'\-5\ F4- $B SE^ SED XLbc TSXnivctBit^ of Cbfcago FOUNDBD BY JOHN D. ROCKEFELLER DEVELOPMENT OF THE PROFILE OF EQUILIBRIUM OF THE SUBAQUEOUS SHORE TERRACE A DISSERTATION SUBMITTED TO THE FACULTIES OF THE GRADUA'J SCHOOLS OF ARTS, LITERATURE, AND SCIENCE, IN CANDIDACY FOR THE DEGREE OF DOCTOR O?" PHILOSOPHY (department of geology) BY N. M. FENNEMAN PRINTED BY Zbc 'dntpetdits ot Cbicaao t>tes9 CHICAGO Ft Digitized by the Internet Archive in 2008 with funding from Microsoft Corporation http://www.archive.org/details/developmentofproOOfennrich DEVELOPMENT OF THE PROFILE OF EQUILIBRIUM OF THE SUBAQUEOUS SHORE TERRACE. The profile of a shore as seen at any one time is a compro- mise between two forms. One of these is the form which it possessed when the water assumed its present level ; from this form it is continually departing. The other is the form which the water is striving to give to it ; toward this form it is continu- ally tending. There is a profile of equilibrium which the water would ultimately impart, if allowed to carry its work to comple- tion. The continual change of shore line and the supply of new drift are everchanging conditions with which no fixed form can be in equilibrium. There are, however, certain adjustments of current, slope and load which, when once attained, are maintained with some constancy. The form involved in these adjustments is commonly known as the profile of equilibrium. When this pro- file has once been assumed the entire form may slowly shift its position toward or from the land, but its slope will change little or not at all. It may be compared to a stream channel which has reached grade but not base level. The force which the water exerts is derived ultimately from the wind. The immediate agencies in the work are waves and currents. It will be convenient to consider these first as acting independently of the wind which caused them, and second, as acting under its continuous influence. It is also desirable to 13331H 2 N. M. FENNEMAN consider waves first in their free forms, while meeting no resist- ance and hence doing no external work. This condition is found in deep water. The various ways in which the bottom or shore may offer resistance and be subject to work may then be dis- cussed. WAVES IN WATER OF INFINITE DEPTH . When wave agitation does not reach the bottom of a body of water it is customary to speak of the depth as infinite, because the wave is not influenced by the existence of a bottom. PURE OSCILLATION. Orbits. — In simple oscillatory waves each water particle moves in a circular and closed orbit. The water body itself, therefore, has no onward motion. These orbits diminish rapidly with depth, but so long as agitation does not reach the bottom, the orbits are circles at all depths.* The particles on the crest are moving in the direction in which the wave is traveling and particles in the trough are moving with the same velocity in the opposite direction. Difierential movement. — -On a line in the direction of the wave movement (hence crossing the waves at right angles) each par- ticle is subject to a gliding between its neighbors. The amount of this gliding is of molecular dimensions, hence not infinitely small. It will be spoken of here as the differential movement of particles. For diagrammatic purposes it is convenient to consider this differential as a considerable arc of the orbit, hence particles are chosen which are removed from one another by a consider- able fraction of the length of the diameter. General form of wave. — In a series of particles moving in equal orbits each particle is more advanced in its orbit (has a more advanced phase) than the one in front of it. If a series *This principle was clearly elucidated by Gerstner in 1804. This and other fun- damental principles of oscillatory wave motion are clearly set forth in the report of the brothers Weber on their experiments conducted in the early part of the last century. See " Wellenlehre, Ernst Heinrich und Wilhelm Weber, Leipzic, 1825. This report also summarizes Gerstner's work and all other previous studies on waves from the time of Newton to 1820. PROFILE OF THE SUBAQUEOUS SHORE TERRACE 3 of orbits be drawn and the positions of the several particles be connected by a curved line, that line will show the wave form (Figs. I to 6). The curve is a trochoid/ It may be produced by rolling a circle on the under side of a straight horizontal line. riff t Via- 2. fia 3 Fie 6 ^ First recognized by Gerstner, Iheorie der Wellen, Prague, 1804. See p. 343 of reprint in Weber's Wellenlehre. Discussed mathematically by W. J. M. Rankine, " On the Exact Form and Motion of Waves at and near the Surface of Deep Water," Philosophical Transactions, 1863, page 127. 4 N. M. FENNEMAN The path generated by any point within the circle is a trochoid. This line will be sharply curved or broadly curved, approaching straightness, according as the point which generates it is chosen near the circumference or near the center. The distance from the center to the generating point is called the tracing arm. When the point is at the circumference — that is, when the trac- ing arm equals the radius of the rolling circle — the curve is cusped at the top and is the common cycloid (see Fig. 8). This is the steepest and shortest form which a true wave can have. If the tracing arm be longer that the radius of the moving circle the curve is looped instead of cusped (^Fig. 4). The failure of the water surface to assume these looped forms results in breakes. Steepness dependent on differential movement. — If the same series of particles in their orbits be represented in several diagrams, assuming for each diagram a different amount of differential movement, the wave will be found to be long when the differen- tial movement is small, and short when the differential movement is large (compare Figs, i and 2). If the size of the orbits be increased while the distance between the particles remains the same, and at the same time the differential movement continues to be a certain arc of the orbit, the wave-length remains the same, but its height and steepness increase (compare Figs, i and 3). If the size of the orbit be increased and the differential movement remain the same in absolute amount, instead of the same in arc, the shape of the wave will be preserved and its dimensions increased with the