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 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1420 
 
 ON THE CONTRIBUTION OF TURBULENT BOUNDARY 
 
 LAYERS TO THE NOISE INSIDE A FUSELAGE 
 
 by 
 
 G. M. Corcos 
 
 and 
 
 H. W. Liepmann 
 
 Douglas Aircraft Company, Inc. 
 
 UNIVERSITY OF FLORIDA 
 DOCUMENTS DEPARTMENT 
 120 MARSTON SCIENCE LIBRARY 
 RO. BOX 117011 
 GAINESVILLE. FL 32611-7011 USA 
 
 Washington 
 December 1956 
 
TECHNICAL MEMORANDUM 114-20 
 
 ON THE CONTRIBUTION OF TURBULENT BOUNDARY 
 
 LAYERS TO THE NOISE INSIDE A FUSELAGE* 
 
 by 
 
 G. M. Corcos** 
 and 
 
 H. W. Liepmann*** 
 
 September, 1956 
 
 "itoedited by the NACA (the Committee 
 takes no responsibility for the 
 correctness of the author's 
 statements . ) " 
 
 Aerodynamics Research Group 
 Doviglas Aircraft Company, Inc. 
 Sstnta Monica, California 
 
 California Institute of Technology 
 Pasadena, California 
 (Consultant, Aerodynamics Research 
 Group, Douglas Aircraft Company, Inc. 
 Santa Monica, California) 
 
NACA TM 11^20 
 
 TABLE OF CONTENTS 
 
 ABSTRACT i 
 
 INTRODUCTION 1 
 
 I. THE ACOUSTIC COUPLING OF A RANDOMLY 
 
 VIBRATING PLATE WITH AIR AT REST 1+ 
 
 II. THE DYNAMIC BEHAVIOR OF THE SKIN 8 
 
 a) The Mean Acceleration 9 
 
 b) The Length Scale X 12 
 
 III. THE FORCING FUNCTION lU 
 
 IV. SPECIAL CASES 20 
 
 1. Convected Turbulence 20 
 
 a) The coupling of the plate with air at 
 
 rest in the case of convected turbulence .... 20 
 
 b) The response of the plate 21 
 
 2. The case of zero scale 23 
 
 V. SUMMARY OF RESULTS AND DISCUSSION 2k 
 
 A Note on Testing 25 
 
 VI. REFERENCES 2^ 
 
 APPENDIX I: THE RANDOM RADIATION OF A PLANE SURFACE: 
 
 A. Four Limiting Cases 28 
 
 B. The Noise Generated by Skin Ripples 
 
 of Fixed Velocity 35 
 
 C. The Generation of Noise by Plate 
 
 Deflections of Zero Scale 38 
 
 APPENDIX II: THE SIMPLIFICATION OF THE PLATE RESPONSE INTEGRAL . . Uo 
 
 APPENDIX III: THE EVALUATION OF THE INTEGRAL SCALE k2 
 
Digitized by tlie Internet Arcliive 
 
 in 2011 witli funding from 
 
 University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation 
 
 http://www.archive.org/details/oncontributionofOOunit 
 
NACA TM 114-20 
 
 ABSTRACT 
 
 The following report deals in preliminary fashion with the transmission 
 through a fuselage of random noise generated on the fuselage skin by a tur- 
 bulent boundary layer. The concept of attenuation is abandoned and instead 
 the problem is formulated as a sequence of two linear couplings: the turbu- 
 lent boundary layer fluctuations excite the fuselage skin in lateral vibra- 
 tions and the skin %d.brations induce sound inside the fuselage. The techniques 
 used are those required to determine the response of linear systems to random 
 forcing fimctions of several variables. A certain degree of idealization has 
 been resorted to. Thus the boundary layer is assumed locally homogeneous, 
 the fuselage skin is assumed flat, unlined and free from axial loads and the 
 "cabin" air is bounded only by the vibrating plate so that only outgoing 
 waves are considered. Some of the details of the statistical description 
 have been simplified in order to reveal the basic features of the problem. 
 
 The results, strictly applicable only to the limiting case of thin 
 boundary layers, show that the sound pressure intensity is proportional to 
 the square of the free stream density, the square of cabin air density 
 and inversely proportional to the first power of the damping constant and 
 to the second power of the plate density. The dependence on free stream 
 velocity and boundary layer thickness cannot be given in general without a 
 detailed knowledge of the characteristics of the pressure fluctuations in 
 the boundary layer (in particular the frequency spectrum). For a flat 
 spectrum the noise intensity depends on the fifth power of the velocity 
 and the first power of the boundary layer thickness. This suggests that 
 boundary layer removal is probably not an economical means of decreasing 
 cabin noise. 
 
 In general, the analysis presented here only reduces the determination 
 of cabin noise intensity to the measurement of the effect of any one of 
 four variables (free stream velocity, boundary layer thicltness, plate 
 thickness or the characteristic velocity of propagation in the plate). 
 
 The plate generates noise by vibrating in resonance over a wide 
 range of frequencies and increasing the damping constant is consequently 
 an effective method of decreasing noise generation. 
 
