1.783 .9002 .4774 1.023 17.55 .09500 .05081 .5969 .5348 .6330 1.184 .8450 1.194 1.498 1.801 .8984 .4805 1.020 17.26 .09600 .05177 .6032 .5393 .6404 1.188 .8421 1.206 1.521 1.820 .8966 .4835 1.017 16.97 .09700 .05275 .6095 .5438 .6479 1.192 .8392 1.219 1.544 1.840 .8947 .4865 1.014 16.69 .09800 .05372 .6158 .5482 .6554 1.196 .8364 1,232 1.567 1.859 .8929 .1*894 1.011 16.42 .09900 .05470 .6220 .5526 .6629 1.200 .8335 1.244 1.591 1.879 .8910 .4923 1.008 16.16 .1000 .05569 .6283 .5569 .6705 1.204 .8306 1.257 1.615 1.899 .8892 .4952 1.005 15.91 .1010 .05668 .6346 .5612 .6781 1.208 .8277 1.269 1.636 1.920 .8873 .4980 1.002 15.67 .1020 .05768 .6409 .5655 .6857 1.213 .8247 1.282 1.663 1.940 .8854 .5007 .9993 15.43 .1030 .05869 .6472 .5698 .6933 1.217 .8218 1,294 1.687 1.961 .8836 .5034 .9966 15.20 .lOljO .05970 .6535 .5740 .7010 1.221 .8189 1.307 1.712 1.983 .8817 .5061 .9940 14.98 .1050 .06071 .6597 .5782 .7087 1.226 .8159 1.319 1.737 2.004 .8798 .5087 .9914 14.76 .1060 .06173 .6660 .5824 .7164 1.230 .8129 1.332 1.762 2.026 .8779 .5113 .9891 14.55 .1070 .06276 .6723 .5865 .7241 1.235 .8100 1.345 1.788 2.049 .8760 .5138 .9865 14.35 .1080 .06378 .6786 .5906 .7319 1.239 .8070 1.357 1.814 2.071 .8741 .5163 .9841 14.15 .1090 .06482 .6849 .5947 .7397 1.244 .8040 1.370 1.840 2.094 ,8722 .5187 .9818 13.95 D-18 Tablf 11-2 Corit'd dA dAo 2ird/L TANH SINH COSH K 8rTdA SINH COSH n H/H' M 2TTd/L 2^d/L 2tl d/L 877dA hfTd/L lx 0 0 .1100 .06586 .6912 .5987 .71*75 1.21*9 .8010 1.382 1.867 2.118 ,8703 .5211 .9191 13.77 .1110 .06690 .6971* .6027 .7551* 1.253 .7980 1.395 1.893 2.1a .8688 .5238 .9775 13.58 .1120 .06795 .7037 .6067 .7633 1.258 .791*9 1.807 1.920 2.165 .8665 .5257 .9753 13.a .1130 .06901 .7100 .6107 .7712 1.263 .7919 1.820 1.988 2,189 ,8685 .5279 .9731 13.23 .imo .07006 .7163 .611*6 .7791 1.268 .7888 1.833 1.975 2,218 .8626 .5301 .9711 13.06 .1150 .07113 .7226 .6185 .7871 1.273 .7850 1.885 2.003 2.239 .8607 .5323 .9691 12,90 .1160 .07220 .7289 .6221, .7951 1.278 .7827 1.858 2.032 2.268 .8587 .5388 .9672 12.78 .1170 .07327 .7351 .6262 .8032 1.203 .7797 1.870 2.060 2.290 .8568 .5365 .9658 12.59 .1180 .071*31* .71*11* .6300 .8112 1.288 .7766 1.883 2.089 2.316 -8589 .5386 .9635 12.83 .1190 .0751*2 .71*77 .6338 .8193 1.293 .7735 1.895 2.118 2.383 .8529 .5806 .9617 12.29 .1200 .07650 .751*0 .6375 .827 5 1.298 .7701* 1.508 2.188 2.369 .8510 .5825 .9600 12.18 .1210 .07759 .7603 .61*12 .8357 1.303 .7673 1.521 2.178 2.397 .8891 .5888 .9583 12.00 .1220 .07868 .7666 .61*1*9 .81*39 1.309 .761*2 1.533 2.208 2.828 .8871 .5863 .9567 11.87 .1230 .07978 .7728 .61*86 .8521 1.311* .7612 1.586 2.239 2.852 .8852 .5882 .9551 11.73 .12U0 .08085 .7791 .6520 .8601* 1.319 .7581 1.558 2.270 2.880 .8832 .5500 .9535 11.61 .1250 .08198 .785U .6558 .8687 1.325 .751*9 1.571 2.301 2.509 .8813 .5517 .9520 11.88 .1260 .06308 .7917 .659U .8770 1.330 .7518 1.583 2.333 2.538 .8393 .5538 .9505 11.35 .1270 .o8ia9 .7980 .6629 .8851* 1.336 .71*87 1.596 2.365 2.568 .8378 .5551 .9890 11.23 .1280 .08530 .801*3 .6661* .8930 1.3a .71*56 1.609 2.390 2.598 .8358 .5568 .9876 11.11 .1290 .0861*2 .8105 .6699 .9022 1.3a 71*21* 1,621 2.830 2.628 .8335 ,5588 .9863 11.00 .1300 .08753 .8168 .6733 .9107 1.353 .7393 1.638 2,868 2.659 .8316 .5599 .9850 10.89 .1310 .08866 .8231 .6768 .9192 1.358 .7362 1.686 2.897 2.690 .8296 .5618 .9837 10.70 .1320 .08978 .8291* .6801 .9278 1.361* .7331 1.659 2.531 2.722 ,8277 .5629 .9828 • 10.67 .1330 .09091 .8357 .6835 .9361* 1.370 .7299 1.671 2.566 2.758 .8257 .5688 .9812 10.56 .13U0 .092di .81*20 .6868 .91*50 1.376 .7268 1.688 2.600 2.786 .8238 .5650 .9801 10.86 .1350 .09317 .81*82 .6902 .9537 1.382 .7237 1,696 2.636 2.819 ,8218 .5672 .9389 10.36 .1360 .091*31 .851*5 .6931* .9621* 1.388 .7205 1.709 2,671 2.852 .8199 .5685 .9378 10.26 .1370 .0951*1* .8608 .6967 .9711 1.391* .7171* 1.722 2.707 2.886 .8179 .5698 ,9367 10.17 .1380 .09659 .8671 .6999 .9199 1.1*00 .711*2 1.738 2.788 2.920 .8160 .5711 -9357 10.07 .1390 .09773 .8731* .7031 .9087 1.1*06 .7111 1.787 2.781 2.955 .8181 .5728 .9387 9.983 .lUOO .09888 .8797 .7063 .9976 I.a2 .7080 1.759 2.818 2.990 .8121 .5736 .9337 9.898 -lUio .1000 .6859 .7091* 1.006 I.a9 .701*8 1.772 2.856 3.026 .0102 .5788 .9327 9.806 .llj20 .1012 .8922 .7125 1.015 i.a25 .7017 1.788 2.898 3.062 .8083 .5759 .9318 9.721 .11430 .1023 .8985 .7156 1.021* 3.1*32 .6985 1.797 2.933 3.099 .8068 .5770 .9309 9.638 .lUljO .1035 .901*8 .7186 1.033 1.1*38 .6951* 1.810 2.972 3.136 .8088 .5781 .9300 9.556 .11*50 .101*6 .9111 .7216 1.01*2 1.1* 1*5 .6923 1.822 3.012 3.173 .0025 .5791 .9292 9.876 .11*60 .1058 .9171* .721*7 1.052 1.1*51 .6891 1,835 3.052 3.211 .8006 .5801 .9288 9.398 .11*70 .1070 .9236 .7276 1.061 1.1*58 .6860 1.887 3,092 3.250 .7987 .5811 .9276 9.321 .11*80 .1081 .9299 .7306 1.070 1.1*61* .6829 1.860 3.133 3.289 .7968 .5821 .9268 9.286 .11*90 .1093 .9362 .7335 1.079 i.ai .6797 1.872 3.175 3.329 .7989 .5830 .9261 9.173 .1500 .1105 .91*25 .736U 1.088 1.U78 .6766 1.885 3.217 3.369 .7930 .5839 .9258 9.101 .1510 .1116 .91*86 .7392 1.098 1.1*85 .673a 1.898 3.260 3.ao .7911 .5888 ,9287 9.031 .1520 .1128 .9551 .71*21 1.107 1.U92 .6703 1.910 3.303 3.851 .7892 .5856 .9280 8.962 .1530 .111*0 .9613 .71*1*9 1.116 1.1*99 .6672 1.923 3.386 3.893 .7873 .5868 .9238 8.698 .151*0 .1151 .9676 .71*77 1.126 1.506 .66a 1,935 3.391 3.535 .7858 .5872 .9228 8.828 .1550 .1163 .9739 .7501* 1.135 1.513 .6610 1.988 3.835 3,578 .7835 .5080 .9222 8,763 .1560 .1175 .9802 .7531 1.11*5 1.520 .6579 1.960 3.881 3.621 .7816 .5807 .9216 8.700 .1570 .1187 .9865 .7558 1.151* 1.527 .6587 1.973 3.526 3.665 .7797 .5893 .9211 8.638 .1580 .1199 .9928 .7585 1.161* 1.535 .6516 1.985 3.573 3.710 .7779 .5900 .9205 0.577 .1590 .1210 .9990 .7612 1.171* 1.5a2 .-6805 1.998 3.620 3.755 .7760 .5907 .9200 8.517 .1600 .1222 1.005 .7638 1.183 1.51*9 .6858 2.011 3.667 3.001 .77a .5913 .9196 8.859 .1610 .1231* 1.012 .7661* 1.193 1.557. .6823 2.023 3.715 3.087 .7723 .5919 .9191 8.801 .1620 .121*6 1.018 .7690 1.203 1.561* .6392 2.036 3.768 3,698 .7708 -5925 .9106 8.385 .1630 .1258 1.02U .7716 1.213 1.572 .6361 2,088 3.813 3.982 .7686 ,5930 .9182 8.290 .161*0 a27o 1.030 .77a 1.223 1.500 .6331 2.061 3.063 3.990 .7667 .5935 .9179 8.236 .1650 .1281 1.037 .7766 1.233 1.587 .6300 2.073 3.913 8,039 .7689 .5980 .9175 8.183 .1660 .1293 1.01*3 .7791 1.21*3 1.595 .6269 2.086 3.968 8.088 .7631 .5985 .9171 8.131 .1670 .1305 1.01*9 .7815 1.253 1.603 .6239 2.099 8.016 8.138 .7613 .5950 .9167 8.079 .1680 .1317 1.056 .781*0 1.263 1.611 .6208 2.111 8.068 8.189 .7595 .5958 .9168 8.029 .1690 .1329 1.062 .7861* 1.273 1.619 .6177 2.128 8.121 8.2a .7576 .5958 .9161 7.900 D-19 Table D-? Cont'd dA dAo 2TT dA TANH SINK COSH K UffdA SINH COSH n Cq/Co H/Hi H 2irdA 2fldA 2nd/L l*rrdA l*?7d/L .1700 .131*1 1.068 .7887 1.283 1.627 .611,7 2.136 1,.175 1*.293 .7558 .5962 .9158 7.932 .1710 .1353 1.071* .7911 1.293 1.635 .6117 2.11,9 h.229 1,.31,6 .751*0 .5965 .9155 7.885 .1720 .1365 1.081 .7935 1.301, 1.61,3 .6086 2,161 h.26h 1*.399 .7523 .5969 .9153 7.838 .1730 .1377 1.087 .7958 1.311* 1.651 .6056 2.171* h.3h0 1,.1,51* .7505 .5912 .9150 7.793 .I7I4O .1389 1.093 .7981 1.325 1.660 .6026 2.187 1*.396 1,.508 .71*87 .5975 .911*8 7.71*8 .1750 .11*01 1.100 .8001* 1.335 1.668 .5995 2.199 1,.1,53 1*.561, .0*69 .5978 .911*6 7.701, .1760 .11*13 1.106 .8026 1.31*5 1.676 .5965 2.212 l,.5ll l,.62o .71*51 .5980 .911*1* 7.661 .1770 .11*25 1.112 .801,8 1.356 1.685 .5935 2.221, 1,. 569 1.677 .71.31* .5983 .911*2 7.619 .1780 .11*37 1.118 .8070 1.367 1.693 .5905 2.237 L.628 U.735 .71*16 .5985 .911*0 7.577 .1790 .11*1*9 1.125 .809? 1.377 1.702 .5875 2.21*9 h.6ad 1*.793 .7399 .5987 .9138 7.536 .1800 .11,60 1.131 .8111, 1.388 1.711 .58U5 2.262 l*.7l,9 1,.853 .7382 .5989 .9137 1.1*96 .1810 .11,72 1.137 .8135 1.399 1.720 .5816 2.275 1,.810 1,.918 .7361* .5991 .9136 7.1,57 .1820 .11,81, l.U*l* .6156 1.100 1.728 .5786 2.287 1,.872 l*.97l* .731*7 .5992 .9135 1.1*19 .1830 .11*96 1.150 .8177 1.1,20 1.131 .5757 2.300 1*.935 5.035 .7330 .5993 .9131* 7,381 .I8i*0 .1508 1.156 .8198 1.1,31 1.71*6 .5727 2.312 h.999 5.098 .7313 .5995 .9133 7.3U3 .1850 .1520 1.162 .8218 l.U*2 1.755 .5697 2.325 5.063 5.161 ,7296 .5996 .9132 7.307 .1860 .1532 1.169 .8239 1.1*51* 1.761* .5668 2.337 5.129 5.225 .7279 .5997 .9131 7.271 .1870 .151*1, 1.175 .8259 1.1*65 1.773 .5639 2.350 5.195 5.290 .7262 .5997 .9131 7.235 .1880 .1556 1.181 .8278 1.1*76 1.783 .5610 2.362 5.262 5.356 .721,5 .5998 .9131 7.201 .1890 .1568 1.188 .8298 1.1*87 1.792 .5581 2.375 5.329 5.1*22 .7228 .5998 .9130 7.167 .1900 .1580 I.I9I* .8318 1.1*98 1.801 ,5551 2.388 5.398 5.1*90 .7212 .5998 ,9130 7.133 .1910 .1592 1.200 .8337 1.510 1.8U .5522 2.1,00 5.1*67 5.558 .7195 .5998 .9130 7.100 .1920 .1601, 1.206 .8356 1.521 1.820 .51,93 2.1,13 5.538 5.625 .7179 .5998 .9130 7.068 .1930 .1616 1.213 .8375 1.533 1.830 .51*65 2.1*25 5.609 5.697 .7162 .5998 .9130 7.036 .191*0 .1628 1.219 .8393 1.510 1.81,0 .51*36 2.1,38 5.681 5.768 .711*6 .5998 .9131 7.005 .1950 .161*0 1.225 .8102 1.556 1.81,9 .51*08 2.1*50 5.751* 5.81,0 .7129 .5997 .9131 6.971, .1960 .1652 1.232 .31,30 1.567 1.859 .5379 2.1,63 5.827 5,913 .7113 .5997 .9131 6.91*1* .1970 .1661, 1.238 .8lil,8 1.579 1.869 .5350 2.1,76 5.902 5.988 .7097 .5996 .9132 6.911* .1980 .1676 1.21,1 .81,66 1.591 1.879 .5322 2.1,88 5.978 6.061 .7081 .5995 .9133 6.885 .1990 .1688 1.25( .61,81, 1.603 1.889 .5291* 2.501 6.055 6.137 .7065 .5991* .9133 6.856 .2000 .1700 1.25'’ .8501 1.611, 1.899 .5266 2.513 6.132 6.213 .701,9 .5993 .9131* 6.828 .2010 .1712 1.263 .8519 1.626 1.909 .5238 2,526 6.211 6.291 .7033 .5992 .9135 6.801 .2020 .1721, 1.269 .8535 1.638 1.920 .5210 2.538 6.290 6.369 .7018 .5990 .9137 6.771, .2030 .1736 1.276 .8552 1.651 1.930 .5182 2.551 6.371 6.U1,9 .7002 .5986 .9138 6.71,7 .20I4O .171,8 1.282 .8570 1.663 1.91*0 .5151* 2.561, 6.1*52 6.529 .6987 ,5987 .9139 6.720 .2050 .1760 1.288 .8586 1.675 1.951 .5127 2.576 6.535 6.6U .6971 .5986 .911*0 6.69I, .2060 .1772 1.291* .8602 1.687 1.961 .5099 2.589 6.619 6.691* .6956 .5981, .9IUI 6.669 .2070 .1781, 1.301 .8619 1.700 1.972 .5071 2.601 6.703 6.777 .691*1 .5982 .9U2 6.6iJ, .2080 .1796 1.307 .8635 1.712 1.983 .501,1, 2.611* 6.789 6,862 .6925 .5980 .911*1* 6.619 .2090 .1808 1.313 .8651 1.725 I.99J* .5016 2.626 6.876 6.91*8 .6910 .5978 .911*6 6.591* .2100 .1820 1.320 .8667 1.737 2.OOI, .1*989 2.639 6.963 7.035 .6895 .5976 .911*7 6.570 .2110 .1832 1.326 .8682 1.750 2.015 .1*962 2.652 7.052 7.123 .6880 .5973 .911*9 6.51,7 .2120 .781J, 1.332 .8697 1.762 2.026 ,1*935 2.661* 7.11*3 7.219 .6865 .5971 .9151 6.521, .2130 .1856 1.338 .8713 1.775 2.037 .1*906 2.677 7.23I1 7.302 .6850 .5969 .9153 6.501 .21UO .1868 1.31*5 .8728 1.788 2.01*9 .1,881 2.689 7.326 7.391* .6835 .5966 .9155 6.1,79 .2150 .1880 1.351 .871*3 1.801 2.060 .1*851* 2.702 7.1*20 7.1*87 .6821 .5963 .9157 6.1*57 .2160 .1892 1.357 .8757 1.811, 2.071 .1,828 2.711, 7.51L 7.580 .6806 .5960 .9159 6.1,35 .2170 .1901* 1.361, .8772 1.827 2.083 .1*801 2.727 7.610 7.675 .6792 .5958 .9161 6.1a3 .2180 .1915 1.370 .8786 1.81,0 2.091, .1*775 2.739 7.707 7.772 .6777 .5955 .9161, 6.393 .2190 .1927 1.376 .8801 1.853 2.106 .1*71*9 2.752 7.805 7.869 .6763 .5952 .9166 6.372 .2200 .1939 1.382 .8815 1.867 2.118 .1*722 2.765 7.905 7.968 .671*9 .591*9 .9168 6.351 .2210 .1951 1.389 .8829 1.880 2.129 .1*696 2.777 8.006 8.068 .6735 .591*6 .9170 6.331 .2220 .1963 1.395 .881,2 1.893 2 . 