dimensions of the orbit (compare Figs. 2 and 3). If the differential movement exceeds a certain limit the curve will loop (see Fig. 4). This condition corre- sponds to that of breaking waves as noted above. Movement of particles below the surface. — If a series of equidis- tant particles be considered which lie in a vertical line in still water, the movement of the topmost or surface particle is repre- sented by any one of the orbits considered above. That of the second one is similar in every way except in size of orbit and hence in velocity. Its orbit is smaller and described in the same PROFILE OF THE SUBAQUEOUS SHORE TERRACE 5 time. The two particles have always the same phase and hence their movements are parallel at a given instant. The same is true of the third particle and all below it, the orbits decreasing in a descending geometrical progression (Fig. 7).' This fact is to be taken with one above stated, namely, that if orbits be decreased while the angular differential movement remains constant, the sharpness of the trochoid curve is reduced. It results from these properties that in a breaker where the curve of the surface would intersect itself, and is therefore impossible, the trochoids below the surface would show less of looping until a level is reached where normal wave motion is going on (compare Figs. 4, 5 and 6). Lines of like phase. — If the orbits of a vertical series of par- ticles be represented in diagram (see Fig. 7) and the correspond- ing points on the circles be connected with lines, then the line connecting the highest points and that connecting the lowest points of the several orbits are seen to be straight and vertical. The remaining lines are curved and inclined. In Fig. 8 these lines of like phase are shown in the positions where they occur in the wave. The particles ranged along any one of these lines would be in a vertical line if the water were at rest, just as all particles on one of the trochoid curves would lie in a horizontal line.* Consequences of the trochoidal form and of decreasing orbits below. — If a horizontal plane be passed midway between the level of the crests and that of the troughs it will pass through the centers of the orbits described by the surface particles. All the water at the surface above this plane will then have a for- ward component, and all the water at the surface below this plane will have a backward component. An inspection of the dia- grams will show that the crests are steeper and shorter than ^Rankine, loc. cit. p. 131. Russell, " Report on Waves made to the meeting of the British Association 1842-3," reprinted in The Wave of Translation, London, 1885, gives formulae adapted from Gerstner for the rate of orbitical diminution with depth. »RANKINE,/<7f. «V., p. 129. N, M, FENNEMAN ^N. \ \ 1 1 1 \ X \ \ X 1 \ \ \ » , Vv \ * « \' \ • \ \ \ \ « \ /•^ /ii \ V ^vU^ -t? 6 ^ ""^ 1 1 J 1 ' 1 ' /lit* / / 1 1 ' y / ' . ' y^ »' f I ' ^y / : .' < >v \ 1 1 >V \ ' 1 1 ^V. ^ * 1 ^V. ^ \ ' vvjx \ \ \ ^*^^"'~~~-^— — Ji^ \'n \ \ \ ^ 'n ' 1 . i \ 1 ' 1 _ a. />'; ; , /,' .' j_-^ --| / / >- 1 1 //■''' / ' /oil • - f ' L- --' T~ 1 / l---^- — * ' A*''''''^' ' ' _i yv--'--'-'r' y^-/ ' ' ' 1 -+- — (U -a O p- o u o PROFILE OF THE SUBAQUEOUS SHORE TERRACE 7 the troughs. This contrast increases as the wave shortens (com- pare Figs. I, 2, and 5). The crests have not. a sufficient volume of water to fill the troughs, and hence the level of the water at rest is lower than the level of the centers of the orbits which the surface particles describe. The lifting of the mean position of particles above their normal level gives a store of potential energy in the wave in addition to the kinetic energy of the motion of the water. It may be shown that this energy of position is exactly equal in amount to the energy of the water's motion.' This lifting and this store of potential energy are at a maximum when the wave has its greatest steepness (when it has the cycloid form). The surface layer is thus divided into strips, in one-half of which the water is moving forward, while in the other half it is moving backward at the same rate. The peculiarity of the case lies in the fact that the backward moving strips are broader than those moving forward. Fig. 8 shows that the same is true in less degree of layers below the surface. Nevertheless, the amounts of water moving in the two directions are equal because of the greater thickness of the layers in the forward moving strips. The contrast of thickness and also of breadth disappears with depth. MOVEMENTS DURING WIND. Effect on size and form. — The immediate effect of wind in the direction of wave movement is to accelerate the movement of the particles on the crest. It also retards the backward motion of those in the trough, but this effect is smaller because these particles are largely protected from the wind. The result is (i) an increase in the size of the orbits; (2) an increase in the differential movement of the particles at the surface; (3) more rapid traveling of crest than trough, hence greater steepness in front. The first would result in increasing both height and length of wave in the same proportion. The second results in greater steepness, that is, a shortening in proportion to their height. The increased differential movement is accompanied by ^Ibid., p. 132. 8 N. M. FENNEMAN increased friction which comes at length to consume all the energy derived from the wind which cannot then further increase the height of the waves. The opposite effects are seen when the wind has ceased. Friction gradually diminishes the differential movement of particles and the size of the orbits. Waves then become lower and at the same time longer in proportion to their height. Periodical large waves, — The change of wave-length must be propagated downward gradually. If such propagation were immediate, the wave-length at the surface would always be equal to that below. Not being immediate, there may be at times considerable differences in length. The periodical large waves always seen in a storm, may result from composition of lower and upper waves having different periods, as well as by composition of surface waves of different systems. WAVES IN WATER OF FINITE DEPTH. Wave base. — The extent of orbital movement decreases in geometrical progression with depth. A point is therefore reached where the force