 One of the main features of the results is that the relevent quantities 
 upon which noise intensity depends are non-dimensional numbers in which boundary 
 layer and plate properties enter as ratios. This is taken as an indication 
 that in testing models of structvires for boundary layer noise it is not 
 sufficient to duplicate in the model the struct\xral characteristics of the 
 fuselage. One must match properly the characteristics of the exciting 
 pressure fluctuations to that of the structure. 
 
TECHNICAL MEMORANDUM ll|20 
 
 INTRODUCTION 
 
 In his efforts to minimize the noise levels for which he is responsible, 
 the airplane designer has had to pay increasing attention to a source of 
 noise which until recently had been ignored. This is the boundary layer. 
 The boundary layer will generate noise whenever it is the seat of any 
 fluctuating phenomen®n. In particular it will nurture randem pressure 
 fluctuations whenever it is turbvilent . 
 
 The designer's interest will naturally center on the characteristics 
 of that part of the boundary layer noise which has been transmitted through 
 the fuselage slvin and into the cabin rather than on that part which is 
 radiated into the free -stream. This is so because the radiation intensity 
 is, as we shall see, a rapidly increasing function of the velocity of the 
 boundary with respect to the air and to a likely observer outside a fuselage 
 either the relative velocity of the plane is low (as near a take-off or 
 lEindlng) or the plane is considerably distant. 
 
 As a consequence the practical question which has to be raised concerns 
 the effects on a fuselage skin, and on the air which it encloses, of the 
 boundai^ layer pressure fluctuations acting directly on the skin.* 
 
 Two features of this problem are worth noting: To toegin with, the 
 fuselage will transmit noise only by deflecting laterally. The thickness 
 of the skin is very small when compared to the wave length in metal of 
 audible sound waves so that, effectively there are no fluctuating pressure 
 gradients within the skin and hence the latter will not oscillate in lateral 
 compression. 
 
 In the second place , the turbulent pressure fluctuations in the boundary 
 layer axe random both in space and in time. The fluctuations are generated 
 locally. If they are measured simultaneously at two different points of 
 the boundary layer, say on the skin Itself, they are found to have no 
 relationship with each other lonless the two points are separated by a very 
 short distance. Mike the value of the pressure fluctiiatlon at a given 
 point soon loses correlation with itself.** 
 
 Noise intensity is defined in this report as pi /po^-i- I't is assumed that 
 this is the physical quantity of interest. It has the dimensions of an 
 energy flux; but it is not necessarily equal to the energy flux at some 
 point in the field, nor is it necessarily equal to the density of 
 energy radiated away (lost) by the fuselage. 
 
 **Recently two authors (Refs. 5 & 6) have s\iggested that the randomness in 
 time is not Independent of the randomness in space; i.e., that the pressure 
 fluctuations at the wall are created by the convection at a single speed 
 of a "frozen" pattern of pressure disturbances. Some attention is paid 
 later to this eventuality which is treated as a special case. 
 
NACA TM 11+20 
 
 V/e are thus led to visualize the process of transmission of "boundary 
 layer noise through the metal slcin of the fuselage as follows : A multitude 
 of external pressure pulses push the elastic slcin in and out and the skin, 
 in turn, not iinlike a set of distributed pistons, creates inside the fuse- 
 lage pressure waves which propagate and superimpose. This constitutes cabin 
 noise. 
 
 It is of course desirable to determine the characteristics of this 
 noise. One should point out that the data for the problem are not complete 
 and are not likely to be so in the near future. Specifically the structure 
 of the turbulent mechanism within the boundary layer and, in particular 
 of the coupling between pressure fluctuations, velocity fluctuations and 
 temperature fluctuations is not enough explained or measured to define 
 wholly our forcing function. As a conseq uenc e it is not now possible to 
 define say average cabin noise intensity pj2 as a function of say, free 
 stream Mach number, Reynolds number and plate characteristics. It Is 
 however possible and it is the pixrposeof this report to indicate the 
 approximate functional dependence of pj^ on these quantities and thus 
 to give similarity rules which will reduce to a minimum the amount of 
 testing reqmred. 
 
 We assume at the outset that the boundary layer unsteady pressure 
 field is known and that it induces small deflections in the skin. As a 
 consequence 
 
 (a) The skin dynamics are described by a linear equation 
 
 (b) The generation of a random pressure field Inside the cabin 
 Is a linear radiation problem. 
 
 Thus the mathematical techniques used are those required to obtain 
 the response of linear systems to stochastic forcing functions of several 
 variables . 
 
 We also assume the fuselage to be a large flat plate. This assumption 
 is not necessary but it simplifies the discussion and allows us to present 
 more clearly the new features of the problem. 
 
 The material in this report is presented as follows: First we study 
 the radiation of sound from a randomly vibrating plate . It is found that 
 the sound levels in the cabin are defined by the intensity and the scales 
 of the plate normal accelerations . 
 
 Second, generalized Fourier analysis is put to \ise in order to relate 
 the normal acceleration of the plate to the forces exerted on it by the 
 boundary layer flow. 
 