1 ia. .1*670 2.790 8.108 8.169 .6720 .591*3 .9173 6.312 .2230 .1975 1.1,01 .8856 1.907 2.153 .1*61J, 2.802 8.211 8.272 .6706 .5939 .9175 6.292 .221,0 .1987 1.1,07 .8869 1.920 2,165 .1,619 2.815 8.316 8.375 .6692 .5936 .9178 6.27 3 .2250 .1999 1.101, .8883 1.931* 2.177 .1*593 2.827 8.1,22 8.1,81 .6679 .5933 .9181 6.251, .2260 .2011 1.1,20 .8896 1.91*8 2.189 .U567 2.81,0 8.529 8.587 .6665 .5929 .9183 6.236 .2270 .2022 1.1,26 .8909 1.962 2.202 .1*51*2 2.853 8.637 8.695 .6651 .5925 .9186 6.218 .2280 .2031. 1.1*33 .6922 1.975 2.2A .1,516 2.865 8.756 8.800 .6637 .5921 .9189 6.200 .2290 .201,6 1.U39 .8935 1.989 2.227 .1*1*91 2.878 8.859 8.915 .6621, .5918 .9191 6,182 D-20 Table D-2 Cont'd d/L d/Lo 2 O’d/L TANK SINH COSH K aiTdA SINH COSH n H/H^ u 2 IT d/L 2frd/L 2td/L 1* ffd/L hrrd/L ,2300 .2058 1.1*145 .891*7 2.003 2.239 .1*1*66 2.890 8.971 9.027 .6611 .5915 .9198 6.165 ,2310 .2070 1.1*51 .8960 2.017 2.252 .i*i*ia 2.903 9.085 9.11*0 .6597 .5911 .9197 6.188 ,2320 .2082 1.1*58 .8972 2.032 2.261* .1*1*16 2.915 9.201 9.255 .6581* .5907 .9200 6.1U .2330 .2093 1.1*61* .8981* 2.01*6 2.277 .1*391 2.928 9.318 9.372 .6571 .5901* .9203 6.118 ,23liO .2105 1.1*70 .8996 2.060 2.290 .1*366 2.91*1 9.1*37 9.1*89 .6558 .5900 .9206 6.097 .2350 .2117 1.1*77 .9008 2.075 2.303 .1*31*2 2.953 9.557 9.609 .651*5 ,5896 .9209 6.081 .2360 .2129 1.1*83 .9020 2.089 2.316 .1*318 2.966 9.678 9.730 .6532 .5892 .9212 6.066 .2370 .211*1 1.1*89 .9032 2.101* 2.329 .1*293 2.978 9.801 9.852 .6519 .5888 .9215 6.050 ,2380 .2152 1.1*95 .90L; 2.118 2.31*3 .1*269 2.991 9.926 9.976 .6507 .S88U .9218 6.038 .2390 .2161* 1.502 .9055 2.133 2.356 .1*21*1* 3.003 10.05 10.10 .61*91* .5880 .9221 6.019 ,21j00 .2176 1.508 .9066 2.11*8 2.370 .1*220 3.016 10.18 10.23 .61*81 .5876 .9225 6.008 .2L10 .2188 1.511* .9077 2.163 2.383 .1*196 3.029 10.31 10,36 .61*69 .5872 .9228 5.990 ,21i20 .2199 1.521 .9088 2.178 2.397 .1*172 3 .Ola 10.1*1* 10.1*9 .61*56 .5868 .9231 5,976 .2h30 .2211 1.527 .9099 2.193 2.1*10 .1*11*9 3.051* 10.57 10.62 .61*1*1* .5863 ,9238 5.961 ,2Ui0 .2223 1.533 .9110 2.208 2.1*21* .ia25 3.066 10,71 10.75 .61*32 .5859 .9238 5.987 ,2U50 .2231* 1.539 .9120 2.221* 2.1*38 .1*101 3.079 10.81* 10.89 .61*20 .5055 .9281 5.933 ,2U60 .221*6 1.51*6 .9131 2.239 2.1*52 .1*078 3.091 10,98 11.03 .61*08 .5851 .9288 5.919 ,2 WO .2258 1.552 .911*1 2.255 2.1*66 .1*055 3.101* 11.12 11.17 .6396 .5886 .9288 5.906 ,21*80 .2270 1.558 .9151 2.270 2.1*80 .1*032 3.116 11.26 11.31 .6381* .5882 .9251 5.893 ,21*90 .2281 1.565 .9162 2.286 2.1*95 .1*008 3.129 11.1*0 11.1*5 .6372 .5838 .9255 5.880 2500 .2293 1.571 .9172 2.301 2.509 .3985 3.11*2 11.55 11.59 .6360 .5833 .9258 5.867 ,2510 .2305 1.577 .9182 2.317 2.521* .3962 3.15U 11.70 11.71* .631*8 ,5829 .9262 5.858 .2520 .2316 1.583 .9191 2.333 2.538 .391*0 3.167 11.81* 11.89 .6337 .5828 .9265 5.881 2530 .2328 1.590 .9201 2.31*9 2.553 .3917 3.179 11.99 12,01* .6325 .5820 .9269 5.829 ,251*0 .2339 1.596 .9210 2.365 2.568 .3891* 3.192 12.15 12.19 .6311* .5815 .9273 5.817 ,2550 .2351 1.602 .9220 2.381 2.583 .3872 3.20I* 12.30 12.31* .6303 .5811 .9276 5.805 ,2560 .2363 1.609 .9229 2.398 2.598 .381*9 3.217 12.1*6 12.50 .6291 .5807 .9280 5.793 ,2570 .2371* 1.615 .9239 2.1*11* 2.613 .3827 3.230 12.61 12.65 .6280 .5802 ,9283 5.782 2580 .2386 1.621 .921*8 2.1*30 2.628 .3805 3.21*2 12.77 12.81 .6269 .5797 .9287 5.770 2590 .2398 1.627 .9257 2.1* 1*7 2.61*3 .3783 3.255 12.9I* 12.98 .6258 .5793 .9291 5.759 ,2600 .21*09 1.631* .9266 2.1*61* 2.659 .3761 3.267 13.10 13.11* .621*7 .5788 .9298 5.788 .2610 .21*21 1.61*0 .9275 2.1*80 2.671* .3739 3.280 13.27 13.31 .6236 .5788 .9298 5.737 .2620 .21*32 1.6U6 .9283 2.1*97 2.690 .3717 3.292 13.1*1* 13.1*7 .6225 .5779 .9301 5.726 .2630 .21*1*1* 1.653 .9292 2.511* 2.706 .3696 3.305 13.61 13.61* .6215 .5775 .9305 5.716 .261*0 .21*55 1.659 .9301 2.531 2.722 .3671* 3.318 13.78 13.81 .6201* .5770 .9309 5.705 .2650 .21*67 1.665 .9309 2.51*8 2.737 .3653 3.330 13.95 13.99 .6193 .5765 .9313 5.595 .2660 .21*78 1.671 .9317 2.566 2.751* .3632 3.31*3 ll*.13 11*.17 .6183 .5761 .9316 5.685 .2670 .21*90 1.678 .9326 2.583 2.770 .3610 3.355 11*.31 ll*.31* .6172 .5756 .9320 5.675 .2680 .2501 1.681* .9331* 2.600 2.786 .3589 3.368 11*.1*9 llj.53 .6162 .5752 .9328 5.665 .2690 .2513 1.690 .931*2 2.618 2.803 .3568 3.380 11*.67 11*.71 .6152 .5787 .9328 5.655 .2700 .2521* 1.697 .9350 2.636 2.819 .351*7 3.393 11*.86 ll*.89 .611*2 .5782 .9331 5.685 .2710 .2536 1.703 .9357 2.653 2.835 .3527 3.1*05 15.05 15.08 .6132 .5737 .9335 5.636 .2720 .25W 1.709 .9365 2.671 2.852 .3506 3.ia8 15.21* 15.27 .6122 .5733 .9339 5.627 .2730 .2559 1.715 .9373 2.689 2.869 .31*85 3.1*31 15.1*3 15.1*6 .6112 .5728 .9383 5.617 .271*0 .2570 1.722 .9381 2.707 2.886 .31*65 3.1*1*3 15.63 15.66 .6102 .5728 .9386 5.608 .2750 .2582 1.728 .9388 2.726 2.903 .31*1*1* 3.1*56 15.83 15.86 .6092 .5719 .9350 5.599 .2760 .2593 1.731* .9396 2.71*1* 2.920 .31*21* 3.1*68 16.03 16.06 .6082 .5718 .9358 5.590 .2770 .2605 1.71*0 .91*03 2.762 2.938 .31*01* 3.1*81 16.23 16.26 .6072 .5710 .9358 5.582 .2780 .2616 1.71*7 .9iao 2.781 2.955 .3381* 3.1*93 16.1*3 16,1*7 .6063 .5705 .9362 5.573 .2790 .2627 1.753 .9la7 2.799 2.973 .3361* 3.506 16.61* 16.67 .6053 .5701 .9366 5.565 ,2800 .2639 1.759 .9l*2U 2.818 2.990 .331*1* 3.519 16.85 16.88 .601*1* .5696 .9369 5.556 .2810 .2650 1.766 .91*31 2.837 3.008 -3321* 3.531 17.07 17.10 .6035 .5691 ,9373 5.588 .2820 .2662 1.772 .91*38 2.856 3.026 .3305 3.51*1* 17.28 17.31 .6025 .5637 .9377 5.580 .2830 .2673 1.778 .91*1*5 2.875 3.01*1* .3285 3.556 17.50 17.53 .6016 .5682 .9381 5.532 ,281*0 .2681* 1.781* .91*52 2.891* 3.062 .3266 3.569 17.72 17.75 ,6007 .5677 .9388 5.528 ,2850 .2696 1,791 .91*58 2.913 3.080 .321*7 3.581 17.95 17.90 .5998 .5673 .9338 5.516 ,2860 .2707 1.797 .91*65 2.933 3.099 .3227 3.591* 18.18 18,20 .5989 .5668 .9392 5.509 ,2870 .2718 1,803 .91*72 2.952 3.117 .3208 3.607 18.1*0 18.1*3 .5900 .5668 .9396 5.501 ,2880 .2730 1.810 .91*78 2.972 3.136 .3189 3.619 10.61* 18.67 .5971 .5659 .9800 5.893 ,2890 .27 la 1.816 .91*81* 2.992 3.151* .3170 3.632 18.08 18.90 .5962 .5658 .9808 5.886 D-21 Table D-2 Cont’d d/L dAo 2vd ~Tr tanh 2iTd L sinh 2trd I cosh 2itd L~ K 4iTd ~ sinh 4nd L cosh 4Tid L n c„/c G o 0 M .2900 .2752 1.822 .9491 3.012 3.173 .3151 3.644 19.11 19.14 .5953 .5650 .9407 5.479 ,2910 .2764 1.828 .9497 3.032 3.192 .3133 3.657 19.36 19.38 .5945 .5645 .9411 5.472 ,2920 .2775 1.835 .9503 3.052 3.211 .3114 3.669 19.60 19.63 .5936 .5641 .9415 5.465 ,2930 .2786 1.841 .9509 3.072 3.231 .3095 3.682 19.85 19.87 .5927 .5636 .9419 5.458 ,2940 .2797 1.847 .9515 3.093 3.250 .3077 3.695 20.10 20.13 .5919 .5632 .9422 5.451 ,2950 .2809 1.854 .9521 3.113 3.269 .3059 3.707 20.36 20.38 .5911 .5627 .9426 5.444 ,2960 .2820 1.860 .9527 3.133 3.289 .3040 3.720 20.61 20.64 .5902 .5622 .9430 5.437 ,2970 .2831 1.866 .9532 3.154 3.309 .3022 3.732 20.87 20.90 .5894 .5618 .9434 5.431 ,2980 .2842 1.872 .9538 3.175 3.329 .3004 3.745 21.14 21.16 .5886 .5614 .9437 5.424 ,2990 .2854 1.879 .9544 3.196 3.349 .2986 3.757 21.41 21.43 .5878 .5610 .9441 5.418 ,3000 .2865 1.885 .9549 3.217 3.369 .2968 3.770 21.68 21.70 .5870 .5605 .9445 5.412 ,3010 .2876 1.891 .9555 3.238 3.389 .2951 3.782 21.95 21.97 .5862 .5601 ,9449 5.405 .3020 .2887 1.898 .9560 3.260 3.410 .2933 3.795 22.23 22.25 .5854 .5596 .9452 5.399 ,3030 .2898 1.904 .9566 3.281 3.430 .2915 3.808 22.51 22.53 .5846 .5592 .9456 5.393 .3040 .2910 1.910 .9571 3.303 3.451 .2898 3.820 22.80 22.82 .5838 .5587 .9459 5.387 ,3050 .2921 1.916 .9576 3.325 3.472 .2880 3.833 23.08 23.11 .5830 .5583 .9463 5.381 ,3060 .2932 1.923 .9581 3.347 3.493 .2863 3.845 23.38 23.40 .5823 .5579 .9467 5,376 ,3070 .2943 1.929 .9586 3.368 3.514 .2846 3.858 23.67 23.69 .5815 .5574 .9471 5.370 ,3080 .2954 1.935 .9592 3.391 3.535 .2829 3.870 23.97 23.99 .5807 .5570 .9474 5.364 ,3090 .2965 1.942 .9597 3.413 3.556 .2812 3.883 24.28 24.30 .5800 .5566 .9478 5.359 ,3100 .2977 1.948 .9602 3.435 3.578 .2795 3.896 24.58 24.60 .5792 .5562 .9482 5.353 ,3110 .2988 1.954 .9606 3.458 3.600 .2778 3.908 24.89 24.91 .5785 .5557 .9485 5.348 ,3120 .2999 1.960 .9611 3.481 3.621 .2761 3.921 25.21 25.23 .5778 .5553 .9489 5.342 ,3130 .3010 1.967 .9616 3.503 3.643 .2745 3.933 25.53 25.55 .5770 .5549 .9493 5.337 ,3140 .3021 1.973 .9621 3.526 3.665 .2728 3.946 25.85 25.87 .5763 .5545 .9496 5.332 ,3150 .3032 1.979 .9625 3.549 3.688 .2712 3.958 '26.18 26.20 .5756 .5540 .9500 5.327 ,3160 .3043 1.986 .9630 3.573 3.710 .2695 3.971 26.51 26.53 .5749 .5536 .9504 5.321 ,3170 .3054 1.992 .9634 3.596 3.733 .2679 3.984 26.84 26.86 .5742 .5532 .9508 5.316 ,3180 .3065 1.998 .9639 3.620 3.755 .2663 3.996 27.18 27.20 .5735 .5528 .9511 5.311 ,3190 .3076 2.004 .9643 3 M3 3.77g .2647 4.009 27.53 27.55 .5728 .5524 .9514 5.307 ,3200 .3087 2.011 .9648 3.667 3.801 .2631 4.021 27.88 27.89 .5721 .5520 .9518 5.302 ,3210 .3098 2.017 .9652 3.691 3.824 .2615 4.034 28.23 28.25 .5714 .5516 .9521 5.297 ,3220 .3109 2.023 .9656 3.715 3.847 .2599 4.046 28.59 28.60 .5708 .5512 .9525 5.292 ,3230 .3120 2.030 .9661 3.739 3.871 .2583 4.059 28.95 28.97 .5701 .5508 .9528 5.288 ,3240 .3131 2.036 .9665 3.764 3.894 .2568 4.072 29.31 29.33 .5694 .5504 .9532 5.283 ,3250 .3142 2.042 .9669 3.788 3.918 .2552 4.084 29.69 29.70 .5688 .5500 .9535 5.279 ,3260 .3153 2.048 .9673 3.813 3.942 ,2537 4.097 30.06 30.08 .5681 .5496 .9539 5.274 ,3270 .3164 2.055 .9677 3.838 3.966 .2521 4.109 30.44 30.46 .5675 .5492 .9542 5.270 ,3280 .3175 2.061 .9681 3.863 3.990 .2506 4.122 30.83 30.84 .5669 .5488 .9545 5.266 ,3290 .3186 2.067 .9685 3.888 4.015 .2491 4.134 31.22 31.23 .5662 .5484 .9549 5.261 ,3300 .3197 2.074 .9689 3.913 4.039 .2476 4.147 31.61 31.63 .5656 .5480 .9552 5.257 ,3310 .3208 2.080 .9692 3.939 4.064 .2461 4.159 32.01 32.03 .5650 .5476 .9555 5.253 ,3320 .3219 2.086 .9696 3.964 4.088 .2446 4.172 32.42 32.43 .5644 .5472 .9559 5.249 ,3330 .3230 2.092 .9700 3.990 4.114 .2431 4.185 32.83 32.84 .5637 .5468 .9562 5.245 ,3340 .3241 2.099 .9704 4.016 4.139 .2416 4.197 33.24 33.26 .5631 .5464 .9566 5.241 ,3350 .3252 2.105 .9707 4.042 4.164 .2402 4.210 33.66 33.68 .5625 .5461 .9569 5.237 ,3360 .3263 2.111 .9711 4.069 4.189 .2387 4.222 34.09 34.10 .5619 .5457 .9572 5.233 ,3370 .3274 2.117 .9715 4.095 4.215 .2373 4.235 34.52 34.53 .5613 .5453 .9576 5.229 ,3380 .3285 2.124 .9718 4.121 4.241 .2358 4.247 34.96 34.97 .5608 .5449 .9579 5.225 .3390 .3296 2.130 .9722 4.148 4.267 .2344 4.260 35.40 35.41 .5602 .5446 .9582 5.222 ,3400 .3307 2.136 .9725 4.175 4.293 .2329 4.273 35.85 35.86 .5596 .54^2 .9585 5.218 ,3410 .3317 2.143 .9728 4.202 4.319 .2315 4.285 36.30 36.31 .5590 .5438 .9589 5.214 ,3420 .3328 2.149 .9732 4.229 4.346 .2301 4.298 36.76 36.77 .5585 .5435 .9592 5.211 ,3430 .3339 2.155 .9735 4.256 4.372 .2287 4.310 37.22 37.24 .5579 .5431 .9595 5.207 ,3440 .3350 2.161 .9738 4.284 4.399 .2273 4.323 37.70 37.71 .5573 .5427 .9598 5.204 ,3450 .3361 2.168 .9742 4.312 4.426 .2259 4.335 38.17 38.19 .5568 .5424 .9601 5.200 ,3460 .3372 2.174 .9745 4.340 4.454 .2245 4.348 38.65 38.67 .5562 .5420 .9604 5.197 ,3470 .3383 2.180 .9748 4.368 4.481 .2232 4.361 39.14 39.16 .5557 .5417 .9608 5.193 ,3480 .3393 2.187 .9751 4.396 4.509 .2218 4.373 39.64 39.65 .5552 .5413 .9611 5.190 ,3490 .3404 2.193 .9754 4.424 4.536 .2205 4.386 40.14 40.15 .5546 .5410 .9614 5.187 May 1961 D-22 T«ble D-2 Cont’d tanh sinh cosh sinh cosh 2 Trd 2 iTd 2 nd 2 nd 4 nd 4 nd 4 nd ^ g/^o dA rf/Lo L L L L K L L L n H/Ho' M .3500 .3415 2,199 .9757 4.453 4,564 .2191 4.398 40.65 40.66 .5541 .5406 .9617 5.184 .3510 .3426 2.205 .9760 4.482 4.592 .2178 4.411 41.16 41.17 .5536 ,5403 .9620 5.181 .3520 .3437 2.212 .9763 4.511 4.620 .2164 4.423 41.68 41.70 .5531 .5400 .9623 5.177 .3530 .3447 2.218 .9766 4.540 4.649 .2151 4.436 42.21 42.22 .5525 .5396 .9626 5.174 .3540 .3458 2.224 .9769 4.569 4.678 .2138 4.449 42.74 42.76 .5520 .5393 .9629 5.171 ,3550 .3469 2.231 .9772 4.600 4.706 .2125 4.461 43.28 43.30 .5515 .5389 .9632 5.168 .3560 .3480 2.237 .9774 4.628 4.735 ,2112 4.474 43.83 43.84 .5510 .5386 .9635 5.165 ,3570 .3491 2.243 .9777 4.658 4.764 .2099 4.486 44.39 44.40 .5505 .5383 .9638 5.162 ,3580 .3501 2,249 .9780 4.688 4,794 .2086 4.499 44.95 44.96 .5500 .5379 .9641 5.159 ,3590 .3512 2.256 ,9783 4.719 4.823 .2073 4.511 45.52 45.53 .5496 .5376 .9644 5.156 ,3600 .3523 2.262 .9785 4.749 4.853 .2060 4.524 46.09 46.10 .5491 ,5373 .9647 '' 5,154 ,3610 .3534 2.268 .9788 4.779 4.883 .2048 4.536 46.68 46.69 .5486 .5370 .9650 5.151 , 3620 ' .3544 2.275 .9791 4.810 4.913 .2035 4.549 47.27 47.28 .5481 .5367 .9652 5.148 ,3630 -.3555 2.281 .9793 4.840 4.943 .2023 4.562 47.86 47.87 .5477 .5363 .9655 5.145 ,3640 .3566 2.287 .9796 4.872 4.974 .2010 4.574 48.47 48.48 .5472 .5360 .9658 5.143 ,3650 .3576 2.293 .9798 4.904 5.005 ,1998 4.587 49.08 49.09 .5467 .5357 .9661 5.140 ,3660 .3587 2.300 .9801 4.935 5.035 .1986 4.599 49.70 49.71 .5463 ,5354 ,9664 5.137 ,3670 .3598 2,306 .9803 4.967 5.067 .1974 4.612 50.33 50.34 .5458 .5351 .9667 5.135 ,3680 .3609 2.312 .9806 4.999 5.098 .1962 4.624 50.97 50.98 .5454 .5348 .9670 5.132 ,3690 .3619 2.319 .9808 5.031 5.129 .1950 4.637 51.61 51.62 .5449 .5345 .9672 5.130 ,3700 .3630 2.325 .9811 5.063 5.161 .1938 4.650 52.27 52.28 .5445 .5342 .9675 5.127 ,3710 .3641 2,331 .9813 5.096 5.193 .1926 4.662 52.93 52.94 .5440 .5339 .9678 5.125 ,3720 .3651 2.337 .9815 5,129 5.225 .1914 4.675 53.60 53.61 .5436 .5336 .9680 5.122 ,3730 .3662 2,346 .9817 5.161 5.257 .1902 4.687 54.27 54.28 .5432 .5333 .9683 5.120 ,3740 .3673 2.350 .9820 5.195 5.290 .1890 4.700 54.99 54.97 .5427 .5330 .9686 5.118 ,3750 .3683 2.356 .9822 5.228 5.322 .1879 4.712 55.66 55.66 .5423 .5327 .9688 5.115 ,3760 .3694 2.363 .9824 5.262 5.356 .1867 . 4.725 56.36 56.37 .5419 .5324 .9691 5,113 ,3770 .3705 2.369 .9826 5.295 5.389 .1856 4.738 57.07 57.08 .5415 .5321 .9694 5.111 ,3780 .3715 2.375 .9829 5.329 5.422 .1844 4.750 57.79 57.80 .5411 .5318 .9696 5.109 ,3790 .3726 2.381 .9831 5.363 5.456 .1833 4.763 58.53 58.53 .5407 .5315 .9699 5.106 ,3800 .3736 2.388 .9833 5.398 5.490 .1822 4.775 59.27 59.27 .5403 .5313 .9702 5.104 ,3810 .3747 2.394 .9835 5.432 5.524 .1810 4.788 60.01 60.02 .5399 .5310 .9704 5.102 ,3820 .3758 2.400 .9837 5.467 5.558 .1799 4.800 60.77 60.78 .5395 .5307 .9707 5.100 ,3830 .3768 2.407 .9839 5.502 5.593 .1788 4.813 61.54 61.55 .5391 .5304 .9709 5.098 ,3840 .3779 2.413 .9841 5.537 5.627 .1777 4.826 62.32 62.33 .5387 .5301 .9712 5.096 ,3850 .3790 2.419 .9843 5.573 5.662 .1766 4.838 63.11 63.12 .5383 .5299 .9714 5.094 ,3860 .3800 2.425 .9845 5.609 5.697 .1755 4.851 63.91 63.91 .5380 .5296 ,9717 5.092 ,3870 .3811 2.432 .9847 5.645 5.732 .1744 4,863 64.72 64.72 .5376 .5293 ,9719 5.090 ,3880 .3821 2.438 .9849 5.681 5.768 .1734 4.876 65.53 65.54 .5372 .5291 .9721 5.088 ,3890 .3832 2.444 .9850 5.717 5.804 .1723 4.889 66.40 66.40 .5368 .5288 .9724 5.086 ,3900 .3842 2.450 .9852 5.753 5.840 .1712 4,901 67.20 67.21 .5365 .5285 .9726 5.084 ,3910 .3853 2.457 .9854 5.790 5.876 .1702 4.913 68.05 68.06 .5361 .5283 .9729 5.082 ,3920 .3864 2.463 .9856 5.827 5.913 .1691 4.926 68.91 68.92 .5357 .5280 .9731 5.080 ,3930 .3874 2.469 .9858 5.865 5.949 .1681 4.939 69.78 69.79 .5354 .5278 .9733 5.078 3940 .3885 2.476 .9860 5.902 5.988 .1670 4.951 70.67 70.67 .5350 .5275 .9736 5.077 3950 .3895 2.482 .9861 5.940 6.024 .1660 4.964 71.56 71.57 .5347 .5273 .9738 5.075 ,3960 .3906 2.488 .9863 5.978 6.061 .1650 4.976 72,47 72.47 .5343 .5270 .9740 5.073 ,3970 .3916 2.494 .9865 6.016 6.099 .1640 4.989 73.38 73.39 .5340 .5268 .9743 5.071 ,3980 .3927 2.501 .9866 6.054 6.137 .1630 5.001 74.31 74.32 .5337 .5265 .9745 5.070 ,3990 .3937 2.507 .9868 6.093 6.175 .1619 5.014 75.25 75.26 .5333 .5263 .9747 5.068 ,4000 .3948 2.513 .9870 6.132 6.213 .1609 5.027 76.20 76.21 .5330 .5260 .9749 5.066 ,4010 .3958 2.520 .9871 6.172 6.252 .1600 5.039 77.16 77.17 .5327 .5258 .9752 5.064 ,4020 .3969 2.526 .9873 6,210 6.290 .1590 5.052 78.14 78.15 .5323 .5256 .9754 5.063 ,4030 .3979 2.532 .9874 6.250 6.330 .1580 5.064 79.13 79.14 .5320 .5253 .9756 5.061 ,4040 .3990 2.538 .9876 6.290 6.369 .1570 5.077 80.13 80.14 .5317 .5251 .9758 5.060 ,4050 .4000 2.545 .9878 6.330 6.409 .1560 5.089 81.14 81.15 .5314 .5249 .9760 5.058 ,4060 .4011 2.551 .9879 6.371 6.449 .1551 5.102 82.17 82.18 .5310 .5246 .9763 5.056 ,4070 .4021 2.557 .9881 6.412 6,489 .1541 5.115 83.21 83.21 .5307 .5244 .9765 5.055 ,4080 .4032 2.564 .9882 6.452 6.529 .1532 5.127 84.25 84.26 .5304 .5242 .9767 5.053 ,4090 .4042 2.570 .9883 6.493 6.571 .1522 5.140 85.33 85.33 .5301 .5239 .9769 5.052 May 1961 D-23 Table D-2 Cont'd tanh sinh cosh 4nd sinh cosh dA d/Lo 2nd 2nd 2nd 2nd K L 4nd 4nd n Cg^o H/Ho* M L L L L L L ,4100 .4053 2.576 .9885 6.535 6.611 .1513 5.152 86.41 86.41 .5298 .5237 .9771 5.050 ,4110 .4063 2.582 .9886 6.577 6.653 .1503 5.165 87.50 87.50 .5295 .5235 .9773 5.049 ,4120 .4074 2.589 .9888 6.619 6.694 .1494 5.177 88.61 88.61 .5292 .5233 .9775 5.048 ,4130 .4084 2.595 .9889 5.661 6.736 .1485 5.190 89.73 89.73 .5289 .5231 .9777 5.046 ,4140 .4095 2.601 .9891 6.703 6.777 .1476 5.202 90.87 90.87 .5286 .5228 .9779 5.045 ,4150 .4105 2.608 .9892 6.746 6.819 .1466 5.215 92.02 92.02 .5283 .5226 .9781 5.043 ,4150 .4116 2.614 .9893 6.789 6.862 .1457 5.228 93.18 93.18 .5281 .5224 .9783 5.042 ,4170 .4126 2.620 .9895 6.832 6.905 .1448 5.240 94.36 94.36 .5278 .5222 .9785 5.041 ,4180 .4136 2.626 .9896 6.876 6.948 .1439 5.253 95.55 95.55 .5275 .5220 .9787 5.039 ,4190 .4147 2.633 .9897 6.920 6.992 .1430 5.265 96.76 96.76 .5272 .5218 .9789 5.038 ,4200 .4157 2.539 .9899 6.963 7.035 .1422 5.278 97.98 97.98 .5269 .5216 .9791 5.037 ,4210 .4158 2.645 .9900 7.008 7.079 .1413 5.290 99.22 99.22 .5267 .5214 .9793 5.035 ,4220 .4178 2.652 .9901 7.052 7.123 .1404 5.303 100.5 100.5 .5264 .5212 .9795 5.034 ,4230 .4189 2.658 .9902 7.097 7.167 .1395 5.316 101.7 101.7 .5261 .5210 .9797 5.033 ,4240 .4199 2.664 .9903 7.143 7.212 .1387 5.328 103.0 103.0 .5259 .5208 .9799 5.032 ,4250 .4210 2.670 .9905 7.188 7.257 .1378 5.341 104.3 104.3 .5256 .5206 .9801 5.030 ,4260 .4220 2.677 .9906 7.233 7.302 .1370 5.353 105.7 105.7 .5253 .5204 .9803 5.029 4270 .4230 2.683 .9907 7.280 7.348 .1361 5.366 107.0 107.0 .5251 .5202 .9804 5.028 ,4280 .4241 2.689 .9908 7.326 7.394 .1352 5.378 108.3 108.3 .5248 .5200 .9806 5.027 .4290 .4251 2.696 .9909 7.373 7.440 .1344 5.391 109.7 109.7 .5246 .5198 .9808 5.026 ,4300 .4262 2.702 .9910 7.420 7.487 .1336 5.404 111.1 111.1 .5243 .5196 .9810 5.025 ,4310 .4272 2.708 .9912 7.467 7.534 .1327 5.416 112.5 112.5 .5241 .5194 .9811 5.023 ,4320 .4282 2.714 .9913 7.514 7.580 .1319 5.429 113.9 113.9 .5238 .5193 .9813 5.022 ,4330 .4293 2.721 .9914 7.562 7.628 .1311 5.441 115.4 115.4 .5236 .5191 .9815 5.021 ,4340 .4303 2.727 .9915 7.610 7.673 .1303 5.454 116.8 116.8 .5233 .5189 .9817 5.020 ,4350 .4313 2.733 .9916 7.659 7.723 .1295 5.466 118.3 118.3 .5231 .5187 .9818 5.019 ,4360 .4324 2.740 .9917 7.707 7.772 .1287 5.479 119.8 119.8 .5229 .5185 .9820 5.018 ,4370 .4334 2.746 .9918 7.756 7.821 .1279 5.492 121.3 121.3 .5226 .5183 .9822 5.017 ,4380 .4345 2.752 .9919 7.805 7.869 .1271 5.504 122.8 122.8 .5224 .5182 .9823 5.016 ,4390 .4355 2.758 .9920 7.855 7.918 .1263 5.517 124.4 124.4 .5222 .5180 .9825 5.015 ,4400 .4365 2.765 .9921 7.905 7.968 .1255 5.529 126.0 126.0 .5219 .5178 .9827 5.014 ,4410 .4376 2.771 .9922 7.955 8.018 .1247 5.542 127.6 127.6 .5217 .5177 .9828 5.013 ,4420 .4386 2.777 .9923 8.006 8.068 .1239 5.554 129.2 129.2 .5215 .5175 .9830 5.012 ,4430 .4396 2.784 .9924 8.057 8.119 .1232 5.567 130.8 130.8 .5213 .5173 .9831 5.011 ,4440 .4407 2.790 .9925 8.107 8.169 .1224 5.579 132.6 132.6 .5210 .5171 .9833 5.010 ,4450 .4417 2.796 .9926 8.159 8.220 .1217 5.592 134.1 134.1 .5208 .5170 .9835 5.009 ,4460 .4427 2.802 .9927 8.211 8.272 .1209 5.605 135.8 135.8 .5206 .5168 .9836 5.008 ,4470 .4438 2.809 .9928 8.263 8.322 .1202 5.61? 137.6 137.6 .5204 .5166 .9838 5.007 ,4480 .4448 2.815 .9929 8.316 8.376 .1194 5.630 139.3 139.3 .5202 .5165 .9839 5.006 ,4490 .4458 2.821 .9929 8.369 8.428 .1186 5.642 141.1 141.1 .5200 .5163 .9841 5.005 ,4500 .4469 2.827 .9930 8.421 8.480 .1179 5.655 142.8 142.8 .5198 .5162 .9842 5.004 ,4510 .4479 2.834 .9931 8.475 8.534 .1172 5.667 144.7 144.7 .5196 .5160 .9844 5.003 ,4520 .4489 2.840 .9932 8.529 8.587 .1165 5.680 146.5 146.5 .5194 .5159 .9845 5.003 ,4530 .4500 2.846 .9933 8.583 8.641 .1157 5.693 148.3 148.3 .5192 .5157 .9847 5.002 ,4540 .4510 2.853 .9934 8.638 8.695 .1150 5.705 150.2 150.2 .5190 .5156 .9848 5.001 ,4550 .4520 2.359 .9935 8.692 8.750 .1143 5.718 152.1 152.1 .5188 .5154 .9850 5.000 ,4560 .4531 2.865 .9935 8.747 8.804 .1136 5.730 154.0 154.0 .5186 .5153 .9851 4.999 ,4570 .4541 2.871 .9936 8.803 8.859 .1129 5.743 156.0 156.0 .5184 .5151 .98 52 4.999 ,4580 .4551 2.878 .9937 8.859 8.915 .1122 5.755 158.0 158.0 .5182 .■5150 .9854 4.998 ,4590 .4561 2.884 .9938 8.915 8.971 .1115 5.768 159.9 159.9 .5180 .5148 .9855 4.997 ,4600 .4572 2.890 .9938 8.972 9.022 .1108 5.781 162.0 162.0 .5178 .5147 .9857 4.996 ,4610 .4582 2.897 .9939 9.029 9.084 .1101 5.793 164.0 164.0 .5177 .5145 .9858 4.995 ,4620 .4592 2.903 .9940 9.085 9.140 .1094 5.806 166.1 166.1 .5175 .5144 .9859 4.995 ,4630 .4603 2.909 .9941 9.143 9.197 .1087 5.318 168.2 168.2 .5173 .5142 .9861 4.994 ,4640 .4613 2.915 .9941 9.201 9.255 .1080 5.831 170.3 170.3 .5171 .5141 .9862 4.993 ,4650 .4523 2.922 .9942 9.260 9.313 .1074 5.843 172.5 172.5 .5169 .5140 .9863 4.992 ,4660 .4633 2.928 .9943 9.318 9.372 .1067 5.856 174.7 174.7 .5168 .5138 .9865 4.992 ,4670 .4644 2.934 .9944 9.378 9.431 .1060 5.869 176.9 176.9 .5166 .5137 .9866 4.991 ,4680 .4654 2.941 .9944 9.436 9.489 .1054 5.881 179.1 179.1 .5164 .5136 .9867 4.990 ,4690 .4664 2.947 .9945 9.496 9.549 .1047 5.894 181.4 181.4 .5162 .5134 .9868 4.990 May 1961 D-24 Table D-2 Cont'd tanh sinh cosh sinh cosh dA d/Ljj 2nd 2nd 2nd 2nd 1C 4nd 4nd 4nd n Cg/Co H/Ho* M L L L L L L L ,4700 .4675 2.953 .9946 9.557 9.609 .1041 5.906 183.7 183.7 .5161 .5133 .9870 4.989 ,4710 .4685 2.959 .9946 9.617 9.669 .1034 5.919 186.0 186.0 .5159 .5131 .9871 4.988 4720 .4695 2.966 .9947 9.678 9.730 .1028 5,931 188.3 188.3 .5157 .5130 .9872 4.987 ,4730 .4705 2.972 .9948 9.740 9.791 .1021 5.944 190.7 190.7 .5156 .5129 .9873 4.987 ,4740 .4716 2.978 .9948 9.301 9.852 .1015 5.956 193.1 193.1 .5154 .5128 .9875 4.986 ,4750 .4726 2.985 .9949 9.863 9.914 .1009 5.969 195.6 195.6 .5153 .5126 .9876 4.986 4760 .4736 2.991 .9950 9.926 9.976 .1002 5.982 198.0 198.0 .5151 .5125 .9877 4.985 4770 .4746 2.997 .9950 9.989 10.04 .09961 5.994 200.5 200.5 .5149 .5124 .9878 4.984 4780 .4757 3.003 .9951 10.05 10.10 .09899 6.007 203.1 203.1 .5148 .5123 .9880 4.984 4790 .4767 3.010 .9952 10.12 10.17 .09838 6.019 205.6 205.6 .5146 .5121 .9881 4.983 4800 .4777 3.016 .9952 10.18 10.23 .09776 6.032 208.2 208.2 .5145 .5120 .9882 4.983 4810 .4787 3.022 .9953 10.24 10.29 .09715 6.044 210.9 210.9 .5143 .5119 .9883 4.982 4820 .4798 3.029 .9953 10.31 10.36 .09655 6.057 213.5 213.5 .5142 .5118 .9884 4.981 4830 .4808 