is too small to overcome the viscosity of the water. Before this point is reached and at comparatively small depths the movement is so slight that it cannot affect the small- est solid particles resting on the bottom. This level, below which the largest waves are inoperative, has been called wave- base. Its depth for any given lake or part of the ocean is a func- tion of the height and length of the largest waves. Behavior of water above wave-base in pure oscillation. — Before considering the action of water on a bottom which lies above wave-base it will be convenient to examine its behavior at any horizontal plane passed through a system of waves. Referring to Fig. 9, let AB be such an ideal plane. Being above wave- base it is in the region where the "planes of continuity" (planes including always the same particles which are in a horizontal plane when at rest) are in trochoid curves. The lines of like phase are inclined toward the crests ; hence the layer of water included between two planes of continuity is not onl}) thinner PROFILE OF THE SUBAQUEOUS SHORE TERRACE 9 but broader under the troughs than under the crests. In any one such layer the water is moving forward under the crests (point D, Fig. 9) at the same velocity with which it moves backward under the troughs (point E of same layer) . Of two adjacent layers the lower one is composed of slower-moving water. The line AB, drawn in a horizontal plane, traverses higher layers of water under troughs (at point C) and lower layers under crests (point D). Therefore the backward moving water along this plane has a more rapid motion than that moving forward. The area covered by it on the horizontal plane is also more than that covered by the forward moving water. This excess of backward movement below is the necessary correlative of the excess of forward movement above, for above the plane traversing the centers of the topmost orbits the movement on all planes is forward. The same when waves are wind-driven. — If now the water be conceived to be driven by a wind, the current movement pro- duced at any given depth must be added to the forward move- ment in the corresponding strips which lie below the crests, and subtracted from the backward movement in those under the troughs. The forward and backward velocities in any one layer between two trochoidal planes are now no longer equal. When a certain rate of current is reached, the forward move- ment in the lower layer traversed by the horizontal plane under the crests (point D, Fig. 9) will equal the backward movement in the upper layer which the plane traverses under the troughs (point C). A certain force of wind will therefore cause a bal- ance of to-and-fro movements at a horizontal plane below the surface. Any greater force will cause an excess of forward motion. Film representing surface of continuity, — If in one of the sur- faces of continuity in a system of waves of pure oscillation, a film could be introduced which is perfectly flexible and friction- less, this film would show alternate depressions and elevations corresponding to those on the surface of the water, but less sharply curved. The curves would progress after the manner of 10 N. M. FENNEMAN surface waves. Any one point in the film would rise and fall vertically ; any particle of water adjacent to it would continue to describe its normal circle, gliding to-and-fro on the friction- less film and tracing a straight line upon its surface. The diameter of this orbit is represented by the vertical distance through which any point in the film swings. If the water above the film be viewed in cross section, the area in which it is mov- ing forward would equal that in which it moves backward. Action on a solid horizontal plane surface. — If the film be sup- posed now to be stretched to a horizontal plane and to become a solid bottom of the ordinary kind, several changes become necessary in the behavior of the adjacent water particles. The up-and-down movement in their orbits becomes impossible, but the to-and-fro movement, tracing straight lines on the surface, can be continued. Observation shows that this does occur, that particles near a shallow bottom move back and forward in straight lines, and that vertical movement gradually appears in the paths of higher particles, these paths being at first very flat ovals, but becoming higher and more nearly circular as the surface is approached.* The energy of the vertical movement thus interfered with is partly expended in friction on the bottom, though it is quite possible that a part of it may be used in an increased horizontal amplitude.'' It is a matter of observation that this flattening of orbits affects the movements of surface particles as well as of those below. 3 This effect on the topmost orbits is in proportion to the degree of interference at the bottom. Very much elon- gated orbits indicate large friction, just as circular orbits indi- cate that there is no appreciable interference at the bottom. ^ Effect on wave-length, etc. — The immediate effect of retarda- tnn of particles in contact with the bottom must be an increased » Weber, Wellenlehre, p. 124. * The observations of the brothers Weber, as recorded in the table given in Wel- lenlehre,^. 124, seem to show that the horizontal motion on a shallow bottom, while less than at the surface, is actually greater than a certain intermediate point. 3 Weber, loc. cit. PROFILE OF THE SUBAQUEOUS SHORE TERRACE II differential movement of adjacent water particles. The laws of fluids require that this differential movement be distributed throughout the series of particles reaching to the surface, though experienced to a less degree as the distance from the bottom increases. It has been shown that increased differential move- ment implies decreased wave-length. This shortening, accom- panied by steepening, may or may not be sufficient to cause breaking. Since these effects are greater at the bottom than at the surface, the lines of like phase will incline forward. These effects — the increased differential movement, the shortening waves, and the forward inclination of lines of like phase — fol- low from friction on the bottom, but it is immaterial whether this friction be that of the forward-moving or that of the back- ward-moving water. The forward inclination of lines of like phase indicates nothing as to