 Third, the boundary layer forces are defined in terms of flow character- 
 istics, and dimensional similarity is used to determine the significant 
 parameters . 
 
 FineLLly, the functional fonii of the noise intensity In the cabin Is 
 given save for an -jnknown function of one non-dimensional parameter. This 
 function depends on the frequency spectrum of the pressure fluctuations 
 In the boundary layer. No measvirements yielding this spectrum have been 
 
NACA TM 1U20 
 
 reported to date and speculations conceminc it would introduce in the 
 analysis both complication and uncertainty. 
 
 A summary of results is given at the end of the report. Some 
 derivations and some of the longer arguments have been presented as 
 appendices to the text. 
 
MCA TM IU2O 
 
 I. THE ACOUSTIC COUPLING OF A RAITOOMLY 
 
 VIBRATING PLATE WITH AIR AT REST 
 
 We start with Rayleigh's well knovm solution of the acoustic equations 
 when the sound is generated in an otherwise unbounded stationary gas by a 
 large flat plate or disc oscillating normally to its plane (Ref. 1 page IO7), 
 
 Ztt J ^t ^ ^ ' ^ ai / \ii 
 
 where : 
 
 the static pressure p has been broken into a steady part p and 
 a f luctiiating part , pj^ : 
 
 P = p" (x,y,z) + pj^(x,y,z,t) 
 
 Vjj = the normal velocity of the plate 
 
 d<r = an element of plate surface area 
 
 a-j^ = the speed of sound in the air 
 
 S = the vector position over which the integration is carried out 
 
 = (x',o,z') 
 
 lr| = the distance between source ajid field points 
 
 (x-x' ) + y + (z-z') = r 
 
 A = the total area of the plate 
 
 The subscript i refers to properties inside the cabin or on the side of the 
 plate on which air is not flowing. 
 
 The normal acceleration cVn/^t is a random function of time and space 
 and so is p-j^. Vfe wish to evaluate the mean square of the pressure (a quantity 
 to which sound intensity is directly proportional). 
 
 We notice that for a sum 
 
 alike, for the integral in (Ij- 
 
 *This discussion follows closely the arguments set forth in ref. 3 and the 
 background of information on statistical methods can be found in ref. 2. 
 
NACA TM 11+20 
 
 
 and since ax = ax if a does not depend on time , we can write 
 
 
 ^r 
 
 
 avn 
 
 
 2) 
 
 Now the expression 
 
 avn / r 
 
 r(^'''-feOf?(^-t-^^:) 
 
 is the correlation (i.e. the time average of the jiroduct) of the normal 
 
 acceleration of the plate surface at two points S-, and So and at two 
 
 different times (t- '^•/cii ) and (t- '^/4^- 1^ 'tbe two points vibrate 
 completely independently from each other, this expression is zero and 
 
 ^2' 
 
 therefore r-. 
 
 >) the 
 
 if the two points are brought to,'-;ether ( Sj- 
 
 correlation fimction is simply (3 V-n/ Qt")i . Now we assume that the average 
 
 properties of the plate motion are the same anywhere on its surface and 
 
 at any time (we assimie statistical homogeneity and stationarity) . Then the 
 
 expression above for the correlation becomes merely a function of the 
 
 distance ( jSi- So\) between the two points and of ( ^.- 1i ) We call this 
 
 function llS '■ '-••L 
 
 b\Jn 
 -hi 
 
 then 
 
 (^.r-^^)5_^(r,t-.).^|(,s%s-0,^^^ 
 
 sT- si 
 
 
 cr. 
 
 CTz. 
 
 (3) 
 
 fL, 1.1 
 
 ft A " ' 
 
 Now we can evaluate pj^2 (y) under a variety of assumptions forTD , for the 
 plate area A and for the distance Y between the plate and the observer. V/e 
 will consistently hold the view that the normal acceleration at most points 
 of the plate surface are not correlated (Wj = O ) and that two points of 
 the plate, in order to show appreciable correlation, must be a small 
 fraction of the total plate size away from each other. 
 
 We define as A^ the mean distance over which^ /dZ) is strongly 
 correlated (i -e .^^^f (dVo^^A^ ) and call it the integral (length) scale*. 
 
 * The integral scale is given, say in the x-direction by 
 
NACA TM ll|20 
 
 Our hypothesis can then be expressed as 
 
 \ ^< X 
 
 where R is the average linear dimension of the plate. 
 
 A representative case 
 
 Suppose that the observer distance Y to the plate is such that 
 
 \ « Y « '^ 
 
 and that no appreciable phase difference can airise at Y between two strongly 
 correlated signals (i.e., between sound pxilses originating within \ of 
 each other) . Then, if we examined eq^uation (3) 
 
 h^ = < 
 
 4Tr^ 
 
 
 
 we see that the inner integral contributes very little except when the 
 
 point Sj. is appr oximate ly within a distance X of S, • Then, approximately 
 
 rj_ _ ^l_i and '\l/' s I ^^"/J • Thus the inner integral is approximately 
 
 [-dt-] 
 
 euad equation (3) caa be rewritten 
 
 We should note that 
 
 The integral in (U) is not finite if the plate area A is infinite 
 
 The length scale /\ plays no role in the geometry of the problem. 
 