3.035 .9954 10.37 10.42 .09595 6.070 216.2 216.2 .5140 .5117 .9885 4.981 4840 .4818 3.041 .9954 10.44 10.49 .09535 6.082 219.0 219.0 .5139 .5115 .9887 4.980 4850 .4828 3.047 .9955 10.51 10.55 .09475 6.095 221.7 221.7 .5137 .5114 .9888 4.980 4860 .4838 3.054 .9956 10.57 10.62 .09416 6.107 224.5 224.5 .5136 .5113 .9889 4.979 4870 .4849 3.060 .9956 10.64 10.69 .09358 6.120 227.4 227.4 .5135 .5112 ..9890 4.978 4880 .4859 3.066 .9957 10.71 10.75 .09300 6.132 230.3 230.3 .5133 .5111 .9891 4.978 4890 .48 69 3.073 .9957 10.77 10.82 .09241 6.145 233.2 233.2 .5132 .5110 .9892 4.977 4900 .4879 3.079 .9958 10.84 10.89 .09183 6.158 236.1 236.1 .5130 .5109 .9893 4.977 4910 .4890 3.085 .9958 10.91 10.96 .09126 6.170 239.1 239.1 .5129 .5108 .9894 4.976 4920 .4900 3.091 .9959 10.98 11.03 .09069 6.183 242.1 242,1 .5128 .5107 .9895 4.976 4930 .4910 3.098 .9959 11.05 11.10 .09013 6.195 245.2 245.2 .5126 .5106 .9896 4.975 ,4940 .4920 3.104 .9960 11.12 11.17 .08957 6.208 248.3 248.3 .5125 .5104 .9897 4.975 4950 .4930 3.110 .9960 11.19 11.24 .08901 6.220 251.4 251.4 .5124 .5103 .9898 4.974 4960 .4941 3.117 .9961 11.26 11.31 .08845 6.233 254.6 254.6 .5122 .5102 .9899 4.974 ,4970 .4951 3.123 .9961 11.33 11.38 .08790 6.246 257.8 257.8 .5121 .5101 .9900 4.973 ,4980 .4961 3.129 .9962 11.40 11.45 .08736 6.258 261.1 261.1 .5120 .5100 .9901 4.973 4990 .4971 3.135 .9962 11.48 11.52 .08681 6.271 264.4 264.4 .5119 .5099 .9902 4.972 5000 .4981 3.142 .9963 11.55 11.59 .08627 6.283 267.7 267.7 .5117 .5098 .9903 4.972 5010 .4992 3.148 .9963 11.62 11.67 .08573 6.296 271.1 271.1 .5116 .5097 .9904 4.971 5020 .5002 3.154 .9964 11.70 , 11.74 .08519 6.308 274.5 274.5 .5115 .5096 .9905 4.971 5030 .5012 3.160 .9964 11.77 11.81 .08466 6.321 278.0 278.0 .5114 .5095 .9906 4.971 5040 .5022 3.167 .9965 11.84 11.89 .08413 6.333 281.5 281.5 .5112 .5094 .9907 4.970 5050 .5032 3.173 .9965 11.92 11.96 .08361 6.346 285.1 285.1 .5111 .5093 .9908 4.970 ,5060 .5043 3.179 .9965 11.99 12.03 .08309 6.359 288.7 288.7 .5110 .5092 .9909 4.969 ,5070 .5053 3.186 .9966 12.07 12.11 .08257 6.371 292.4 292.4 .5109 .5092 .9910 4.969 5080 .5063 3.192 .9966 12.15 12.19 .08205 6.384 296.1 296.1 .5108 .5091 .9911 4.968 5090 .5073 3.198 .9967 12.22 12.26 .08154 6.396 299.8 299.8 .5107 .5090 .9911 4.968 5100 .5083 3.204 .9967 12.30 12.34 .08103 6.409 303.6 303.6 .5106 .5089 .9912 4.967 5110 .5093 3.211 .9968 12.38 12.42 .08053 6.421 307.4 307.4 .5104 .5088 .9913 4.967 5120 .5104 3.217 .9968 12.46 12.50 .08002 6.434 311.3 311,3 .5103 .5087 .9914 4.967 ,5130 .5114 3.223 .9968 12.53 12.57 .07952 6.447 315.4 315.4 .5102 .5086 .9915 4.966 5140 .5124 3.230 .9969 12.62 12.65 .07903 6.459 319.2 319.2 .5101 .5085 .9916 4.966 ,5150 .5134 3.236 .9969 12.70 12.74 .07853 6.472 323.3 323.3 .5100 .5084 .9917 4.965 5160 .5144 3.242 .9970 12.77 12.81 .07804 6.484 327.4 327.4 .5099 .5084 .9917 4.965 5170 .5154 3.248 .0970 12.86 12.89 .07756 6.497 331.5 331.5 .5098 .5083 .9918 4.965 ,5180 .5165 3.255 .9970 12.94 12.98 .07707 6.509 335.7 335.7 .5097 .5082 .9919 4.964 5190 .5175 3.261 .9971 13.02 13.06 .07659 6.522 339.9 339.9 .5096 .5081 .9920 4.964 ,5200 .5185 3.267 .9971 13.10 13.14 .07611 6.535 344.2 344.2 .5095 .5080 .9921 4.964 5210 .5195 3.274 .9971 13.18 13.22 .07564 6.547 348.2 348.2 .5094 .5079 .9921 4.963 ,5220 .5205 3.280 .9972 13.27 13.30 .07517 6.560 353.0 353.0 .5093 .5079 .9922 4.963 ,5230 .5215 3.286 .9972 13.35 13.38 .07469 6.572 357.5 357.5 .5092 .5078 .9923 4.963 ,5240 .5226 3.292 .9972 13.44 13.47 .07422 6.585 362.0 362.0 .5091 .5077 .9924 4.962 ,5250 .5236 3.299 .9973 13.52 13.56 .07376 6.597 366.6 366.6 .5090 .5076 .9925 4.962 ,5260 .5246 3.305 .9973 13.61 13.64 .07330 6.610 371.2 371.2 .5089 .5076 .9925 4.962 ,5270 .5256 3.311 .9974 13.69 13.72 .07284 6.622 375.9 375.9 .5088 .5075 .9926 4.961 ,5280 .5266 3.318 .9974 13.78 13.81 .07239 6.635 380.3 380.3 .5087 .5074 .9927 4.961 ,5290 .5276 3.324 .9974 13.86 13.89 .07194 6.648 385.5 385.5 .5086 .5073 .9928 4.961 May 1961 D-25 Table D-2 Cont'd tanh sinh cosh sinh cosh dA dAo 2ird 2nd 2nd 2vd K 4nd 4Trd 4nd n Cg/Co H/Hq* M L L L L L L L i 5300 .5286 3.330 .9974 13,95 13.99 .07149 6.600 390.3 390.3 .5085 .5072 .9929 4.960 .5310 .5297 3.336 .9975 14.04 14.08 .07104 6.673 395.3 395.3 .5084 .5072 .9929 4.960 .5320 .5307 3.343 .9975 14.13 14.17 .07059 6.685 400.3 400.3 .5084 .5071 .9930 4.960 .5330 .5317 3.349 .9975 14.22 14,25 .07016 6.698 405.3 405.3 ,5083 .5070 .9931 4.959 .5340 .5327 3.355 .9976 14.31 14.34 .06972 6.710 410.5 410.5 .5082 .5069 .9931 4.959 .5350 .5337 3.362 .9976 14.40 14.43 .06928 6.723 415.6 415.6 .5081 .5069 .9932 4.959 .5360 .5347 3.368 .9976 14.49 14.52 .06885 6.736 420.9 420.9 .5080 .5068 .9933 4.958 .5370 .5357 3,374 .9977 14.58 14.62 ,06842 6.748 426.2 426.2 .5079 .5067 .9933 4.958 .5380 .5368 3.380 .9977 14.67 14.71 .06799 6.761 431.6 431.6 .5078 ,5067 .9934 4.958 .5390 .5378 3.387 .9977 14.77 14.80 .06757 6.773 437.1 437.1 .5077 .5066 .9935 4.958 .5400 .5388 3.393 .9977 14.86 14.89 .06715 6.786 442.6 442.6 .5077 .5065 .9935 4.957 .5410 .5398 3.399 .9978 14.95 14.99 .06673 6.798 448.2 448.2 .5076 .5065 .9936 4.957 .5420 .5408 3.405 .9978 15.05 15.08 .06631 6.811 453.9 453.9 .5075 .5064 .9937 4.957 .5430 .5418 3.412 .9978 1$.14 15.18 .06589 6.824 459.6 459.6 .5074 ,5063 .9937 4.956 .5440 .5428 3.418 .9979 15.25 15.27 .06548 6.836 465.4 465.4 .5073 .5063 .9938 4.956 .5450 .5438 3.424 .9979 15.34 15.37 .06507 6.849 471.2 471.2 .5073 .5062 .9939 4.956 .5460 .5449 3.431 .9979 15.43 15.46 .06467 6.861 477.2 477.2 .5072 .5061 .9939 4.956 .5470 .5459 3.437 .9979 15.53 15.56 .06426 6.874 483.3 483.3 .5071 .5061 .9940 4.955 .5480 .5469 3.443 .9980 15.63 15.66 .06386 6.886 489.4 489.4 .5070 .5060 .9941 4.955 .5490 .5479 3.449 .9980 15.73 15.76 .06346 6.899 495.6 495.6 .5070 .5059 .9941 4.955 .5500 .5489 3.456 .9980 15.83 15.86 .06306 6.912 501.9 501.9 .5069 .5059 .9942 4.955 .5510 .5499 3.462 .9980 15.93 15.96 .06267 6.924 508.2 508.2 .5068 .5058 .9942 4.954 .5520 .5509 3.468 .9981 16.03 16.06 .06228 6.937 514.6 514.6 .5067 .5058 .9943 4.954 .5530 .5519 3.475 .9981 16.13 16.16 .06189 6.949 521.1 521.1 .5067 .5057 .9944 4.954 .5540 .5530 3.481 .9981 16.23 16.26 .06150 6.962 527.7 527.7 .5066 .5056 .9944 4.954 .5550 .5540 3.487 .9981 16.33 16.36 .06112 6.974 534.4 534.4 .5065 .5056 .9945 4.953 .5560 .5550 3.493 .9982 16.44 16.47 .06074 6.987 541.2 541,2 ,5065 .5055 .9945 4.953 .5570 .5560 3.500 .9982 16,54 16.57 .06036 6.999 548.0 548.0 .5064 .5055 .9946 4.953 .5580 .5570 3.506 .9982 16.64 16.67 .05998 7.012 554.9 554.9 .5063 .5054 .9946 4.953 .5590 .5580 3.512 .9982 16.75 16.78 .05960 7.025 561.9 561.9 .5063 .5054 .9947 4.953 .5600 .5590 3.519 .9982 16.85 16.88 .05923 7.037 569.1 569.1 .5062 .5053 .9948 4.952 .5610 .5600 3,525 .9983 16,96 16.99 .05886 7.050 576.3 576.3 .5061 .5052 .9948 4.952 .5620 .5610 3.531 .9983 17.07 17.10 .05849 7.062 583.5 583.5 .5061 .5052 .9949 4.952 .5630 .5621 3.537 .9983 17.17 17.20 ,05813 7.075 590.9 590.9 .5060 .5051 .9949 4.952 .5640 .5631 3.544 .9983 17.28 17.31 .05776 7.087 598.4 598.4 .5059 .5051 .9950 4.951 .5650 .5641 3.550 .9984 17.39 17.42 .05740 7.100 606.0 606.0 .5059 .5050 .9950 4.951 .5660 .5651 3.556 .9984 17.50 17.53 .05704 7.113 613.6 613.6 .5058 .5050 .9951 4,951 .5670 .5661 3.563 .9984 17.61 17.64 .05669 7.125 621.4 621.4 .5057 .5049 .9951 4.951 .5680 .5671 3.569 .9984 17.72 17.75 .05633 7.138 629.2 629.2 .5057 .5049 .9952 4.951 .5690 .5681 3.575 .9984 17.84 17.86 .05598 7.150 637.3 637.3 .5056 .5048 .9952 4.950 .5700 .5691 3.581 .9985 17.95 17.98 .05563 7.163 645.2 645.2 .5056 .5048 .9953 4.950 .5710 .5701 3.588 .9985 18.06 18.09 .05528 7.175 653.4 653.4 .5055 .5047 .9953 4.950 .5720 .5711 3.594 .9985 18.18 18.20 .05494 7.188 661.7 661.7 .5054 .5047 .9954 4.950 .5730 .5722 3.600 .9985 18.29 18.32 .05459 7.201 670.0 670.0 .5054 .5046 .9954 4.950 .5740 .5732 3.607 .9985 18.41 18.43 .05425 7.213 678.5 678.5 .5053 .5046 .9955 4.949 .5750 .5742 3.613 .9986 18.52 18.55 .05391 7.226 687.1 687.1 .5053 .5045 .9955 4.949 .5760 .5752 3.619 .9986 18.64 18.67 .05358 7.238 695.8 695.8 .5052 .5045 .9956 4.949 .5770 .5762 3.625 .9986 18,76 18.78 .05324 7.251 704.6 704.6 .5051 .5044 .9956 4.949 .5780 .5772 3.632 .9986 18.88 18.90 .05291 7.263 713.5 713.5 .5051 .5044 .9957 4.949 .5790 .5782 3.638 .9986 18.99 19.02 .05258 7.276 722.5 722.5 .5050 .5043 .9957 4.949 .5800 .5792 3.644 .9986 19.11 19.14 .05225 7.289 731.6 731.6 .5050 .5043 .9957 4.948 .5810 .5802 3.651 .9987 19.23 19.26 .05192 7.301 740.9 740.9 .5049 .5043 .9958 4.948 .5820 .5812 3.657 .9987 19.36 19.38 .05160 7.314 750.3 750.3 .5049 .5042 .9958 4.948 .5830 .5822 3.663 .9987 19.48 19.50 .05127 7.326 759.8 759.8 .5048 .5042 .9959 4.948 .5840 .5832 3.669 .9987 19.60 19.63 .05095 7.339 769.4 769.4 .5048 .5041 .9959 4.948 .5850 .5843 3.676 .9987 19.73 19.75 .05063 7.351 779.1 779.1 .5047 .5041 .9960 4.948 .5860 .5853 3.682 .9987 19.85 19.87 .05032 7.364 788.9 788.9 .5047 .5040 .9960 4.947 .5870 .5863 3.688 .9988 19.97 20.00 .05000 7.376 798.9 798.9 .5046 .5040 .9960 4.947 .5880 .5873 3.695 .9988 20.10 20.13 .04969 7.389 809.0 809.0 .5046 .5039 .9961 4.947 .5890 .5883 3.701 .9988 20.23 20.25 .04938 7.402 819.3 819.3 .5045 .5039 .9961 4.947 Moy 1961 D-26 Table D-2 Cont'd dA d/Lo 2trd tanh 2nd sinh 2nd cosh 2nd sinh K 4nd 4nd' cosh 4nd n H/H„* M L L L L L L L .5900 .5893 3.707 .9988 20.36 20.38 .04907 7.414 829.6 829.6 ,5045 .5039 .9962 4.947 .5910 .5903 3.713 .9988 20.48 20.51 .04876 7.427 840.1 840.1 .5044 .5038 .9962 4.947 .5920 .5913 3.720 .9988 20.61 20.64 .04846 7.439 850.7 850.7 .5044 .5038 .9962 4.946 .5930 .5923 3.726 .9988 20.74 20.77 .04815 7.452 861.5 861.5 .5043 .5037 .9963 4.946 .5940 .5933 3.732 .9989 20.87 20.90 .04785 7.464 872.4 872.4 .5043 .5037 .9963 4.946 .5950 .5943 3.739 .9989 21.01 21.03 .04755 7.477 883.4 883.4 .5042 .5037 .9964 4.946 .5960 .5953 3.745 .9989 21.14 21.16 .04725 7.490 894.6 894.6 .5042 .5036 .9964 4.946 .5970 .5963 3.751 .9989 21.27 21.30 .04696 7.502 905.9 905.9 .5041 .5036 .9964 4.946 .5980 .5974 3.757 .9989 21.41 21.43 .04667 7.515 917.3 917.3 .5041 .5036 .9965 4.946 .5990 .5984 3.764 .9989 21.54 21.55 .04639 7.527 929.0 929.0 .5041 .5035 .9965 4.945 .6000 .5994 3.770 .9989 21.68 21.70 .04609 7.540 940.7 940.7 .5040 .5035 .9966 4.945 .6100 .6094 3.833 .9991 23.08 23.11 .04328 7.666 1067. 1067. .5036 .5031 .9970 4.944 .6200 .6195 3.896 .9992 24.58 24,60 .04065 7.791 1210. 1210. .5032 .5028 .9972 4.943 .6300 .6295 3.958 .9993 26.18 26.20 .03817 7.917 1371, 1371. .5029 .5025 .9975 4.942 .6400 .6396 4.021 .9994 27.88 27.89 .03585 8.043 1555. 1555. .5026 .5023 .9978 4.941 .6500 .6496 4.084 .9994 29.69 29.70 .03367 8.163 1754. 1754. .5023 .5020 .9980 4.941 .6600 .6597 4.147 .9995 31.61 31.63 .03162 8.294 1999. 1999. .5021 .5018 .9982 4.940 .6700 .6697 4.210 .9996 33.66 33.68 .02969 8.419 2267. 2267, .5019 .5016 .9984 4.939 .6800 .6797 4.273 .9996 35.85 35.86 .02789 8.545 2571. 2571. .5017 .5015 .9985 4.939 .6900 .6898 4.335 .9997 38.17 38.18 .02619 8.671 2915. 2915. .5015 .5013 .9987 4.938 .7000 .6998 4.398 .9997 40.65 40.66 .02459 8.796 3305. 3305. .5013 .5012 .9988 4.938 .7100 .7098 4.461 .9997 43.29 43.30 .02310 8.922 3748. 3748. .5012 .5011 .9989 4.938 .7200 .7198 4.524 .9998 46.09 46.10 .02169 9.048 4250. 4250. .5011 .5010 .9990 4.937 .7300 .7299 4.587 .9998 49.08 49.09 .02037 9.173 4819, 4819. .5010 .5009 .9991 4.937 .7400 .7399 4.650 .9998 52.27 52.28 .01913 9.299 5464. 