the movement of the water as a body. The inclination of these lines may be arrived at in another way. The retarded particles below may be thought of as having a decreased angular velocity, and hence a less advanced phase than the upper particles in the same vertical line. This would require that lines of like phase should connect them with upper particles in advance of them in the direction of wave movement. Comparison of friction in forward and backward movement. — Looked at in cross-section, the area of the backward-moving water above the line AB (Fig. 9) is less than the area of for- ward-moving water. The areas are equal when bounded below by one of the trochoid curves. The area of backward-moving water is made smaller by the substitution of a rigid plane for the depressed part of the trochoid, and that of the forward-moving water is made larger by the substitution of a flat bottom for the curve bulging upward. This constriction and consequent greater friction of the backward-moving water makes itself felt in the form of the wave and in the bodily movement of the water. Asymmetrical form. — The velocity of propagation of wave crests depends purely upon the behavior of particles in the upper halves of their orbits, while the propagation of troughs 12 N. M. FENNEMAN takes account of the lower halves only. It results from a greater orbital velocity in the upper halves that crests are propa- gated more rapidly than troughs.' The necessary accompani- ment of this is the asymmetrical form, steeper in front than behind. Resulting currents. — The constriction of backward-moving water mentioned above may be compensated either by greater velocity or by broadening the area of backward flow. Upon either of these assumptions, or upon the assumption of no com- pensation, certain conclusions follow from a geometrical inspec- tion of the diagram, and these conclusions agree with observed phenomena. Assume first that the deficiency in backward movement is uncompensated. This assumption involves an excess of forward movement which would be observed as a current, a well-known phenomenon where waves enter shallow water. On this same supposition of no compensation the area of the bottom covered by the backward-moving water is greater than that covered by the forward-moving water, and the velocity of that moving back- ward on the bottom is greater than of that moving forward at the same depth, because the former, being under the trough, is nearer to the surface. A current of this type would therefore be distinctly a surface feature which would not wash the bottom in the direction of its flow. It would, in fact, involve a certain amount of counter-current at the bottom, independent of any of the conditions which give rise to undertow. Assume next that the deficiency of area of backward mov- ing water is compensated in one of the ways above mentioned, either by greater velocity or by broadening the area. In either of these cases the backward movement on the bottom will be in excess, and will suffer more interference by friction than the forward movement will. This greater interference with the backward movement will favor, with each oscillation, a residual advance of the water as a whole, causing a progression in the * Compare also C. S. Lyman, "A New Form of Wave A.^i^^xz.ins,,'" Journal of the Franklin Institute, Vol. LXXXVI, p. 187. PROFILE OF THE SUBAQUEOUS SHORE TERRACE 1 3 direction of wave movement, by a process which has something in common with walking. In this way also, pure oscillation would give rise to a current. It is evident then, that when a system of waves of pure oscillation advances over a shallow bottom, any supposition that may be made regarding the adjustment of internal movements will result in a forward flow of water above, and a dominance of movement in the opposite direction below. Owing to friction, the latter alone is never equal to the former. The resulting movement of water in the direction of wave propagation, whether it be viewed as a current or as an increase of the positive over the negative parts of ordinary waves, is not the same as waves of translation, technically so called.^ These latter obey differ- ent laws and move with different velocities. They may be occasioned by breakers, or may perhaps grow out of oscillatory waves by gradual transition, but their movements are character- ized by certain features to be mentioned later. The return current. — As soon as a current is initiated a return of the water becomes necessary. If the process described above be supposed to take place on a shoal without shores this return may take place by another route. In this case the current may proceed as described for an indefinite time. If there is no return over another area by horizontal circulation, then the return must be over the same area by vertical circulation; that is, either above or below the original current. If the forward orbital movement above exceed the backward orbital movement below, as seems necessary, and no lateral escape is at hand, the pressure due to increased height of the water would cause a counter cur- rent which would appear below as undertow. Action on bottom materials. — The essential value of the consid- eration of these currents, springing from waves of pure oscilla- tion, is in the necessary conclusion that the work of such waves is backward at the bottom, and 7iot forward. The advance of the water described is due to interference with its backward flow. The same friction which impedes the backward movement of the * J. Scott Russell, The Wave of Translation. 