 The integral is a function of Y, the distance to the plate, and the plate 
 dimensions only. For instance if the plate is circular and of radius R 
 
 * This case is treated more rigorously in Appendix lA 
 
 w 
 
NACA TM ll|20 
 
 ^-"^h'^f ^5(-^:) 
 
 2-7^ \Jt~J ^ ^ ^^ ^ (5) 
 
 It is appaxent from this result that the distance Y from the plate to 
 the observer is measured in terms of plate diameters, and not in terms 
 of average correlation lengths or wave lengths ^ . This result holds for 
 all the cases considered (see Appendix I) aind depends only on the 
 assumption that at a given time the plate vibrations are largely un- 
 correlated or incoherent.* 
 
 Here, in effect, /\ loses its identity ajid combines with V^ '"'^t J 
 
 to define a strength. 
 
 Eqiiation (k) indicates that in order to evaluate the pressure — —jt 
 intensity for the case considered above, one must first determine (^""/Ot) 
 the mean square normal acceleration of the plate and A , the length 
 scale for the plate deflections. Other cases (i.e. cases for which 
 either the plate deflects differently or the observer moves closer to 
 it) are treated in Appendix lA. For some of them the time scale or 
 mean period »©< is required as well as \ . 
 
 We rewrite Equation {h) 
 
 where <P I ^ - Y J is a fimction of the plate geometry and of the distance Y 
 between the observer and the plate. It is defined from equation (k) as 
 
 Notice that if the integral scale is not the same in the x' and in 
 
 the z' direction we may simply substitute in equation (6) Xx' and A%' 
 
 for X 
 
 This result holds, as Appendix IB shows, even when the pressure is 
 generated by the travel through air at rest of a "randomly bumpy plate" 
 i.e., when the time-wise and space-wise variations of upwash are not 
 independent. The latter example is therefore quite distinct from the 
 flow of an infinite (periodically) wavy wall for which the only 
 characteristic length is the wave length. 
 
8 KACA TM li+20 
 
 II. THE DYNAMIC BEHAVIOR OF THE SKIN 
 
 The response of the bare fuselage skin to the random pressure field 
 of the boundary layer is given, in the absence of axial loads, by a plate 
 equation. For a flat plate this equation is 
 
 Eh* 
 
 ^,-5^^.^H. = |(-^'^^ 
 
 (7) 
 
 where x and z are the coordinates along the plate surface (x being the 
 free stream direction), y the deflection of the plate at a point, E the 
 modiilus of elasticity of the plate material, r its density, ^4poisson's 
 ratio, 2h the thickness of the plate, /i a damping constant which has 
 dimensions (1/T). Damping may be present because energy is absorbed either 
 within the skin or by the air. For air damping (o^^ P.Oil /^\^ 
 
 Notice that to elLIow for air damping is to provide for a feedback in 
 the coupling between the plate and the air at rest. On the other hand 
 we exclude feedback between the plate and the boundary layer. In other 
 terms we are not considering the possibility that the plate vibrations 
 are large enough to induce time -dependent pressure gradients of the same 
 order of magnitude as our forcing function. Such a feedback would amount 
 to panel flutter. It cannot be handled by the present method. 
 
 f(x,z,t) is the random force/unit mass exerted by the pressure 
 fluctuations on the plate surface. It is characterized by a power spectral 
 density F(kj_,k2,^) which is a continuous function of the wave numbers k3_ 
 (in the x direction), kg (in the z direction) and of the frequency CjJ . 
 The coefficient E/3cr(l-/*- ) has the dimensions of a velocity squared 
 and it is defined as c^. 
 
 *The skin construction may be such that the damping it causes is primarily 
 viscous or primarily flexural. In the latter case it seems more appropriate 
 to write with Ribner (ref . 5) 
 
 where /3j is the flexural dajnping constant due to the plate and/^^ the 
 damping due to the energy radiated to the air. As is shown in Appendix IB 
 the noise intensity within the fuselage may or may not be related to the 
 acoustic energy radiated by the plate and thus ft^ may or may not be 
 zero . \ 
 
NACA TM 11+20 
 
 r.i«t« f ? ° specify the space-wise boundary conditions on the 
 plate and we are led, for the sake of simplicity, to either one o^two 
 linutxng cases. In the first case, the forcing function (the random 
 
 scS:":: SSe ti; J^\^°-^-^ l-^-) - characterized hy an inL^r" 
 scale so large that at a given time, a skin area between two stiffeners 
 (assumed rxgid) is very likely to be subjected to a pressure load St he 
 