5464, .5009 .5008 .9992 4.937 .7500 .7499 4.712 .9998 55.66 55.66 .01796 9.425 6195. 6195. .5008 .5007 .9993 4.936 .7600 .7599 4.775 .9999 59.26 59.27 .01687 9.550 7025. 7025. .5007 .5006 .9994 4.936 .7700 .7699 4.838 .9999 63.11 63.12 .01584 9.676 7966. 7966. .5006 .5005 .9995 4.936 .7800 .7799 4.901 .9999 67.20 67.21 .01488 9.802 9032. 9032. .5005 .5005 .9995 4.936 .7900 .7899 4.9 64 .9999 71.56 71.56 .01397 9.927 10240. 10240. .5005 .5004 .9996 4.936 .8000 .7999 5.027 .9999 76.21 76.21 .01312 10.05 11610. 11610. .5004 .5004 .9996 4.936 .8100 .8099 5.089 .9999 81.14 81.14 .01232 10.18 13170. 13170. .5004 .5004 .9997 4.936 .8200 .8199 5.152 .9999 86.40 86.40 .01157 10.30‘ 14930. 14930. .5003 .5003 .9997 4.936 .8300 .8300 5.215 .9999 92.01 92.01 .01087 10.43 16930. 16930. .5003 .5003 .9997 4.935 .8400 .8400 5.278 1.000 97.98 97.98 .01021 10.56 19200. 19200. .5003 .5003 .9998 4.935 .8500 .8500 5.341 1.000 104.3 104.3 .009585 10.68 21770. 21770. .5002 .5002 .9998 4.935 .8600 .8600 5.404 1.000 111.1 111.1 .009000 10.81. ,24680. 24680'. .5002 .5002 .9998 4.935 .8700 .8700 5.466 1.000 118.3 118.3 .008453 10.93 27990. 27990. .5002 .5002 .9998 4.935 .8800 .8800 5.529 1.000 126.0 126.0 .007939 11.06 31730. 31730. .5002 .5002 .9998 4.935 .8900 .8900 5.592 1.000 134.1 134.1 .007455 11.18 35980. 35980. .5002 .5002 .9999 4.935 .9000 .9000 5.655 1.000 142,8 142.8 .007001 11.31 40800. 40800.. .5001 .5001 .9999 4.935 .9100 .9100 5.718 1.000 152.1 152,1 .006575 11.44 46260. 46260. .5001 .5001 .9999 4.935 .9200 .9200 5.781 1.000 162.0 162.0 .006174 11.56 52460. 52460. .5001 .5001 .9999 4.935 .9300 .9300 5.843 1.000 172.5 172.5 .005798 11.69 59480. 59480. .5001 .5001 .9999 4.935 .9400 .9400 5.906 1.000 183.7 183.7 .005445 11.81 67450. 67450. ,5001 .5001 .9999 4,935 .9500 .9500 5.969 1.000 195.6 195.6 .005114 11.94 76480. 76480. .5001 .5001 .9999 4.935 .9600 .9600 6.032 1.000 208.2 208.2 .004802 12.06 86720. 86720. .5001 .5001 .9999 4.935 .9700 .9700 6.095 1.000 221.7 221.7 .004510 12.19 98340. 98340. .5001 .5001 .9999 4.935 .9800 .9800 6.158 1.000 236.1 236,1 .004235 12.32 111500. ,111500. .5001 .5001 .9999 4.935 .9900 .9900 6.220 1.000 251.4 251.4 .003977 12.44 126400. 126400. .5000 .5000 1.0000 4.935 .000 1.000 6.283 1.000 267.7 267.7 .003735 12.57 143400. 143400. .5000 .5000 1.0000 4.935 Moy 1961 D-27 D -28 WAVE LENGTH-FEET PLATE D-l. RELATIONSHIP BETWEEN WAVE PERIOD. LENGTH, AND DEPTH. PLATE D-I a. RELATIONSHIP BETWEEN WAVE PERIOD, LENGTH,AND DEPTH Moy 1961 D-29 WAVE VELOCITY- FEET/SEC. 60 DEPTH - FEET PLATE D-2 D-30 WAVE PERIOD TANH 10 D-31 TABLE D-3 CONVERSION TABLE FOR WIND FORCE. BEAUFORT SCALE Beaufort Statute Miles per Terms used Explanatory 1 Number Hour Knots in Forecasts Terms 0 Less than 1 Less than 1 Calm 1 1-3 1-3 Light Light air 2 4-7 4-6 Light breeze 3 8-12 7-10 Gentle Gentle breeze 4 13-18 11-16 Moderate Moderate breeze 5 19-24 17-21 Fresh Fresh breeze 6 25-31 22-27 Strong Strong breeze 7 32-38 28-33 High wind 8 39-46 34-40 Gale (moderate gale) Gale (fresh gale) 9 47-54 41-47 Strong gale 10 55-63 48-55 Whole gale Whole gale 11 64-72 56-63 Storm 12 73-82 64-71 Hurricane 13 83-92 72-80 Hurricane 14 93-103 81-80 Hurricane Hurricane 15 104-114 90-99 Hurricane 16 115-125 100-108 Hurricane 17 126-136 109-118 Hurricane Code Figure Approximate height of sea TABLE D-4 WIND WAVE (SEA) HEIGHT CODE (U.S. Navy H.O. Pub. #607) Seaman's descript ion 0 1 2 3 4 5 6 7 8 9 0 Less than 1 foot 1 to 3 feet 3 to 5 feet 5 to 8 feet 8 to 12 feet 12 to 20 feet 20 to 40 feet 40 feet and over Calm - Sea like mirror. Smooth - Small wavelets or ripples with the appearance of scale but without crests. Slight - The waves or small rollers are short and more pronounced, when capping the foam is not white but more of a glassy appearance. Moderate - The waves or large rollers become longer and begin to show white caps occasionally. The sea produces short rustling sounds. Rough - Medium waves that take a more pronounced long form with ex¬ tensive whitecapping and white foam crests. The noise of the sea is like a dull murmur. Very rough - The medium waves become larger and begin to heap up, the whitecapping is continuous, and tie seas break occasionally; the foam from the capping and breaking waves begins to be blown along in the direction of the wind. The breaking and capping seas produce a perpetual murmur. High - Heavy, whitecapped waves that show a visible increase in height and are breaking extensively. The foam is blown in dense streaks along in the direction of the wind. The sea begins to roll and the noise of the breaking seas is like a dull roar, audible at greater distance. Very high - High, heavy waves developed with long overhanging crests that are breaking continuously, with a perpetual roaring noise. The whole surface of the sea takes on a white appearance from the great amount of foam being blown along with the wind. The rolling of the sea becomes heavy and shocklike. Mountainous - The heavy waves become so high that ships within close distances drop so low in the wave troughs that for a time they are lost from view. The rolling of the sea becomes tumultuous. The wind beats the breaking edge of the seas into a froth, and the whole sea is covered with dense streaks of foam being carried along with the wind. Owing to the violence of the wind the air is so filled with foam and spray that relatively close objects are no longer visible. Note - Qualifying condition applicable to the previous conditions, e.g. (5-9), A very rough, confused sea. February 1957 D-32 TABLE D-5 SWELL (X>NDITION CX)DE Code 0 9 Height in feet 0 - no swell 1 to 6 low swell 6 to 12 moderate Greater than 12 high Confused (U.S. Navy H.O. Pub, #607) Description Short or average Long Short Average Long Short Average Long Approximate length in feet 0 0 to 600 Above 600 0 to 300 300 to 600 Above 600 0 to 300 300 to 600 Above 600 TABLE D-6 International Code 69 for Wave Period(138) SYMBOL "P " PmiOD OP THE WAVES SLj Code Figure 2 5 seconds or less 3 5-7 seconds 4 7-9 seconds 5 9-11 seconds 6 11-13 seconds 7 13-15 seconds 8 15-17 seconds 9 17-19 seconds 0 19-21 seconds 1 Over 21 seconds X Calm or period not determined TABLE D-7 International Code 42 for Wave Height^^^B) SYMBOL "H *• w MEAN MAXIMUM HEIGHT OP THE WAVES Code Pigure 0 1 2 3 4 5 6 7 8 9 X Height Less than 1 foot (1/4 m.) 1^ ft. (1/2 m.) 3 ft. dm.) 5 ft. (1^ m.) 6| ft. (2m.) 8 ft. (2| m.) 9§ ft. (3 m.) 11 ft. (3i m.) 13 ft. (4 m.) 14 ft (4| m.) Height* 16 ft. (5m,) 17i ft. (5i m.) 19 ft. (6 m.) 21 ft. (65 m.) 22i ft. (7 m.) 24 ft. (7i m.) 25i ft. (8 m.) 27 ft. (8^ m.) 29 ft. (9 m.) 30i ft. (9| m.) Height not determined If 50 is added to d d or if code figure for the true direction from which the waves come (^[ymbol d d ) is between 50 and 99 inclusive. w w February 1957 D-33 WAVE ENERGY, E, PER FOOT OF CREST, FT LBS./FT. WAVE HEIGHT, H , IN FEET PLATE D-4 RELATIONSHIP BETWEEN WAVE ENERGY, WAVE LENGTH, AND WAVE HEIGHT D-34 function of wave period (T). a V3 a s 0)1 rH x> -p c Q> rH a> to Ttt CD in to rH CD cn t- in CO rH o CD CD 00 t’- ?}■ to 'ey A g C3 o CO 03 f—t rH f-H rH rH rH rH rH rH rH fH rH rH rH rH rH rH rH rH rH rH rH rH rH rH iH rH rH rH CO CO CO CO CO CO CO CO (0 -p u> CO in CO rH t-- o CO to CD CO in CO rH CO rH C- o CO to 03 CO in CO rH C- o CO to to O’- c- C- CO O O • • • • • • • • • • • • • • o o rH rH rH CO CO CO CO to CO CO m in in to to to r- c- c- CO 00 cn CD CD CD n o n CO to CD .CO 1/3 CO in in 1/3 in U3 in m 1/3 1/3 m 1/3 in 1/3 in in lO m in in in in 1/3 1/3 in U3 in in L/3 in in to to to to to to C** Ch CO 4 in UN • o O m O in o to rH to rH to rH to rH to CO C- CO C- CO C’- CO CO to CO CO CO to CO to CO ■O’ in to C- 00 o rH lA o to to b- CO CO CD CD o o rH CO CO CO to in in to to r- C- cn m CD CD o o rH rH CO f- CO CO 00 CO o ;P CO CO 00 CO CO CO CO CO 03 CD CD CD O rH CO CO m to !>• CO CD O rH CO CO in to CO CD o rH CO CO in to C- CO CD o o o o o o o ■ M O to to C- C- t-- C- C- C- C- CO CO CO CO CO 00 CO fiO CO cc CD 03 CD CD 03 CD 03 CD CD C3 o rH CO to in to o exj in o CO to CD CO m Oi CO to CD CO C- in 03 ’'t CO CO CO to to rH t- CO C- to 03 o to to to o to to CD to CO o C- o to to CD CO in CO r- ■O’ t- O CO to 0 o t*- O CO fin CO CD CD CD o o o O rH rH CO CO CO CO to to CO in in in to to to t- C- t- CO 48. 48. CD CD CD O o o o s CO CO CO CO t'J CO CO CO CO ri* O’ ’tf ^ in in w o 0) o in o in O in O in o to rH to rH to rH CO rH to CO c*- CO c*- CO 0 CO C- CO CO to CO CO CO CO CO 03 CD CD ■^ CD ^ o in o C\3 CO CO •O’ •0^ in in to to t- c- CO CO CD CD n o rH rH CO CO to to in in to to t> fiO CD 03 o o rH CO CO CO to ^ in in i to to to to to to to to to to to to to to to C- C- 13- C- C- C- C- C- t- C- t> o C- t> c- CO 00 CO CO CO CO 00 CO CO CO CO m g CO to in to t- CO CD O rH CO to in to C*- 00 03 o rH CO to in to t- CO CD o rH CO to O’ in to C- GO 03 o rH CO to 'i’ in to t- o CM CO CO CO CO CO CO CO CO CO CO to to CO CO CO CO CO in m in in m in in in in in to to to to to to to to (D 75 f—• rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH fH rH rH rH fH- rH o« +> 0 to CO o CO to CO rH o CD 03 to in CO CO CO rH rH CO CO CO CO to m to m o CO in t- rH to m C- O 0 c;> C5 rH CO CO CO U3 to C- t- on CD o rH CO U3 to t- CD CD o CO in tn O’- 03 o CO in to e- 00 o rH CO to m bi CO eto W w V3 C<3 CO to CO CO to to -tjl in in in in in m in m in in to to to to to to to to t- c- 0 . -P 7> to to CD CO in CO rH C*- rH •«i< o to to 03 CO in CO rH o to to CD CO in CO rH o CO to CD CO in 00 ’tf CO rH o C to to CO CO in in in to to to C- c- C- CO 03 CO 03 03 03 o o o O CO CO to -O' 1/5 in t/5 to to to w CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO to CO CO to CO to to to CO to CO CO CO in to CO CO to CO to CO CO CO o 0 < Oi CD CD o in o m o in o to rH to rH to rH CO rH to CO C- CO CO t> CO c- CO CO CO CO CO CO CO CO to CD CD CD ■’i’ CD o CO a> o> O O rH CO CO CO CO in in to to c*- C-' 00 CO CD 03 o o rH CO CO CO in in to to C** CO CD CD o o rH fH o £ CO to in in in m in m in in in in m in m in m m in in in in to to to to (spue to c* CO CD o rH CO to in to c- CO CD o rH CO CO in to GO 03 o rH CO CO in to O’- CO 03 O rH CO to in • to e-“ .8 ,9 O rH o c- c- c- C- CO 00 CO CO CO CO CO CO CO CO CD 03 03 03 CD CD 03 CD CD CD o o O o o o o o O o fH rH rH rH rH r- rH rH r~ CO CO 0 CO rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH CO CO CO C- rH CD CD CD rH CO c*- o 0 • • • • • « • • • • • • • • « 0 to O) CO in CD CO to o CO t- rH to n CD 00 to CO CO cn to CD 03 in to CO CO t- o to o C*- CO m CO o GO fa in in in to to t- 00 CO 03 03 CD o O rH rH CO CO W to in to to C- f- 00 CD CD o rH CO to CO in in to C- cn CD rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH CO CO CO CO CO CO CO CO CO CO CO CO CO CO p rH t*- o CO to CD CO in 00 rH t- o to to CD CO in CO CO m CO rH C- o CO to CD CO m CO fH ti’ C*- O to to CD CO in 00 rH ■^ C- o C O) CD o o o o rH rH rH CO CO CO CO to CO to in in in to to to C- C- CO CO CO CD CD CD o o o O rH rH CO CO CO rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH rH fH CO CO CO CO CO CO CO CO CO CO o 0 o> CD CD CD O in o m O in o m o CO rH to f-H to rH to rH o. CO CO C* CO t- CO 00 CO CO CO CO CO CO CO 00 ^ CD o in in to to C- CO CO CD o rH rH CO CO CO to in in to to C- C*- CO 03 03 o o CO to in to to t** t- CO £ rH rH rH rH rH rH rH rH rH CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO CO to to CO to CO CO to CO CO CO CO CO V> CO to CO CO (0 TJ C o rH CO CO in to C- CO CD o rH CO CO in to C- 00 CD O rH CO to in to t«- CO CD o CO CO -'i’ in to C*- CO CD o rH CO ^5 in 0 0 to to CO to to CO CO to to CO lO in in in in in in in in in to 'O to to to to to to to to C*- C- 0- C- CO D-35 D-36 I XJ ^ g O w O VI X +> O hO (D 6h >. 