14 N. M. FENNEMAN water causes the motion which the water loses to be communi- cated to the materials of the bottom. The case is roughly analogous to the wheels of a locomotive, which in "flying the track" brush the sand on the track backward. The case of wind- driven waves, — ^The above case is applicable only to waves of pure oscillation, which have of necessity been generated in deep water and are advancing over a shallow bottom. If the wind is blowing at the same time in the direc- tion of wave movement, the result will be similar to that found in considering a mathematical plane above wave-base, provided, of course, that the return of the water is by horizontal circula- tion. The action of the wind increases the forward motion under crests and diminishes the backward motion under troughs. When the effect of this action reaches a certain amount, the influences named above, which give dominance to the backward movement at the bottom, will be counterbalanced, and any greater effect of the wind will give, at the bottom, an excess of forward movement. A moderate effect of the wind is probably usually sufficient to overcome the backward brushing due to oscillation alone. If the return is by vertical circulation, any increase in current above involves an increased reverse current below. The case of breaking waves. — When waves generated in deep water advance over a bottom sufficiently shallow to cause breaking, a new factor is introduced. In this case there is a tendency to the formation of positive waves of translation, which may sometimes develop typically, though doubtless more often their motion enters in merely as a component. It is in the nature of these that all the particles in and under the wave form move forward and not backward, and the forward motion is the same at all depths.' To the extent that this factor enters, the effect on the bottom will of course be to urge material in the direction of wave movement. » See Russell, The Wave of Translation, p. 42 ; Report on Waves, p. 307 ; also D'AURIA, "A New Theory of the Propagation of Waves in Liquids," yi?«r«a/ of the Franklin Institute, 1890, p. 460. The last named is a mathematical discussion. PROFILE OF THE SUBAQUEOUS SHORE TERRACE 1$ WAVES IN SHALLOWING WATER. Tendency to enlargement of orbits. — When a system of waves generated in deep water reaches shallow water, certain forces operate to increase the sizes of the orbits, while others produce the opposite tendency. In general the increase of orbital motion is due to the transmittal of the motion of a larger amount of water to a smaller amount.^ If the shallow water be separated from the deep water by a vertical face i^BC in Fig. lo), the change may operate in some manner similar to the following : The deep water on the right side of the figure is agitated to the depth of C by waves travel- ing toward the left. The motion of particles below the level of B is influenced by the vertical face BC, this influence being greater in proportion to their nearness. Those in contact with the surface must move in straight lines up and down, while those farther away describe ovals whose longer diameters are vertical, and whose shapes become more circular with distance from BC. The energy of the horizontal motion thus lost is, of necessity, partly expended in friction on the vertical face. That which remains must be devoted to increasing the vertical move- ment. By this means it is again communicated to the particles above the level of B. If the change from deep to shallow water be gradual, the analysis of the process is essentially the same. In this case^ however, the circular orbits below will give way to straight line movement, not vertical, but parallel to the sloping bottom DC. As before, friction will consume a part of the energy which orbital motion has lost, the remainder being expended in increased movement parallel to the sloping bottom. Of this movement the vertical component will go to increasing the ver- tical axis of the orbits above. Tendency to diminishing orbits, — Along with the above tendency to increased orbits come two tendencies toward diminution. The first of these is the influence of the flatter orbits of the lower particles. It tends to diminish the vertical movement * Compare C. S. Lyman, loc. cit., p. 193. l6 N. M. FENNEMAN above, but not the horizontal. The second influence toward diminution is the friction on the bottom which is shared by the particles above. Opposite tendencies simultaneous. — On a sloping surface the opposing tendencies act at the same time. It is evident that in proportion as the slope is steep, sudden enlargement will be favored, and that slow shallowing favors reduction in size because of the long continued action of friction. Theoretically, there should be a grade on which an incoming wave should suffer no change of height, but since the form and internal movements would change, this ideal grade is not of importance in considering the work of water on the bottom. Tendency to decreased wave-length. — If the supposed tendency toward orbital increase be balanced by the opposite tendency arising from friction, there will, of course, be no increase in the length or height. However, when waves do increase in height, showing that the orbits have enlarged, they are still very com- monly diminished in length and of necessity increased in steep- ness. This is readily explained by the increased differential movement of particles, initiated by friction on the bottom. TendeTicy to steepening due to wind. — The largest on-shore waves usually act in conjunction with the wind blowing in the approximate direction of their movement. The effect of wind on waves in deep water was seen to be similar to the effect of a shallow bottom, namely, (i) increase of orbits; (2) increase of steepness ; (3) asymmetrical form. These effects may be carried to the point of breaking, even in water of infinite depth (white- caps). On a shallow bottom the effects are increased by the concurrent action of the two factors. Where there is no wind waves are commonly supposed to break in water whose depth is equal to or a little greater than the height of the waves above the level of repose.* When waves advancing on a shallow bot- tom are already strained by the wind, they may break with much regularity in much greater depths of water, equal to perhaps two, three, or four times the height of the wave. Thus while the 'Russell, Report on Waves, p. 245. PROFILE OF THE SUBAQUEOUS SHORE TERRACE I J breaker line for waves without wind is far up the slope from wave-base, it may move down indefinitely near to wave-base when the wind is active. Tendeficy of wave to recover form. — Suppose a system of oscillatory waves to advance toward a shelving shore until the wave-base intersects the bottom. One effect must be produced here regardless of qualifying conditions. Bottom friction begins and that involves increased differential movement of particles, which is accompanied by shortening and steepening of waves. This implies increased internal friction, which in turn, operates to decrease the orbital motion and therefore wave dimensions. In so