 
 Jmalf .-n '^'^' ^^^ integral scale of the forcing function is very 
 small m comparison with the distance between two stiffeners and the 
 behavior of the skin is in the average very much as though thHupports 
 were removed to infinity (see Fig. lb). The real casein in geS 
 be intermediate between these two limiting examples. H^e" Se Sat 
 
 reasonahT.' ^""l"" '° ''"""'^^ ^^^'^' °' excessive thickness A 
 reasonable guess for the average correlation length might be one dis- 
 
 l^TfTr^rsTof^lT'^/' 'f^l '\^' ^^'^'^ ^^ *^^ spacing between 
 Stf to be if th^ °H f .^ '°°'-^ ^^" boundary layer thickness / would 
 S reference (7)! °' ''"" '^"^ °" "°"^' ^^^ -^^^^^^ --« i- treated 
 
 to ..°°-H^^ °*^^'" ^^^' ^^^ ^^^^"'^ limiting case (Fig. lb) would seem 
 
 one'^fo^i ?h^r""t^'' "f ?' '°" '^^"^^ ^^^^^ ^hi'^-^ -t exce'lng 
 one loot. This is the model discussed now, 
 
 a) The Mean Acceleration 
 
 According to our assumption, the average motion 
 
 tL(I^)- 
 
MCA TM 1^4-20 
 10 
 
 is not sensibly aiffected. by the presence of stiffeners. A large number 
 of pulses act on the skin at a given time between two consecutive 
 stiffeners. The ramdom pvilses may be positive or negative axid thus 
 there will be a large number of load reversals between supports . Then 
 the effect of the boimdary conditions can be expected to become small, 
 in the average . Consequently one can define a generalized admittance and 
 use it in much the same way as is often done in one -dimensional problems.* 
 For instance, the mean square plate displacement is given by 
 
 7-tIi 
 
 
 Here the mean square of the forcing function f is related to the 
 spectrum (p by: 
 
 CD 00 00 
 
 1/^is the generalized admittance, and k-[_, kQ and CO are respectively 
 the wave number in the x direction, the wave number in the z direction 
 and the frequency. The determination of l/ji^is easy once it is realized 
 that this expression is the square of the Fourier transform of the 
 fundamental solution and so can be written by inspection. Thus an 
 average solution of (7) is: 
 
 and 
 
 
 Equation (9) gives the mean square response of an unboimded plate to 
 a random forcing function. One should notice that the plate will always 
 exhibit resonance no matter what value the dar.iping constant /3 may have. 
 This resonance occurs, not at a given frequency or at a set of discrete 
 
 * See in particular reference (2), 
 
 (9) 
 
MCA TM 1420 
 
 11 
 
 \o-t 
 
 9- 
 
 OJ 
 
 8- 
 
 7- 
 
 6- 
 
 5- 
 
 3- 
 
 2- 
 
 I - 
 
 [c'h'(f,-.ji^iy-cjf(3'cj 
 
 WITH (3-0.\ 
 
 AND !kf-HJk^ = Jb^ 
 
 -r 
 
 .5 
 
 I.O ^/CtTJk 1.5 
 
 a.o 
 
 \ 
 
 S.5 
 
 —I 
 3.0 
 
 THE ADMITTMCE MAP FOR AN UNBOUNDED PLATE 
 
 FIGURE 2 
 
12 NACA TM li<-20 
 
 frequencies but over the whole frequency spectrum, whenever the follow- 
 ing relationship obtains between frequencies aind wave numbers:* 
 
 cj-^dw-n.:*^.--) 
 
 We caji visualize the resonance condition as a crest or ridge in the 
 wave space (see Fig. 2) which originates at the line k^ = (J/ch/^ and 
 which becomes higher and steeper as the wave number and the frequency 
 increase. Thus the effective damping is a fimction of the exciting 
 frequency. 
 
 b) The Length Scale A 
 
 Equation (6) shows that }\ is needed as well as (3Vji/^t)^. 
 Now A is a length scale. It was defined, say in the x direction as 
 
 '^^ 
 
 
 sjid could be termed the equivalent length of perfect correlation. 
 
 There are various ways of evaluating the integral scale . Perhaps 
 the most convenient one for our purpose is that (found for instance in 
 Ref . 2 Eq. 115) which is derived from the relationship between correla- 
 tion functions and spectral functions. Thus if a stationary random func- 
 tion j(t) possesses a correlation function which is sufficiently well 
 behaved. 
 
 f(r) - ^^^^(t^^) 
 
 * Here resonance is defined as the maximum of the response curve l/y(k.) 
 holding^ constant. The locus of maximae holding k constant is given 
 by: 
 
 
 These maximae correspond only for zero damping. 
 
NACA TM 11+20 15 
 
 one cein define a spectral density function 
 Now for the particxilax case ^ = 0, this gives 
 
 Since 
 
 
 we have the resiilt that 
 
 2- 9Lo) 
 
 (10) 
 
 We have already obtained a formal representation for the spectral density 
 function of the plate. It is the integrand of Eq. (9) so we can write, 
 in view of (lO) 
 
 
 
 (11) 
 
 i»0 
 
 and a similar expression for Ay. 
 