'O o u •H © u +> © 0 © cu > © 05 © g O o Oi csj o o lO rH C- CO CO CO CO CD CO CO CO o> c- C- CO CD LO to CO O c- CO rH o> a C7> o 05 05 05 05 05 05 05 05 05 05 CO C- • • • • • • • • o o o o o o O o o o o o o o o o o o o o O « o o o o O 05 to CO 05 05 LO to CO O CO 05 CO cc5 N CNJ CO CO CO CO to to lO CD c- o to CO < o o o o o o o o o • o o O O o • O • o rH i-H rH o d o o d o o o o o o O o o o o o o o o o ji to -p o CNj lO iH LO 05 lO CO 05 to rH CD lO o CO 05 CO t- c- CD CO CO C- c- CD 05 <75 <75 05 05 05 <75 <75 <75 <75 <75 • • • • • • • • o o o o o o o O o o O a> lO a> C75 <75 rH 05 o O CO CO CO C75 co CO c- • • o o o o o fl iJ •» LO CD CO t- <35 CO CO rH CO CO rH CO o CO CD O o cva CM to CO CO to to << LO CO c- o to O o o o o O o o o o o O o o o o o rH rH c (0 -P o rH rH rH rH rH rH rH rH rH rH pH rH rH rH rH pH rH rH rH o bO o 05 C a> n rH t> LO lO 05 CM to C75 lO to CO rH CO CO CD o Ph ^ at CO CO to rH pH CO CO CO to CO c- to o o o O o O O O O o o o o o o o rH rH • • « • • • • • • • 'V o o o O o o o o o o o o o o o o O O o a) Eh o t- •H © o »-i -p © © a > © o« > © 05 © ^ Q 6 o jc m •p to o to D-37 Viave period (T) = 13 seconds Wave period (T) ■ 14 seconds Deep water wave length (Lq) = 865 feet ^^eep water wave length (L^) - 1,003.5 feet * 'O o LO 05 eg eg CO eg eg CD CD LO o c- eg CD 05 <75 O) 05 05 <75 05 <75 CO CO (0 • • o O o O o o o o o o o O o * V3 to c** eg eg CO <75 <75 CO lO >P CO CO lO CO CO O o o o O o o O o C rH f—c eg O rH rH rH rH rH rH rH rH rH rH rH rH o5 to rH CD eg CO o LO LO CO to to LO LO CO o CO o o o C o C o o rH rH to • • • <3 o o o o o o o O o o o o e o X lo -P rH 05 ^3 CO O CO rH rH e- CO eg eg eg o <75 c- o> 05 <75 o> (75 05 (7) u • • o O O O O o o O O o O o Cn 3 f-« r L D-38 Table D-9 Cont'd CO U 296*0 896*0 296*0 0.947 0.941 026*0 0.917 868*0 0.370 0.82S 0.711 * C7) O-i CD to lO o 05 in to is> CO • e- cn rH rH o o o o o o o o rH rH CM TJ O »—1 rM rH rH rH rH rH rH rH rH rH > 0 CO CM CO to to CO CO to o sh3 to lO CO c- CO O CO o o o o o o o o rH rH CM CO « • e • • « -eg o o o o o o o O o o O CO fc; -p O cu X CM »HOO>OOC'“CDLO'^tOCM»-l 0 -P .-1 Q 0 Qh -p 0 0 o to CO c- CM CM CO C7> CO CO rH 05 CO t- CD in CO rH CO C*- CO CO to a> o> cn o a> Oi C3^ 05 05 05 C75 0 • • • * • II o o o o o o o o o o o o o CO T3 ^ c o » o 0 CO c- CO o o o c o o o o o o o 00 X T? -P fH fH rH rH rH rH rH rH rH rH rH CD hD 0 H r-» 0 0 CO LO lO CO Oi in rH C- o to in E-< > tJ} CM to lO to CO CO in CD CO CO CO a o o o o o o o C O o O O .J • « • T3 <3 o o o o o o o o o o o o o u, a> a a* ^ o cu > 0 0 ^ Q O) o o to o to lO CM O CM It T3 CO u 296*0 696*0 0.954 0.948 0.941 026*0 816*0 668*0 0.871 128*0 0.712 » 05 cn CO CO CO CO 05 CM Cp CO lO -P CO ^0 to CO C- 00 rH rH o o O O o o O o rH CM 'O o rH rH rH rH rH rH rH rH rH l-H rH > 0 (T) CO c- to CO O CO CO C-- CO to in CO CO o CO 05 to \ O o o o o o o rH rH rH CO • • • • • • • • • • • o o o o o o o o o o O X -p o X. X CM rHOO5oOt-COlO'0CO CM 0 -P Q 0 Cm « CO CM CO <75 Q CM 05 CO lO CM CM C- CO CO \ <75 05 <75 05 <75 05 05 05 05 <75 09 • • • • • « • o o o O o O O o o o O • 05 o CO CO rH cO -0 <75 CM in CO *<0 CO CO 00 lO CO CM CO to \ o o o O o o O o o o o rH rH rH rH rH rH rH rH rH rH o t- o 05 o o to <75 *0 CD 00 in CO IC CO CM CO to o o o o o o O O o o 1-3 • » • • • o o o o o o o o o o 05 d in O • X X CO QQOiOOt-iO^tO CM -P -p c;5 lO-^tOCMCvJrHrHrHrH CU 0 0 •m o TJ ^ C O O ^ O 0 w X +J in fcX) *H C 0 ^ 0 tH J> ^ 0 , ■« ^ o u i: js 0 cu > 0 0 » Q D-39 Table D-9Cont'd rtave period (t) * 17 seconds uave period (T) = 18 seconds Deep water wave length (L^) - 1,480 feet Deep water wave length (Lq) ■ 1,659 i X3 to m CM CO o CO 05 CO o o x> CO m CV3 to o to rH CO CO o rH 05 CO CM \ o o> 05 05 05 05 00 00 05 00 CJ5 CD GO 00 C- CO • • • • • • • • • • • • • • • o o o o o o o o o o o o o o O o o » (0 CO 00 rH 05 CO CO to CM 05 rH lO CM lO C75 05 CO CO o 05 CO 05 i-H C75 rH lO o \ o o o o rH o o rH rH rH o rH CM XJ « o rH rH rH rH rH rH rH rM rH rH rH rH rH rH rH rH LO C- Oi 05 05 LO lO rH CD CO rH CO 05 CO 05 CM 00 O CM rH o o CO o o o o o o o rH rH rH rH rH rH rH CM • • • • # • • • • • • • • • • • o o o o o o o o o o o o o o O o Si -p a. 0 G * XJ m O * (0 O > OOOOOtOQi^OcOCDiO'^tOCSJf-l Ot*-lOt^ 05 CO to lO 05 05 05 CJ5 05 05 • • • • • • o o o o o o lO CO CO t- CO CO rH to CM o 05 05 05 05 05 05 4 • • • o • o o o o O o CO CO CM 05 c^ CM 05 D- rH rH co CO CO CO CO c*- • • • • • e o o o o o o CO o to CO 05 CM CO CO CO CM to CM O o CO C-- CO LO CO C75 LO CO 00 o rH CM o o o o o o o c O o o o rH rH CM rH rH CM rH rH rH rH rH rH rH i-H rH rH rH rH rH rH rH rH rH rH C^ CO o CM LO rH LO to t- CM to LO CM o 05 C75 o ^0 CO CO LO CO 05 LO CD C-- o to 05 rH 00 CO O O o o o o o o o o o rH r-* r—* rH ^ rH rH CO • • • • • • • • • • • • • • • • • • o o o o o o o o o o o o o o o o o o s: -p (0 rt 6 CO O o o o LO o lO O to lO to o LO to CO CM rH a. 0 Q X CM -P rH 05 cp O rH LO CO CO CM CM 17. rH 12. rH • C- 4 D-40 Plates D-5 thru 0-8 inclusive - Graphic determination of the weight of armor units in terms of wave height, side slopes and for sea water. These curves are graphic solutions of the new W.E.S, stability formula developed by Hudson^162) for determination of the side and end slopes for rubble structures and the weight of armor units on those slopes required to withstand various wave heights. W Wj. H- Kp (Sr - 1) cot a (46) where W = Weight of armor unit in primary cover layer, lbs. Wj. = Unit weight (saturated surface dry) of armor unit, lbs/ft3, H = Design wave height measured at the location of the proposed structure (see Section 4,1) Specific gravity of armor unit, relative to^the water in which structure is situated (S = ^) Unit weight of water, fresh water 62,4 lbs/ft^» sea water 64,0 Ibs/ft^ a = Angle of breakwater slope, measured from horizontal, degrees. Kp s Coefficient that varies primarily with the shape of the armor units, roughness of the surface, sharpness of edges and degree of interlocking. Sr = Ww = Table D-10 shows the provisional values for use in determining the armor unit weights. An example of the use of these curves is as follows: Given: A breakwater composed of 2 layers of rough quarrystone having a unit weight of 155 pounds per cubic foot with a side slope of 1 on 2, founded in 30 feet of sea water and exposed to 10-foot waves. The weight of the quarrystone is determined by using Plate D-6b and Table D-10, From Plate D-6b for 1 on 2 side slope and H = 10 feet, WlCp = 2,7 x 10^, and from Table D-10 for structure trunk, non-breaking waves, Kp = 3,5, Therefore, the weight of the armor unit is determined to be .SjlZ _= 7,720 poinids 3 5 or about 3,9 tons. Should this structure be founded *in fresh water rather than salt or sea water, the weight would then be 0,880 x 7,720 = 6,780 pounds or about 3,4 tons. May 1961 D-41 001 U| m6|3H 3A0M i S m N 8 « $ K Uj m6|3H 9^Dff^ May 1961 D-42 May 1961 D-43 WEIGHT OF ARMOR UNITS x K„ VS WAVE HEIGHT FOR VARIOUS SLOPE VALUES May 1961 D-44 WEIGHT OF ARMOR UNITS x VS WAVE HEIGHT FOR VARIOUS SLOPE VALUES May 1961 D-45 WEIGHT OF ARMOR UNITS x K^VS WAVE HEIGHT FOR VARIOUS SLOPE VALUES PLATE D-8 TABLE D-10 PROVISIONAL Kp VALUES FOR USE IN DETERMINING ARMOR UNIT WEIGHT NO DAMAGE CRITERIA Structure Trunk Structure Head Armor Units n* Breaking Wave^^^ Non-Breaking Wave^^^ Breaking Wave^®^ Non-Breaking Wave^^^ Smooth Quarrystone 2 2.1 2.6 2.0 2.4 tf »t >3 2.6 3.2 - - Rough Quarrystone 2 2.8 3.5 2.7 3.2 »• M >3 3.4 4.3 - - Modif ied Cube 2 6.0 7.5 - 5.0 Tetrapod 2 6.6 00 . OJ 5.0 6.5 Quadripod 2 6.6 8.3 5.0 6.5 Hexapod 2 7.2 9.0 5.0 7.0 Tribar 2 8.0 10.0 5.0 7.5 n* is the number of units comprising the thickness of the armor layer (a) Minor overtopping criteria (b) No overtopping criteria Moy 1961 D-46 TABLE D-11 VALUES OP SLOPE ANGLE (g) AND COT g FOR VARIOUS SLOPES Slope Slope Angle a Cot a (XA) (Y on X) 45° 1.0 1 on 1.0 42° 16* 1.1 1 on 1.5 39° 48« 1.2 1 on 1.2 38° 40 1.25 1 on 1.25 3 70 34* 1.3 1 on 1.3 35 O 32* 1.4 1 on 1.4 330 41* 1.5 1 on 1.5 32° 00* 1.6 1 on 1.6 29° 45* 1.75 1 on 1.75 26® 34* 2.0 1 on 2.0 23® 58* 2.25 1 on 2.25 21® 48* 2.5 1 on 2.5 19® 59* 2.75 1 on 2.75 18® 26* 3.0 1 on 3.0 17® 06* 3.25 1 on 3.25 15® 57* 3.5 1 on 3.5 14® 56* 3.75 1 on 3.75 14® 02* 4.0 1 on 4.0 13® 14* 4.25 1 on 4.25 12® 32* 4.5 1 on 4.5 11® 53* 4.75 1 on 4.75 11® 19* 5.0 1 on 5.0 10° 18* 5.5 1 on 5.5 9° 28* 6.0 1 on 6.0 8® 49* 6.5 1 on 6.5 8® 08* 7.0 1 on 7.0 7® 36* 7.5 1 on 7.5 7® 08* 8.0 1 on 8.0 6® 43* 8.5 1 on 8.5 6® 20* 9.0 1 on 9.0 6® 01* 9.5 1 on 9.5 5® 43* 10.0 1 on 10.0 0 tan 0 TABLE D-12 - VALUES OP tan^ (90 - 0^/2 tan^ 0 tan 0 C» tan 0 0 1.00 45 1,000 0.17 10 0.176 0.70 50 1.192 0.13 20 0.364 0.49 60 1.732 0.07 25 0.466 0.41 70 2.748 0.03 30 0.577 0.33 80 5.671 0.01 35 0.700 0.27 90 00 0 40 0.839 0.22 Moy 1961 D-47 2 Deep Water Wave Length, Lq, In Feet »2^Z — coh-(p lO^rocNj (Ti N iD in ro O in to C\J *aej u|‘Oh ‘-latDM daaa u| ighiaH 3 ad/v\ i: Moy 1961 D-48 Wave Period, T, In Seconds PLATE D-9 DETERMINATION OF WAVE HEIGHT AND DEPTH OF WATER AT POINT OF BREAKING. o Q liJ >- < -I Q. February 1957 D- 49 & a 3 3$ 8 8^ ti 8 3 q « N « « n M n s s : « e f« M N s a : a d 8 s a 8 d ^ s : s d — - S » ii d ■» d 8 : t * = « n ^ Z - u 3 B “ * B M >1 .) S S 8 S o d o o $ 3 8 8 2 O O ffl a ::888$as3 d^ddddddrC 3sa8S8s:s d^dddr^ddfi d«^dddf^d«^d Mq«^<>«0kA«g) dv^ddddddd «MQnr^r»«r<>o«r^ai o«>2r>M«-o*o«»5 aasssssssass d — dddd^ddfJdd ss^82;:ss;; :SS898Xa2 dedd*dd|^«( «^S8S88SS dedddd«^dd s8a8S8aaa •oddddddd S $$888888 do — •^d-*dd*i (O z u O CO = Q y 5 ' o Ei Q Q_ •” o lU ^ to < < I z o h* O u Vi UJ -I bJ O O) (\) I CJ I o PLATE D-l May 1961 D- 50 0 Ot 0 0 *- «D ri «n li. O ^ 5 i? 3 D UJ i: i o j o o z > 0 < o u o UJ u. o I I CVJ I o E o U- May 1961 D-51 PLATE D-12 2904 — r* » r* in N ni « tn 8 S t 1 3 Z o o o « •»> — o o ^ r* O D U) *** O ^ CD 9l in <0 d CM IN N N § * *> * 8 ' —* fj d (N N (N o o « r» O 2 ^ m 9 S Z r in lA i/i in 8 2 In m N IN IN o o - - S m m o m in in 6 d d d o m o o S * d d IN « m 10 oS:n:^933!!i;;8 dcjtfiiniNdddWdd n^iniAi®®h>e d—'WdfJKdivdi^d ®iRr*> i^?S?S»9R 8 9a®«iNr»iN®<*>® — 0^0 ^•'‘ddnlioddddd «Nipa*r^®*'Mr^NiN iiJ«^dm«>*ddiAiNoid S Q®r»®®9iniNn»w or'Oi^NiNW^'fNin ni — di'I--’dind<4d« d d IN »•' d ® ® d d ’ d d d ni ••' d d d d d •* nj •• d d d M O > in I- iO Hi I- ^ Z o S S Ss - 2 o - OJ (X V) UJ D 0. X (0 UJ ID (D D U tf) >- < o a: UJ ID ul ^ o ^ O ID o t Ul z C\J I o PLATE D-14 May 1961 D- 53 2904 o r* o 'T O CT> CO I CO I O 5 LU E o »- u. I _ z 3 O Q PLATE D-15 Moy 1961 D- 54 APPENDIX E MISCELLANEOUS DERIVATIONS E APPENDIX E MISCELLANEOUS DERIVATIONS 1, Refraction Dlagrani,q; derivation of method and design of template. a. Derivation . The parametric equations for an orthogonal given by- Arthur, Munk and Isaacs(3) are: dx „ n dy „ , n d ^ . o 3 C n — = C cos ^ = C sin /i^ ; — - sin - 5 ^ - cos C (E-l) These may be solved by simple separation of variables for a velocity field which is a function of y alone. (Here (B = 90° - CL where a = the angle between a tangent to a contour and