doing it would take away the conditions of bottom friction and its results. The wave would then return to its deep water form. Thus there is a chain of consequences from the original interference at the bottom, which involves at first the change of wave form, but later a restoration, the final result being reduc- tion in dimensions only, suited to the diminished depth. Another decrease of depth must then be assumed if the wave be supposed to continue its contact with the bottom. Thus there is a certain minimum slope for the bottom, upo?i which the waves may be propa- gated as a shallow-water wave. In so far as the wave is affected by increase of orbit due to diminishing amount of water, the effect will be to hasten the deformation and to retard the recov- ery of form. If the wind is active it would retard the decrease of orbital movement and the minimum slope mentioned would be smaller. Limit of tendency to recover form. — The greater the reduc- tion of depth, the greater the increment of internal friction tending to reduce the wave size, and the greater this friction, the more rapidly does it operate to accommodate the wave dimensions to diminished depth. This corrective tendency has, however, a limit. This limit is marked by the breaking of the wave. There is, therefore, a certain maximum slope for the bottom upon which the wave m,ay be propagated without breaking; at or beyond this maxi- mum the wave breaks and other agencies come in. The effect of wind as before, is to diminish the maximum slope ; hence 1 8 N. M. FENNEMAN true breakers (not whitecaps merely) may occur during a wind on a shore where waves of the same size would not break in a calm. Effect of breaking on wave propagation. — Even when the distortion of wave form has been pressed beyond the breaking point, the effort to recover its form and habit does not cease. This effort is now favored by all the tendencies which existed before breaking and re-enforced by one more arising from the falling crests. As shown in the diagram (Fig. 4), breaking is an expression of conflicting orbits. The water above the node of the hypothetical surface does not continue the curve which it has been describing, but falls confusedly on the front of the wave. Here its downward motion is in direct opposition to the upward motion of the water in front of the crest. Thus, to the molecular resistance of friction, is added mass conflict, both of which operate to reduce wave motion. This reduction is there- fore accomplished more rapidly than in the case of unbroken waves. It results from this, that waves often break at some dis- tance from shore, and after traveling a short distance with foam- ing crests, recover their form and advance a long distance with crests entire. There is a certain slope on which waves will advance with nearly uniform shape and continuously foaming crests. On a gentler slope they will recover their unbroken form ; on a steeper slope the first breaking occurs close to shore, and the wave form is speedily lost. Waves of translation. — When waves of oscillation enter shallow water the habit of the water particles changes and becomes a compromise between orbital oscillation and movement of an entirely different nature, belonging to waves of translation.' The essential features of the positive wave of translation, known also as the wave of the first order or the solitary wave are, ( I ) it is initiated by an elevation of the water surface above its normal level; (2) it is propagated without a corresponding trough and without companion crests, being entirely above the undisturbed level of repose; (3) its rate of travel is greater * Russell, Report on Waves and Wave of Translation. PROFILE OF THE SUBAQUEOUS SHORE TERRACE \^ than that of waves of oscillation, when like wave-lengths are assumed, the two rates having about the ratio of three to two;* (4) the water particles move forward and not backward, start- ing from rest as the wave approaches and coming to rest when- the wave has passed ; (5) the forward motion of particles at all- depths is the same and equal to the volume of the wave divided by the depth of the water; (6) the paths of the particles are: semi-ellipses in a vertical plane, the major axis being the dis- tance through which the particle moves forward, and the minor axis varying from zero at the bottom to the height of the wave at the surface. This movement is in no sense the same as that of wind-driven waves or any other oscillatory wave motion com- pounded with current. It usually coexists with the latter on shallow bottoms, resulting in waves of a hybrid kind ; but waves of nearly typical translatory character may sometimes be seen in nature. Whether the waves be of a pure or mixed type, the essential fact here is that a nt.^ factor has entered, whose action at the bottom is different from that of oscillatory waves and from that of currents. The fact of this change to translatory character on a gently sloping beach may be seen in the behavior of floating chips which are seen to move forward on crests but not backward between crests. In place of the trough proper is a wide strip whose surface is almost flat and the water of which is standing still. The laws of translatory waves require that they move more rapidly than the oscillacory. This might be expected to reveal itself in broadening intervals between crests as waves take on the translatory character. It is probable that this may occur under suitable conditions. The tendency is usually more than counterbalanced by two factors. The first is the decreasing depth which is the main factor in controlling the velocity of waves of , translation. The second is the increasing strength of undertow near shore which retards the translatory movement at the bot- tom. As to the manner in which this new habit is developed, it may ^ Ibid., -p. 288. 