■j_j^ MCA TM li^-20 
 
 III. THE FORCING FUNCTION 
 
 We suppose that a turbulent "boundary layer develops on the skin of 
 the airplane (on one side of our plate). The forces which excite the 
 plate are the pressure fluctuations experienced by the plate itself. 
 We assume that all characteristics of the boundary layer are fixed once 
 we have specified the boundary layer thickness S , the free stream 
 velocity U09 and density^ . In terms of pressure fluctuations this 
 implies that at a fixed point of the "wetted" surface of the skin, we 
 have for the mean square pressure fluctuations 
 
 and the integral scales, i.e. 
 
 U. 
 
 Jeo 
 
 ^' 
 
 
 u = ir, \'M..o,^.)\i>^^x.,o,±.-)i.i. 
 
 v^ 
 
 are proportional to O • Also the relative contribution to pressure 
 intensity of the various frequency bands must be a function of Ua9,<J, 
 
 is a random function of three independent 
 related to a three-dimensional spectrum by: 
 
 and J^ only so that if }f>0 is 
 variables, x,z,t, b^ is rel 
 
 K= rr.si'^^'- ,A..^.)i^^.^ 
 
 such that : 
 
 7r(^,4.> ^O'^'^'^fj''^^''^^'^) 
 
MCA TM li|20 
 
 15 
 
 Loosely speaking, this means that a characteristic frequency for pressure 
 fluctuations is proportional to Ujo /j and a characteristic wave length is 
 proportional to O . Now the forcing function of Eq. (7) is a force/unit 
 mass so that according to our similarity liypothesis 
 
 i 
 
 /.*U 
 
 4 
 eo 
 
 <r*h* 
 
 One can thus define a spectral function associated with the forcing 
 function f(x,z,t): 
 
 r° r r {" %(^'>-^-;'^)^-'>^-^ 
 
 " Jo J. 00 J-ao 
 
 and thus 
 
 -^1,^ 
 
 ?.(-(.., 4., co)= illr i'i T.(''.,'<vJl3 
 
 Here F2 is a fxinction of Ki, K2, and JL only and these are non-dimensional 
 variables: 
 
 Jlr UjS/yj„ 
 
 In terms of these non-dimensional units, Eq. (8) becomes: 
 
 (12) 
 
 fo'U* 
 
 30 , to CO 
 
 V ' ^-#. Clj 
 
 dM,(iK7.dJlT^.li<.y^^J^'^ 
 
 ! 
 
 £^(K.S..7."iJiT-^^^V* 
 
NACA TM 1^4-20 
 16 
 
 and Eq. (9) becomes 
 
 ?./^^ f'Ut rr ^■^''^^(K.,K^,s^y(.iJcUJl 
 
 0-)-iw)- ^ 
 
 h^i^ Jo J 
 
 ■ao J-o» 
 
 [^'^'-^^J-f.^'j'-I^A'- 
 
 which can be written 
 
 00 
 
 
 Again, F2 is a fimction of the integration variable only, so that one 
 can write 
 
 3) 
 
 dV^ 
 
 
 2>C / (5"^ 1^-^ I 5ac» (j«, J (Ik) 
 
 Equation (1*+) yields the two-non-dimensional parameters upon which the 
 
 plate dynamics depend. The first one, Ch /S \Jae> is the product of a 
 
 ,, , , , speed of free stream ^ , 
 
 Mach number, ( ^ — s z'- 7 : — TZ , . ) and a 
 
 ' ^ speed of propagation of waves in the plate' , 
 
 thickness ratio ( boundary layer thickness ^^ ^^^ ^^^^^^ ^^ ^6 /UoO 
 
 plate thickness ' 
 
 is a non-dimensional damping parameter which is, alike, a function of 
 plate and free stream properties . 
 
 If we treat the equation for the integral scale (Eq. 11) in the 
 same way, we notice that no new non-dimensional parameter occurs, so 
 that, at most 
 
 A 
 
 - f ^0 ' M.] 
 
MCA TM li^20 
 
 17 
 
 We now wish to Investigate the form of the functions H and L in 
 Eqs. (ih) and (15) respectively. First, we make an assumption which 
 is not strictly necessary but which simplifies the manipulation of 
 Eq.. (12). We take the function F(K, , K,,A) to be symmetric in K, 
 and K« , which leads us to define a new wave number. 
 
 Ks \l K.^-f Ki" 
 
 and to write 
 
 Tx(k.,Kx;/l) = T(K>-^;) 
 
 Thus, Eq. (13) becomes 
 
 iV-n \\ -LTV Jo Ui 
 
 feo 
 
 
 <T-^h' 
 
 rep 
 
 
 Now, the damping parameter / / >Jeo ^^ assumed small and under these 
 circumstances it can be shown (see Appendix II) that 
 
 00 
 
 ^dM. 
 
 Jo 
 
 CO 
 
 \<,dM. 
 