a normal to an orthogonal, see Figure E-l), In particular, for a field which varies linearly with y, C = Cq (1 - ay), the solutions for x and y are X = a sin a, (cos a - cos On) y = -T—— (cos Oq- cos Gt) •' a sin CLq which are the paramet ric equations of a c ircle of radius ^ . sin^ (2^0) / 4 \ a sin (2 do) X = - ; y = and center at 1 a sin (2 do) The solution for y may be put in the form sin d = sin (do - A d) = (1 - Ac ) sin Co (E-2) (E-3) (E-4) (E-5) = p sin do, which is Snell's law. From these, exact values of Ad and X at any point in the field may be found. b. Template Design . Referring to Figure E-l, if from the point of inter¬ section P of the mid-contom* with an incoming orthogonal, a perpendicular is dropped to an arbitrary point R, then the line ^ perpendicular to the tangent to the mid-contour = sin a, x PR. (angle RPQ = a, ), If another line from R equal in length to Cp/C 2 x PR (where C]^ and C 2 are the velocities at contours 1 and 2 respectively) is drawn to intersect the tangent to the mid-contour, the following relationships hold: E-l sinO] X P2 sin (R S P) = --- = C /C sin a, (E- 6 ) C^/C^ X m but 1 “ ® 2’ f- by Snell's law: (R S P) = Cl £ (E-7) c. Template Construction . To construct a template on these principles, a line, the length of which represents RP in Figure E-1, is laid out. This length is entirely arbitrary; any length foiind convenient for use may be utilized. At one end of this line, a perpendicular is constructed. This perpendicular represents the orthogonal in Figure E-1, its intersection with the first line corresponds to point P, and the other end of the first line drawn corresponds to point R, The distance between points R and P is then divided into tenths, and when convenient, into even smaller intervals. These divisions are labelled, starting with zero at point R to 1,0 at point P, and continuing the designations on extended. Figure 17 of the text is one such template. The two perpendicular lines, one marked "orthogonal,” and the other, divided into tenths and himdreds marked '*Cp/C 2 ", are the basic template. For convenience in using the R/J method (See Figure 14), a protractor centered on R ("Turning Point") with radius RP, and a graph showing A a as a function of R/J and C 2 /CP (Note: not Cj^/G 2 ) have been plotted. ( E-2 2 Diffraction (E- 8 ) a. Waves Passing a Singflo Breakwater - The general equation for progressive irrotational waves of small amplitude may be written; AikC V, \ 7) = - cosh (kd) • e . F(x,y; where 17 = the surface elevation at time t, given by the real part of equation E -8 k = 2 7 r/L L = the wave length i = C = the wave velocity X X cosh (kd) = the maximum amplitude of wave motion. For waves travelling in the direction of the positive y axis with no barrier present, F(x,y) = e"^ With a single rigid barrier present, Putnam and Arthur(105) give as a solution ., . , F(x,y) = e o f(u^) / e ^ . g(u 2 ) where f(u^), and g(^ 2 ^ given by i ^4 J U-, -iTTv /2 / N 1 i % u^ and u^ being given by> P -ITTV -/2 dv dv (E-9) (E-10) (E-11) (E-12) ^1 =N Uo = \l 4(r / y) (E- 13 ) (E- 14 ) In Figure E-2 for both x and y positive (protected region) the signs of the upper limit of the integrals in equations E-11 and E-12 are negative. For x positive and y negative both limits are positive. For x negative and y positive the upper limit of the integral in E-11 is positive and that in E-12 is negative. As a simplified solution, Putnam and Arthur give F(x,y) = e"^^ f(u) (u = u^) (E-15) E-3 Using the simplified solution and comparing equations E-9 and E-15 it can be seen that the modiilus of f(u) (written | f (u)| ) determines the relative height of waves with a barrier present to those without a barrier. That is, the ratio diffracted wave height _ ^ |j. ' incident wave height i m Also from equation E-9 and E-15 diffracted waves differ in phase from undiffracted waves by the argument of f(u) (written arg f (u) ). i.e. equation E-15 niay be written _ ar g f(u )x F(x,y) r e ^ k <> | f (u) | (E-17) Both the modulus and argument of f(u) may be determined from tabulated values of the Fresnel integrals C(u) and S(u) since C(u) - iS(u) = f e dv ■'0 and it can be shown that for uc 0, f(-u) for u > 0 l-f(u) (E-18) (E-19) The modulus and argument of f(u) are plotted on Figure E-3 as a function of Uo Equation E-13 may be solved for X/L. X/L = ( + ) 0.707 u \|y/L / 0.125 u" i (E-20) E-4 A diffraction diagram consisting of lines of equal wave height reduction (K*) and wave crest advance positions may be drawn from computations similar to those shown in Table E-1 , Values of K* are chosen (including the maximum and minimum at the points of reversal of the curve lf(u)| = K’ vs, u), and are entered in Column -I. From Figure E-3 corresponding values of u are found and entered in Column 2. Columns 3 and 4 are computed as indicated. With coltimns 3 and 4> for every value of Y/L in the heading of columns 5 through 12 (these values represent distances in wave lengths leeward of the end of the breakwater) corresponding values of X/L are computed with equation E-20 and are entered in the table. Curves of constant K’ may be drawn from these X/L and Y/L values of coliamns 5 through 12. (See Figure 21 of text). Along these curves, since u is constant, arg f(u) will also be constant; which means that the lines of constant K' may be considered to be lines of constant phase lag. The amount of crest lag in percent of wave length along any of these lines is given by Crest lag = in which u is taken to correspond to each value of K’, and arg f(u) is taken from Figvire E-3, and entered in column 13. With arg f(u), the crest lag is computed from equation E-21 and entered in column 14. Since the wave crest lag is constant along any one line of K’, crest positions along these lines after diffraction may be plotted as on Figure 21 by marking points on them, from and normal to the undiffracted position of any wave crest, a distance equal to the calculated values of crest lag. Positive values of crest lag represent lag and negative values represent lead of the new position of wave crest. All linear dimensions on the graph. Figure 21, are divided by L the incident wave length. Note that Figure 21, E-5 Table E-l-~ Calculation of diffraction coefficlenta and crest lag for a single breakwater Q 4» s> •> u D 1-^ o 1-^ O n t> 0 iH > t= tu (4 ct 0 vT t)0 4 00 <£> ir\ ro ^ O ^ ^ KO O I—I • • • • • w iH O O O O K^ ir\ o o o o o o CM O ocpcp<^cp VO ir\ o^ o o ^ CM • • • • • rH ^ CM CM rH o O 00 VO 00 r^vo VO 00 VO CD r<^vo cTi CM ir\3- cm oo 3 ??? I I I I I Q ^ rOCTNCM CTV 0^i^• rH • • • • • rH J- CM iH O O O ^ CTN ^■00r-^00VOr^00CMK^ r^VO rH r4^ O viO rH ??? o r-l Jt lOv cr> CM cr> CTM^ 00 rH iH crv-0- r^MTiOO rH CM t4^J± I I I I I O VO VO 00 ITi iH iH 00 r<^ 00 CM O O O Cp Cp rH CM CM t4M ^ vO vO O ^ » CM CTV'O rH ITv ??? I I I CM K>r<^ I I I 00 CM O ^ irvCM V J- iH I— CO ■ CO CM rH o O O o itn to VO oocncrvcM loioioo CM-0-vOOOfH l^cor^o CpcpcpcpHrHcjIcMcp coj- 00 moo . O 00 CM HO CM CO tH cH I— o r^vo vo m CMCMCJvr^C^CMvOCM;d- cvj^ mc^como^vo 0000). Computations may be made by use of equation E-23 and Figure E-7, which show real and imaginary parts of f(-u). Writing f(-u) as f(-u) = s / iw (E-27) equation E-26 may be written as F(x,y) - e ^ 0 - x —b/2 (E-28) for X ^ b/2 s^ and w^ correspond to u^^ which is defined by F{x,y) = e ^ 1(“ 2 , "l = - — = L ' ' ^L' L and s^ and w^ correspond to u^ whcih is defined by 4(r--y) u^ = 4 z L (E-29) (E-30) If equation E-26 is written F(x,y) = e"^^ (S / iW) (E-3l) Where S and W represent the sums of the real and Imaginary parts respective¬ ly of equation E-28 comparison with equation E-9 shows that a diffraction coefficient K’ may be defined as | f(x.v)I for diffracted wave “ |F(x,y)| for incident wave which is eq\ial to K» = \ls^ / W^ |F(x,y)j for diffracted (E-32) wave (E-33) The flTnmint of crest lag in percent of wave length at any point is given by crest lag = _ tan~^ (^) (E-34) E-8 -e-ihy r fz- f, Equation (E-25) |[e-i''y-f,)t(e-i' being a tangent function, con¬ tains no indication of the position of a wave crest other than giving the amount of lag or lead (phase difference) of a diffracted wave crest over an undiffracted one. (Positive values of phase difference indicate a crest lag, and negative values, a crest lead). There is no way of telling from the solution of equation E-27 alone to which undiffracted wave crest this lag or lead applies. This may be determined however, through the construction of a graph of Y/L vSo complete phase Figure E-8 for various values of X/L, (’’Complete phase” indicates the actual distance in wave cycles of a wave crest from the gap.) A 45° line is first sketched in. The complete phases along the line for X/L = 0 will lie just below this 45° line and successive curves for X/L = 0.5, 1.0, 1.5 etc. will lie above and approximately parallel to each preceding curve. ,, „ , „,-lW, S Phase difference(PD) From equation E-34^ values of tan g and --*— are calculated. For each integral Y/L value, these PD/360° values are added to or subtracted from that value of integral phase which will bring the actual complete phase line to the desired position; slightly above the curve for the next smaller value of X/L. For example; with X/L =2,5 and Y/L = 2; PD/360° - /0,380 which is subtracted from 3, and for Y/L = 3; PD/360° = -0.386 which is added to 3 to give complete phase values of 2.62 and 3.39 respectively. Points for the wave pattern are computed by noting from the curves shown in Figure E-8 the values of Y/L at the points where the lines of con¬ stant X/L intersect the line of integral phase. These values are tabulated in Table E-3. Wave patterns now may be drawn as curves joining the points having the same integral phases. The patterns for the gap width of 2L are shown in Figure 23 of the text together with the contours of equal dif¬ fraction coefficients. E-12 Table E-2. - Calculation of diffraction coefficients and phases for 2 breala’aters; pap widbh 2 \iave lengths . ^ § E 0) <§ o H o a, w w ^ o rH Xh I U) S O) tJ -p ^ CM CO o tl I CM CO w f- I CM CM CO >» MIACnCMrHO O OOsO O OvOnOnOnOnOnOsCOOO «••••••••• HrHCM r^-cyiAvO CO Qs \C -Z! r-\-=T 04 C^CO-CfNO«» CMO-3C^OOQnOCKOOO OOOOOOOOrHiHM • •••••••••• ocpoooooooo + I + + + + ++ + + + vO C'-OvOCMO O-Cj-XAcnC*- f-1 o \A CO CO + I + + + + + + + + + Os O m 'O r*-\ O oo oo -g rH O \AvO vO c*- CO ooooooooooo + I+ + + + + + + + + ^CM<^iHt^COrHsOrHvg-g OO CM ri Os CO c. -» f— r- vc ^ • •••••••••• OiHHrHOOOOOOO S .