20 N M. FENNEMAN be cited that perfect waves of the first order are produced experi- mentally by the sudden addition of water at one end of a rectangular vessel, or by the immersion of a solid, or by a sudden pushing forward of the wall of the vessel, the effect in «ach case being the local raising of the water surface above the level of repose. A corresponding process in lakes or sea where the bottom becomes shallow may be found in the sudden deliv- -ery of the mass of water which falls upon the front of a breaking wave. Observation on the shores of large water bodies, such as the great lakes, would indicate that the area over which waves show a translatory element is somewhat definitely limited by the breaker line. It is probable, however, that there is also a more gradual change by which the waves become increasingly positive as the water shallows and the features of waves of the first order are thereby assumed. If the modifications of oscillatory waves in shallowing water be reviewed while holding in mind the charactertistics of trans- latory waves as given above, it will be observed that these changes are all in the direction which would favor the conver- sion of oscillatory into translatory waves. This is seen in the increase of crests with corresponding disappearance of troughs ; the growing excess of the forward movement of particles over backward movement and the increased horizontal amplitude of the lower orbits, approaching equality with that of the orbits above. For present purposes it may suffice to adopt the con- ception of Mr. Russell^ who thought of the overgrown crest as * The wave of the second order may disappear and a wave of the first order take its place. The conditions under which I have observed this phenomenon are as fol- lows : one of the common sea waves, being of the second order, approaches the shore, consisting as usual of a negative or hollow part and of a positive part elevated above the level ; and as formerly noted, this positive portion gradually increases in height and at length the wave breaks, and the positive part of the wave falls forward into the negative part, filling up the hollow. Now we readily enough conceive that if the positive and negative parts of the wave were precisely equal in height, volume, and velocity, they would by uniting, exactly neutralize each other's motion, and the volume of the one, falling into the hollow of the other, give rise to smooth water ; but in approaching the shore the positive part increases in height and the result of this is to Jeave the positive portion of the wave much in excess above the negative. After a •wave has first been made to break on the shore it does not cease to travel, but if the PROFILE OF THE SUBAQUEOUS SHORE TERRACE 21 falling forward into the diminished trough in the act of break- ing ; the trough is more than filled and the excess of water ini- tiates a wave of translation exactly as in Mr. Russell's experi- ments. Volume of undertow . — It is not necessary to suppose that the loss of velocity of the undertow is as rapid as the increase of its cross-section. This would be the case if all the upper water moving shoreward should reach the shore before turning back. The volume of the undertow would then also be constant throughout its course and its velocity would be inversely as its cross-section. But even if the loss of motion due to friction and interference of the bottom be ignored, not all the shoreward moving water reaches the shore. The on-shore motion causes elevation of level over a belt of considerable width. This broad elevation constitutes a head which is the cause of outward flow below. It may be shown that the average position at which incoming particles turn back and join the undertow, is at the center of mass of the head. This head is greatest at the edge of the water, hence more water turns back at that point than at any other, but the undertow which has its beginning here is constantly being augmented by that which returns toward deeper water without reaching the shore. slope be gentle, the beach shallow and very extended (as it sometimes is for a mile inward from the breaking point, if the wave be large) the whole inner portion of the beach is covered with positive waves of the first order, from among which all waves of the second order have disappeared. This accounts for the phenomenon of breakers transporting shingle and wreck and other substances shoreward after a certain point ; at a great distance from shore or where the shores are steep and abrupt the wave is of the second order, and a body floating near the surface is alternately carried forward and backward by the waves, neither is the water affected to a great depth ; whereas, near the shore the whole action of the wave is inwards, and the force extends to the bottom of the water and stirs the shingle shoreward ; hence the abruptness also of the shingle and sand near the margin of the shore where the breakers generally run. .... The residuary waves given off after breaking are wide asunder from each other, are wholly positive, and the spaces between them, several times greater than the amplitude of the waves, are perfectly flat and in this condition they extend over wide areas and travel to great distances. These residuary positive waves evidently prove the existence, and represent the amount, of the excess of the positive above the negative forces in the wind wave of the second order. — Report on WaveSy p. 292. 22 N. M. FENNEMAN Relation of the phenomena above to agitation on the bottom. — It is to be inferred from what precedes that symmetrical wave form indicates freedom from interference at the bottom, that friction below is great in proportion as crowding, steepening, and asymmetrical form above are prominent, and that where an off-shore breaker line is seen it indicates a maximum of bottom interference. It is understood in all cases that the surface effect will lag a little behind the cause below, and therefore appear a little to shoreward. PROFILES RESULTING FROM FORCES DISCUSSED ABOVE. In the actual operation of the forces discussed above, the resulting action on a sloping bottom may be outward at all places, or inward at all places, or outward over one part and inward over another. Forces in either direction may be gradu- ually augmented or diminished. The different forces are capable of different combinations. Each set of conditions will lead to certain features of profile, if there be no change of condition, a permanent profile of equilibrium may be reached. The con- stant supply of load constitutes an ever shifting condition. Equilibrium as commonly realized depends on the uniformity of this supply. Factors in profile -making. — The agencies which shape the marginal bottom may be treated in three groups, ( i ) oscil- latory wave action and undertow, carrying material from shore ; (2) on-shore