 
 (16) 
 
 The small difference between these two integrals can easily be evaluated 
 for arbitrarily small values of /i*/UeP even though both integrals are 
 lanbounded as /i-*-0. This leads us to believe that for low damping the 
 main contribution to the inner integral comes from the resonance condition 
 
 cJn 
 
 Thus, if the spectral function F(J1,k) is reasonably wide, i.e. if3F/^K«l 
 over a large range of K, Eq. (16) suggests that we write 
 
 n^.^)-T{4W^'^) 
 
 (IT) 
 
18 
 
 MCA TM 114-20 
 
 The requirement that F be flat In K when compared to l/X(K) is equivalent 
 to the requirement that the average correlation distance or integral scale 
 for the boundary layer pressure fluctuations be sina.1 1 coinpared to integral 
 scale of the plate deflection. Translated in physical terms the simplifica- 
 tion sxiggested here is prompted by the following remark: If the plate 
 has some stiffness, it makes little difference whether the forcing 
 function is assumed to be distributed over small distances or made of con- 
 centrated loads (see Fig. 3) 
 
 FIGURE 3 
 
 Thus a satisfactory model for the problem at hand woijld be the impact of 
 rain drops on a metal roof. Equation (l6) allows us to integrate over K, 
 to get 
 
 
 h^cp 
 
 (18) 
 
MCA TM 114-20 19 
 
 The expression 
 
 jVF(\/i^.A)aa 
 
 can be evaluated only >rtien F(K,JI) is known. It may be an increasing 
 or a decreasing function of ^U«/th • In the absence of data on the 
 spectral function F, we will noi attempt to define it. 
 
 The function H defined by Eq. (l4) can be written: 
 
 \ 
 
 17.' u^] " l'~l f"' l^^ ' '^'> 
 
 where f„ is an unspecified function related by (18) to the boundary 
 layer pressure spectrum. 
 
 In order to determine pj^^ ^g need to find out, in addition, what 
 quantities the integral scale \ depends on. Here we make use of con- 
 siderations which are similar to those yielding Eq. (16) (see Appendix III). 
 The result is that 
 
 MS ' .. 
 
 (20) 
 
 -(^)'i-i 
 
 where fo is another function related to the boundary layer pressure 
 spectrum by IIl(i+). Now we are able to write Eq. (6) as 
 
 Si 
 
 (21) 
 
 Here 
 
 h= i'f 
 
 Expression (2l)g.ves the functional dependence of pressxire intensity "inside' 
 on boundary layer parameters for a typical case. The only quantity, not 
 immediately available is h (^ /ct»)< It is probable that we shall have to 
 await experimental data to define its numerical value reliably. 
 
20 NACA TM li+20 
 
 IV. SPECIAL CASES 
 
 1. Convected Turbulence 
 
 Two authors (ref . 5 & 6) have recently suggested that the ■boundary- 
 layer pressure fluctuations at any point of the fuselage skin are caused 
 essentially by the passage over the point of a fixed (i.e. time inde- 
 pendent) pattern of pressure disturbances carried downstream at a fixed 
 (jonvective velocity. So far, experimental evidence in proof or disproof 
 is lacking. However, it is interesting to incorporate this special case 
 in the general formulation which has been presented. Both the response 
 of the plate and the coupling of the plate with the air at rest must 
 then be reconsidered. 
 
 a) The coupling of the plate with air at rest in the case of 
 convected t\xrbulence 
 
 If a fixed spatial pressiore distribution is carried downstream 
 on the surface of the plate, it is easy to show that the (infinite) plate* 
 response will be of the same kind, i.e., that it will consist of ripples 
 which are randomly distributed in space but which travel through the plate 
 at the same comvective velocity as the boundary layer disturbance. The 
 determination of the pressure field inside the fuselage is not in principle 
 different for this case and has been carried out in Appendix IB**. The 
 res\ilt is that for both subsonic moving ripples (with convection velocity 
 (J, < CL'i, ) and moderately supersonic ones: 
 
 K- ^^^T^'1 
 
 (I. 10) 
 
 A a.(a+ML3t'J 
 
 where 
 
 
 For higher supersonic speeds , the function of geometry and Mach number 
 appearing as an integral is more complicated. The eqviation (I. 10) above 
 has the same form as equation (6). On the other hand there is a sharp 
 difference in terms of energy radiated by the plate between the subsonic 
 and the supersonic case, since no energy at all is radiated by subsonic 
 ripples while the supersonic ones do generate some. One must, then, make 
 
 *Here the presence of transversal bulkheads will change the picture 
 because of multiple reflections of the ripple. 
 
 **This problem can also be viewed as a steady (randomly bumpy) wing 
 problem from the standpoint of a stationary observer. 
 
NACA TM 114-20 21 
 
 a distinction between the results in terms of pressure intensity (the 
 quantity of practical interest) and in terms of energy radiation. This 
 distinction stems from the fact that (as is pointed out on page 7) the 
 acoustical field investigated is truly a near field. 
 
 b) The response of the plate 
 
 According to the convective hypothesis, time is not aji independent 
 variable once the convective velocity U[| is fixed. Translated in terms 
 of the spectral density TT {to yk-^ f}^) °^ ^^^ pressure fluctuations, this 
 means that TT (^ Al^^2)i^ zero, except vhenUf ^ Utf^i> or in non- 
 dimensional form, when -^*(^'/(, ^K,■ We rewrite equation (12) for this 
 speciaJ. case . *' 
 
 Here a L UooJ ^^ the Dirac delta fxmction of the variable Jc. 
 