-rCMOsOcocoooCN O -c CM 0^-g C?sco rH r^r^i/sCMOsC^so'*'- - • • • O rH rH Vf \-ij -CJ • •••••• o o o o o o o OCMOO O OCM-crXACOO CM o r'> -g rH Os OC sO sO O C. O CM CM CM CM j—t ri ri rH • •••••••••• ocooooooooo OsCMCM sOcosO-g-gOO QO _g Os O-g GO rH \A sGC^r^Of'-'LfN-g r»“>mcMCM • •••••••••• OrHHrHOOOOOOO ^-gMDoO OsOOsO^r^rH O i-i O -g -g -cf -g -g -g -g- -g O O O + I + oooooooo + + + + + + + + ; -3 CM rH O O J CJ CM CM CM CM CM CM CO^- O O 1-1 N ( *•••••••••• ooooooooooo I + I I I I I I I I I Vfs o < H mCMcc Q-g-gso cmvocmO CO<^rHOOO r-vOvOUiUslA «•••••••••• OHHrHOOOOOOO + + + + + + + + + + + \A lAUS r^CMOO^-g-gr^CM rH C O OOr-lCMCMCMCMCMCMCMCM • •••••••••• OOOOOOOOOOO I + I I I I I I I I I OOMO CsOcO P-\c-- OsCM-gUS OriOOOrHf—irHCMCMCM • •••••••••• OOOOOOOOOOO + I I++ + + + + + + ^ r^sOcc-gcoOOs vgZgvsosCKCMcc-gcM os O'-C o^^O-g^■^r^CM cmcm i— i • •••••••••• ^iHOOOOOOOOO •USVfN CO m O' CM -g US OOrjOOOOOOO I l'^+ + + + + + + ^ nSNOCD-gcoOCN sO3XA0sOsCMCO-gCM 0\ C>'0 0'^C-g^r^<▼^CM CM CMr-l • •••••••••• ^rHOOOOOOCOO O rH CM rs-gus'O r-oo OsO CMvOcOsO-gPSr^CMrHrHO O OsOsOsOv 0s0s0\0\0s0s O O rHCM PS-gUSMD r-QO 0\ -g_gos^-gO\C^O O rH CMvOHOCCOPS-grH-gNO CM rscM-gus t^r-oo c?sOsQs OOOOOOOOOOO • •••••••••• ooooooooooo I + + + ++ + + + H- + OVrH r^USC^ rrssOir\0OaD'>O *••••••«••• OUSMDO CMr*s^ rH rHCMCMCMPSPSPSPS I+ + + ++ + + + + + OsCM-g c>sCO PSOco PSO O PS PS PS US o Qo -g C'- On r-l CM iH CM PS-gXAUsMDsCsO OOOOOOOOOOO I + + + + + + + + + + OCMrHCMCOCMOCM rH CMCO OsOscO COCO r-MjsCsO • ••••••«••• rHOOOOOOOOOO ps-g-gusCM rHooc—-g^-Os -gc--CM^c---r-_grH-gOvO -gsOcoco c^vCsOXfs-g_gps rHOOOOOOOOOO f^-g-gpSMD PSO PSO^ Os Cj PSrHUSCKCMPSPSPSCMrH OOOOOr-ii—irHrHrHrH • ••••••••«• OOOOOOOOOOO sOOOCMCMOOCC-g-grHO rH-grHOsr--grHCOrHCOUS -gsOco C^SCUSUSPSPSCM CM rHOOOOOOOOOO US US US US Us sOOO CM PSrHUS^sOsOUS-g rH rH (-1 CM p■^ PS PS PS PS P'> PS • ••••«••••• O O O O O O I + + + + + p O O O o + + + + + rH ■ X OsOOOsCM-gCMCMvOPSO rHCC OnCOCO r-f*-sOUSUSUS • •••••••••• HOOOCCOOOOO + + + + + + + + + + + US lA US USUS sOso c us^ o CM ps-g-g-g OOr-iOrHCMCMCMCMCMCM • ••••••••♦« OOOOOOOOOOO + + + I I I I I I I I C--rHCMC^PSr^PSrH^COO O rH rH rH r—i O O O O O' rH • •••••••••• OOOOOOOOOOO I + I I I I I + + + + sO iH -g CO H CM^ccUSPSUsC3-g O CMO_g Oco^-'OUsUS-g • •••••••••• sCPSCMrHrHOOOOOO US O-gCMCO r^US-gPSCMrHO rH CM CM rH rH rH r*i rH i—i rH rH OOOOOOOOOOO H- I I I I I I I I I I CM ONCMOOrHPSUSr-OOOsO (HCcmcm •^np’^pspsps p■^-g • ••••»••••• OOOOOOOOOOO I + + + + + + + + + + CMoo-g-gO-gCM-gsOCM r^-.gvo CM OoD r^vOXAUs O —g CM rH rH rS O C C O O • •••••••••• CMOOOOOOOOOO O rHCM PS-gUSMDr-OO OsO cocM_gcMO oor^sous-g OsOOOO OsOsCNOsOn • •••• ••••• O CM PS-rjUSUSMD r^QO CN cc PS O rH C^-g rH CM -3 ^ 0\ rH CMCM_grHO CM PS3USUS O OOOOO OOOOO • ••••• ••••• OOOOOOOOOOO + +I I I+ + + + 4- + USPSPSCO CMC^ COsOcO PSPS • •«••• •«••• satX>r-»-gsOrH C^rHUSOsrH rH rH rH rH CM + +I I !++ + + + + -g PSCD-g OsOvO'OPSrH Os _g-gcMsO OPS PSOcousco rH iHrHCMriO rHCMCMPSP-S • ••••• «•••• OOOOOOOOOOO + + I f !++ + + + ♦ -:a O r-OsXA*gr- r--gcM Oco USP’SU^^O sOM5sOsOUS • •••• ••••• oooooooooo r-o USrHUSOs_g^ OsUSsO OnCM UsO rH-g3HC0'«0 PS rH PS rH PS-H-g -3-3 P> Pi P' • » •••• ••••• OO OOOC OOOOO PSSO PSOUsOcOC^OsO^ OO OCMOO OrHCM-g3 OO OO OO OOOOO • • •••• ••••• OO OOOO OOOOO -g-g CMrHOOSsOC^OUSCM gsrH USCOrH-g PSOssOCMCK rHPSr-»CM-g-g-gPSPSPSCM 0 • •••• ••••• OO OOOO OOOOO us<5)Us-gr-CM qspsc^OiH OO OHOO OrHrHCMCM • •••••••••• CO ooooooooo + + I I I+ + + + + T -gsC Osps-gr^sOPSOc^_g -gus PSUSsOOsOsOsOUSUS • • •••« ••••• ^O OOOO OOOOO + + + + + ++ + + + + UsmXA-gr^CM OsPSr*“OrH O O O rH O C' O rH r—i CM CM «• •••••#••• OO OOOOOOOOO I » + + + I I I I I I soso rH ps-gc^'O PSO r--g O O rH O iH rH rH rH rH O O • •••••••••• OO ooooooooo + 1+11111111 _ USUS CM US rH CM CN-g O CM OD r- Os O Osps_g«ouspsi-l C'CC r- • •••••••••• CC-gpSCMrHrHrHrHOOO OOOOOOOOOOO Us US US US US U\ U' US U\ US US • •••••••••• ooooooooooo OOOOOOOOOOO O rH CM PS-g USSO OO Os O E-13 Table E-2 (cont'd) 0 ) 0 ) i. *2 £ 0) c0 O rA S'. o Q. a r-r I m CO W ^ 0) S O) ITJ -p fo- CO > CM > I iH 5 CO CO I CM < s lA S rAf^cArnOaoiArACMO CMOrHriiHOOOOO • •«•«•••••• OrHCM P^^-LTWO f^GO CNO \r\0 ^ 0\ O Cf\ 0\t^ OsCM C^CM mOCO-ZjCM rH CMOrHrHr^OOOO • ••••••••• oooooooooo -*'111111111 OfAsO r*-Or'\rHaOfACMO *••••«••••• CM ^ vOvOaor-t^r-0-:jr-4 ^CO CM-^-^jr^CMfHrH , y I I I I I I I I I + CM CO O O \0 • - \ o CA rH M CM CACO _ VrHf'-HCMCOr-rH r-\A'U\OrH'0'LAr^rHOO wwwmmwwwmmm OvOOr-irHOOOOOp I I t I I I I I I I ^ & o~\OsfACNCO CAnCvOXAUN r^CM CM P'N.rjXA'LfN'LA'UMA OOOOOOOOOOO r-iHVN'LAr-XA'LACOXArHMD vOcO-^-^f-lOOOO OnCM-iJCM OOCOXAU\CMC^OrHOO OrHOOrHCMCMfAr-^r^NfA • •••••••••• OOOOOOOOOOO 'LfN^OO\ONrHH'00\OsOiH CM'LA'OcO.a^r-COOrHO OOrHCMcCCOlACMrHOO 0r-i00C0000C:0 • •••••••••• OOOOOOOOOOO CMXAnOMDvO-;:? OOnVAIAUN -aCMC^IAC^-^COOCMCMCM O O'O CM'O-njCMcoOOO OOOOOp-lCMCMrAr^\r^\ • *(*•••••••• OOOOOOOOOOO ■LACM rnr-0\CN-:::r^“0^rH OfArHrHCMCMCMrHiHO^' • «•••••••«• OOOOOOOOOOO **■111111111 + •g\ggvp(CM_::lO-^AJ^rH OrHrHrHOOOrH ???? OOvOOO'O^Ot^'nrHCN OgCJiHi-IrHO • •••••••••• i-l iH CM r^_3 'LTV vO OO 0\ O OsO rH r-NO OnTh OsCCXA-^OnnC mOcQ 0-:jCMrHCMCMrHrHrHHO • ••«••••••• OOOOOOOOOOO I + I I I I I I I I I C^O ir\_:jn^O rH C^CMvOO • •• •••••••• r^CNvDsOCACOrH CTsr-COrH O • On CO r-- \A-c? O ca I H rH I I I I I I I I + ' NO O CO >OOcoC?N CMiHcoOnO nOnO r'>CMir\0 ONr*-0 r^so • •••••«•••• OOmCMt>tX>CMHHOO I I + I rH CM I I I I I + I UNO rHOO OnO C^vO •LTN^-CO f-r- C^t^iH'LAt^'O O rHCMiHrH CMr*\-:j-::r-cr-z, OOOOOOOOOOO P^rHCNrHrHCOC^-CfqD-CJCO O rryC'-r^rr, ["-.rr, c>*-0CMrH OOOOC OrHrHCMCMCM • •••••••••• OOOOOOOOOC'O iHcorAvOrHcocAOCMC^-ao O O C'-CMCAC—CM rr, I—tCOUN OOOOO OrHf-lr^OO • •••• •••••• OOOOOOOOOOO CMCA^IAOO-:! nOC^O O CMOOO OrH-:JCKrr»NO OOOOO OO-crOi—ir-i • •••••••••* OOOOOOOOOOO CM LA lALA frvONC^NOC^COlA'O cacTn-J? II OOCM^-Hr^CMrAf^^cr^CMCM Lr^^O^O'OCO r-CALTstALA O CM rH CJ rA-=7LAXA\ALA OOOOOOOOOOO I + + + + + + + + + + OOOOQOOpO + + + I + + + + I \A \A-J:r-^CM'OLACMNO CANOLA OOOrHOOrHrHrHrHrH 1^00000 I I I I I I — OCJrHCM r--vO O fACA^OcA Or'-CCNCcOCAOt'^LArr, CM *•••••••••• Oso -c;rACMCMCMrHrHrHH ■'-A _ 0_=jOcoc^\a-3cacmhO i—iCMCMrHrHrHrHrHrHrHrH OOOOOOOOOOO + I I I I I I I I I I O CO CM OnCM CMCMCAlAr-QO CNO rHOCMOrArArArAn-> CA-^ • •••••••••• OOOOOOOOOOO I + + + + + + + + + + CMoO^- 30 -CJCM^nOCM C^^^CMOOpr-'O'LAlA 0-=)CMr-irHrH00000 • •••••••«•• CMOOOOOOOOOO O O O O CO I + I I I O O O O c o I I I I I I LAlAGOC^rH rHCMrH rHr-O -ZJrHOOO OrHCM P^>fA~:j • •••••••••• OOOOOOOOOOO + I 1+ I + ++ + + + r-^co t^-crcM-ij CM on- o o o o ^ O O O fH rH rH O O O rHCM rA-aiA'O r-CO CNO OOOOOOOOOOO I I + I I + + + + + + -^rHrHvOOOCMLACM OncaIA O O O C O rH O O O rn m • •••••••••• OOOOOOQOOOO + I I I + + + I I I I iACM£^ CMrANOC^LAvO ONO_^ONOCACO-=frHaNC*^ *••«••••••• CMCONO-rj-JUCACMCMCMrHrH LA lALA r-CM 0 O-C)_=j-=JfACM rH O O OOrHCMWCMCMCMCMCMCM *•••••••••• OOOOOOOOOOO I + I I t I I I I I I ONH^CAr-ONCM-aLA rHOOOrHrHi-HCMCMCM • •••••••••• ooopooooooo + I | + + + + + + 4' + ^ CAnOCO-^COOCn nO- 3-3 ONO\CMcO-CfCM On OnO OnvC^CACACMCM CM m • •••••••••• -3rHOOOOOOOOO OrHCMCA.JZTLA^r^COONO rH\DCMONrHf'*“vOrHvOAJO sOO^fACACArAfACM^AJ «•«••••• ••• iH C\l CM CA-^LAsO C-^CO OnO -- OnUAOOOO OMDIaOLACMOn CAO rArAPAACArACMCMf-H • ••«•••• ••• OOOOOOOO OOO + I + I I I 1 I III CM^-G0OC0OC0Cr^N0^-'O • ••••••• ••• OLAOONOCMr*-rH rHCAO -ZJ PAfArHCACMrHOC^C^ fH rH rH rH rH rH rH + I + I I I I I III rAO-iJC^CArHCJNON OiA OOrlONCONOrHCMLAOvAC^ • •••••••ea* COOOCMrHfHCMlAtACM I I I + + + + + CA I I VA^ fAfACMIAOlA rH CM ooO rArHHtACMOlACAH OrHCMCMrHrHCMr^tA • ••••••• ••• OOOOOOOO OOO CMcHlACMNOrHOrALArHOO C^O-aLACMCOCM CACM fAC*- O iHLA-L'rHpHLAON CMLAnO C> C) 0> r^ rH rH • ••••••• ••• ^'OOOOOOO OOO CArHsO O -:JCMLACMCM vOOLAnOiH CMrHCMCOCO OC CMONrHiHCA® CM-Cf-CT O O O fH O O O O rH rH rH • «•••••• ••• OOOOOOOO OOO On ONvO^rH'OrHrHONNO O CCLArHq00NCMO-30N O rH CM CM O O rl rH O O rH OOOOOOOO OOO • ••••••• ••• OOOOOOOO OOO ■LA CM II lA lA \J'\ lA \A CMrHvC-^OOajOO^AOaoO OOrlHr-li-irHCM (ArAiA • ••••••• ••• OCpOOOOO OOO + I + f I I I I III fAO C*-NO~iaN-:jrH rHC*--*H- OrHrHrHPOrHfH OOrH • ••••••• ••• OOOOOOOO OOO I + I I I I I I I ^ ^ NO On-:? C -CJ-XA rH OOOOOrHOOf—t rirH • ••••••*• •• ? OOOOOOOO oo + I + I I I + + + + —3 rH LA i—• On CM rH CM r*- rH -Cl OOOOOOmr-iO OO OOOOOOOOO OO I + + I I + + + + + I COCM-c:XACMOOOrACMCO O VArH -C 3 CM^ r-rAONNCfA • ••••«••* •• ■''i'OCONO'XA-CjfArACM CMCM LA LA \A LALA NO ^ O LA-:? O CM -3 -3 OOrHOrHCMCMCMCM CMCM • •••••••• •• OOOOOOOOO OO + + + I I I I I I II C-'-rHCMA-ANA-CArH^ C rH rH iH rH O O O O OOOOOOO CO I I I + + + + O p O I + I NO iH CO rH CMOnCOLAPALaO-:^ OCM 0-?:0cGC-“^LALr,-3 • •••••••• •• sOrACMrHrHCOOO OO O rH CM CA-ZJ LA vO r- GO On O E-14 Table F-2 (cont'd) 0 0 kt 0 H a x: D a w/s 0 rH 0 1 fA CO w \ o s d) fH S-i I tci § 0 xs +> '+ CVJ CO f- s Cvj 10 C\J 3 < X UN 0\Cp CO -3 O H iH sO rH rH AJ rH r^ w w m w w w w w w w oooooooooo I + t I I I I I I I -C?^0-:jlA OrH C>vC_^fACO OO'-'OOOOrHOOO • ••• •*•«••• OOOOOOOOOOO I I + I + + I I I + + - 4 ; vOvOCMO-^UrHCMON OOOOOOO CrHrHO • •••••••••• OOOO OOOOOOO + ♦ I+ + I I + + + + ^ ^^CMrH’lAXAOOO'NOO O »AOvOvOvOCOCM A-CAO *••• ••••••• ^ CM OtO vO\.A-Cl-3 rACArA \A O O O CM rH rH Al CM OOOOOOOOOOO I I + + + I I I I I I vOvO rH N-^^tCNM^_3_3^QNvO OO-::5 OCM-=/H^n0O cA OOOOrAOOOOOO OOOOOOOOOOO • •••••••••• OOOOOOOOOOO -CIA rHiH XACAnOiHXA O CM O vC O CM-^XA-CfCM OOrH rAOrHOOCM-:J-Np OOOOOOOOOOO • •• ••• •••• • 000 ooc 00000 XA rH CMA“ CMCDA^-A*CO0OrAv0 OOOOi-HOOOOOO • •• ••• •••• • OOOOOOOOOOO l+l I+++ + + + I CM XA O On rH o VTv O Q H rH O fH O A---:? rH XA O rH CM OOOOOOOOOOO I + I + + I I I I I I XA CA'O CM rH >0 A- nA OOOOOOO -Xf O o XA O iH rH O o o + t 000 + I + OOOO + 111 XA CA iHO r^XAXA A> O O O O o o o 00000 I + I + + fAXA O O O H 00000 I I I + + OCAOOOO OvCMXAvO O-C*A“V0O0\O AlA-CMCC • •••• on^-:rrH OnCOvOvO XA-^-:3^fA \AQ:»ONrHCM^O-CtCMA-H OOOOiHrHrHOCOiH • •• *••••••• OOOOOOOOOOO + + I I + + I I I I I XA XA__ - - , XA O O fH O XA-Xt^ CMvCXACMvC --- ^ -vO XA OOOOOOOOOOO I + I + + I I I II I OOfHCM A-OTJ O CACAVO CA OA-covOcorAOr ‘ • •••••• O NO -JH rA CM CM CM A- XA OA CM O rHCM fA-;jXANO C^OOO'. O XAiHCMOvOONOrHCM XAnO -? CM^CC CAOcC A-A-A-A-nO • •••••••••• rArACA_:^»A\ANO A-ooOnO A-0\OvOHOnO H-rt XaOnO OvA--ctCOGOXAONO CM^rHCAOrHCMCMAirArA OOOOOQOOOOCp + l+l I + + + + + + OcaOiHA-CMO CAOCMoo • •••••••••• OnO iH O cm rA-5 rH O <5> O ON-:? ^ -rtrHXAOOONOcA iH fH iH rH iH rH + 1+ I I + + + ++ + NO CM NO Np rA rH CM nO 00 00 XA AJ O fH • •••••• O iH O O H-CrXA + + + I + I I NO XA H CA H I I XA NO CA ^-:j CM CA iH CM XA O fAOXAr-^opXA co O O fH fH fH fH O O O iH rH • •••••••••• OOOOOOOOOOO XA CM O rA^-:J-3^CMcc^ rHCM O OxHO-JjOOvOCMoOO-:? OOfHCMrACMOOOCMrA OOOOOOOOOOO • •••••••••• OOOOOOOOOOO XA CM o OOOOOOOOOOO iH XA iHnO no nO CM nO * ■ ^ “ )QO.._. _ _ '•OOOOOOOOOO H O rH O O CM » On Os Q H H OnXA-^ O H H Q CM --- NO ^ u > —j ' I—I r-% -~-i CM-;t_^OOOQ S HCMOOO^QH 00000000 «•••••• •• OOOOOOO O o o -=T H < X «•••••••••• OOOOOOOOOOO I I + I I + + + + + + XA gv’gi ^ m gN w fH -=y CM O O O O H 000 I + D p O I + + 000 I I o o I I XA fAXA CM H ^ fAXA O 00 O O O ■ 00000099 I +7TTT?‘T7 O O H • • O o ^W^'pArCA^C:l o o ? o b o' o t>' S S' o oooooooooo + I + I + + I I I I UN-rtOOoo-cfHPOooCM OcoirvcoNOCMCJ-St^i-tr- • •••••••••• CnO CAfH CNCO A-nOxaXA- 3 CM fH rH rH -rt A- pN» A*-j O C O O O O CM CM s^ CD O i D^OCD^OO^O : P ^ O O 'OoDCMUnCM Ovr^VA OOrHOOOf-li-l oooopopoooo + I I I + + + I I I I •LTV CM P- CM P^vO r-VTivO 0\C_3 0-. Of’-NCP^i-'O'P' • •••••••••• CMOO'O^-aPDCMCMCMHrH Ot-)CMfn_3Vrv'0p-co0\O E-15 table E -3. - PLOTTING VALUES OF v/L and x/L FOR DIFFRACTION DIAGRAM FOR m w EH O CN2 II X EH O Si W EH < P»; S CQ O to CO o ro| to 02 O 02 to O • (D CO cS "p, Ch o CQ o o cH to O 02 02 CO O C\2 lOi \0 ro O H c\2 02 02 02 • • • o • • • 02 CO CO vO O' to O tOvOOtOvOHcr\cr\ H o^cocoro-<}-'<}'^ rH02'■ to CT^ io002tocot)002v00 ONiOvO'O^^ O-IO-tO 002co-'tc0v00't0cr' C^202'tOO to 02 to O O C\2 CO to^ O \0 otooooo ooo • •••••• ••• Oi—I 02 ro~ oooooooo O H02ro-'^covDC'-tOO rovDtotooooqg OOOOHrHHHiH O I—I02ro^cr\v0!>t00 OiH02co-v}'COv0t>t0og E-16 * U S. GOVERNMENT PRINTING OFFICE ; 1961 0 — 60601 6 I n