currents and translatory wave action, carrying the material toward the shore ; (3) currents alongshore. The tend- ency of the first group is to steepen the slope from the water's edge to the line at which its erosive power ceases, and deposi- tion begins and to reduce the slope beyond that line. There is also for the second group a line of maximum power on the bottom, within which their effect is to steepen the profile by accumulation at the water's edge, and beyond which the slope is reduced by cutting down. Currents alongshore will be intro- •duced later. Conflict between on-shore and ofi-shore action. — The first two UNIVFR< PROFILE OF THE SUBAQUEOUS SHORE TERRACE 23 pairs of agencies are in conflict as to the direction in which bottom materials are to be moved. If all the water which moves shoreward must return over the same area and as a bot- tom current, this current would seem to have greater efficiency than the one above, moving in the opposite direction. This is certainly the case where translatory waves are not favored, as where the off-shore slope is steep. Where slope is gentle and translatory waves are well developed, they have one decided advantage. They are short as compared with the distance from wave to wave, hence all the shoreward movement of the water is concentrated into a small portion of the entire time. Divers are said to feel the passing of one of these waves as a sudden jerk between intervals of quiet. The undertow, on the other hand, has a steady flow except as interrupted by these sudden reverses.* The laws of energy give to these concentrated move- ments a much greater efficiency than to the same amount of motion more evenly distributed in time. On many shores of gentle slope, sand is worked landward, and in this process the agency just mentioned is doubtless important. The effect here referred to is that of waves of translation and is therefore inside the breaker line. It might accumulate sand on-shore but not in off-shore barriers. The dominance of shoreward action is essentially temporary (omitting currents alongshore from con- sideration). Its effect is to steepen by narrowing the slope. This steepening, in turn, is adverse to waves of translation. Laws of equilibrium ; eroding currents. — Ignoring the presence of a bank and the load derived from it, a current of uniform power tends to reduce the bottom to a level surface, that is, to require equal depth throughout. Equilibrium cannot exist on a level bottom where the power of the current is unequal at dif- ferent places. In such cases, the depth must suffer a corre- sponding change until the power of water on the bottom is ^ Henry Mitchell, "On the Reclamation of Tide-Lands and its Relation to Navigation," Report of the U. S. Coast and Geodetic Survey, i86g. Appendix 5, p. 85. In tliis paper Mr. Mitchell takes the extreme view that the sea restores to the conti- nent " all the material washed from its bluffs and headlands." Certain exceptions are made for islands. 24 N. M, FENNEMAN everywhere the same. A current of uniformly increasing power requires a uniformly increasing depth, that is, a plane slope. The opposite is true for a current of uniformly diminishing power. A current whose power is augmented at an increasing rate^ as, for example, in geometrical ratio, requires a descent to deep water on a curve which is convex upward. Increase of power at a diminishing rate requires concavity. Loss of power at increasing rates, and loss at diminishing rates, require con- cavity and convexity respectively. Uniform cuttifig or building. — If a uniform current on a level bottom has eroding power, the whole will be cut down at the same time, and the bottom will remain level while depth increases. In this case the load is furnished at all points equally, and is all carried forward at the same rate. If load be furnished in excess of carrying power, and at all points uniformly (as from top or sides), then the level surface of the bottom would be preserved while depth would decrease. Load derived from the shore. — To make the case applicable to undertow, the excessive load must be supposed to be furnished at the end where the current enters upon the bottom in ques- tion. In this case deposition will first reduce the load at the end upon which it enters and at the same time reduce the depth and thus constrict the current, increasing its power. The latter influence will determine a higher level to which the bottom will be built ; a level at which the power of the water is sufficient to carry the load which before was excessive. Filling will then advance forward over the bottom, the filled and unfilled portions both being level, the former growing while the latter diminishes^ and the two being separated by a slope, mentioned below. It is evident that the depth at which this slope begins is determined jointly by the power of the water, the amount of the load, and the size of the fragments which make up the load. The front. — The shape of the slope which intervenes between the area which has been filled and the bottom beyond, will be determined by the rate at which the power of the current decreases. If the loss of power were instantaneous, the slope PROFILE OF THE SUBAQUEOUS SHORE TERRACE 25 would be simply the subaqueous earth slope. If it be in any arithmetical progression, the slope will be a plane whose steep- ness will vary with the rate of decrease, the slope being steeper when the rate is higher. If the loss of power be in some other manner than by arithmetical difference, the slope will show a curve which will be convex or concave according as the rate of decrease is augumented or diminished. In actual deposition by a current advancing into deep water, the decrease of power is at an increasing rate, as may be seen from the following. If a plane slope be assumed, so that depth increases in arithmetical ratio, then the velocity of the current will decrease in similar ratio, but transporting power varies as the square of the velocity, hence its rate of decrease is progressively augmented. This will require convexity of slope, a feature generally observed at the edge of embankments and subaqueous terraces. The general law of equilibrium, as given above for an eroding current still applies ; current power is uniform over all parts of the bot- tom, if by the term current power is, understood /