 Then the plate response becomes 
 
 
 - ^?o^ f*(TLl.\^ K.^F, (K.,lCv^JLU.<iJCt. 
 
 mu 
 
 (23) 
 
 [i^;(.w).(^jcp(p-j.y 
 
 where 
 
 F,{..,KO= F('^"'^"''''fe.) 
 
 Now if we assume as before that F is symmetric in K^ and K^ and 
 substitute 
 
 I 
 
 K=Vk.*+ Ki 
 
 9c Ur," ill. 
 
22 
 
 NACA TM IU2O 
 
 we finally get 
 
 (fff-^s^r-'^-'^'-'S^'' 
 
 (21^) 
 
 Here 
 
 and 
 
 with 
 
 and 
 
 3 = ^'/U^ 
 
 IC.% iCx^ r K 
 
 SU 
 
 w 
 
 The length scaled is indicated by the following dimensional argument. 
 
 *'= ^ 
 
 The mean correlation length or integral scale is a weighted average 
 of all wave lengths, so that dimensional 1 y 
 
 ^■^ i 
 
 Since resonance dominates the plate response, tt is given from the 
 plate response equation (eqiiation 23) by the resonance condition 
 
 or 
 
 A similar reasoning would have yielded, in the non-convective case, 
 "X/v^Ch^/y \^2, instead of eq. (20). 
 
 Combining {2k) and (25) according to (l.lO) we notice that we can 
 still write as in equation (2l). 
 
 ?iai ai c h*(i> (26) 
 
 Here A is a weak function of the Mach number as seen from (l.lO). 
 
MCA TM 11+20 
 
 25 
 
 2. The case of zero scale 
 
 Under some circumstances it is possible that the space average of 
 the plate motion vanishes, i.e.: 
 
 
 This does not mean that the normal accelerations at two neighboring points 
 
 show no correlation, but that the 
 
 correlation function becomes negative 
 
 as indicated in Fig. {h) and in such 
 
 a way that its space integral vanishes. 
 
 We can then consider the normal 
 
 accelerations as dlpoles rather than 
 
 sources and we are led to a slightly 
 
 different radiation problem. Appendix 
 
 IC shows, however, that if one defines 
 
 a length "^e such that 
 
 ■k; = r y Rji? 
 
 The results are a^ain identical in form with those of equation (6). Here ^ 
 caji be viewed as the mean moment arm of deflection moments. Alternatively 
 one can redefine the integral scale as 
 
 \>=cp|Tlj|Aj 
 
 (27) 
 
 where c is a constant. Equation (27) can thus be used to define the 
 integral scale in any event. 
 
2j^ MCA TM 11+20 
 
 SUMMARY OF RESULTS AND DISCUSSION 
 
 Appendix I discusses in addition to the cases mentioned In the text 
 a few examples which provide different limiting conditions. Thus the 
 observer is brought close to the plate (Y<'< ^ )• A short time scale is 
 
 considered etc The common feature of all these analyses is that the 
 
 resulting mean noise intensity can always be represented, say by eq.uati0n 
 (26). We sheill therefore retain this equation: 
 
 as the most general statement that we can make at the present time. 
 Here 
 
 b^ = mean square noise intensity inside 
 
 (?• = air density inside 
 
 ^\ = speed of sound inside 
 
 y^ = air density in the free stream 
 
 ^ = plate density 
 
 Ueo = free stream velocity 
 
 O = boundary layer thickness 
 
 'Xf) = plate thickness 
 
 jO = viscous damping constant (of units l/time) 
 
 Y = perpendicular distance between observer and fuselage 
 
 (3 = geometry of the plate 
 
 Mi, = I'lach number ^'/^' 
 
 V\ = convective velocity of turbulence pattern 
 
 4- 
 
 ^ = characteristic velocity in the plate 
 
 3(r(.-y40 
 
MCA TM li4-20 25 
 
 For all but high supersonic velocities, the dependence of ^ on M^^ is 
 quite small and can be disregarded. The function 2^ (Y,g), a quantity 
 which does not depend on the dynamics of the problem but only on its 
 geometry should be modified to take into account the fact that the 
 fuselage is a cylinder and not a large flat plate. 
 
 The form of the function S cannot be given here both because no 
 information is yet available on boundary layer pressure spectra and 
 because S d epends too critically on the type of model assiamed. How- 
 ever, if bjl is measured while any one of the four variables defining 
 S {S ,iiao , c or h) is varied, then the functional form of the noise 
 intensity inside a fuselage can be determined. Thus the main contribution 
 of the analysis is to diminish the extent of the testing required. 
 
 One of the conclusions which can be drawn from the foregoing equation 
 is that unless the boundary layer pressure spectrum is a ver^ sharp 
 function of frequency (which would make S very sensitive tooUtt/ch ) it 
 is not practical to decrease cabin noise by boundary layer suction: Since 
 the noise intensity is a weak function of boundary layer thickness, 
 decreasing appreciably cabin noise would involve the removal of a 
 prohibitive amount of air. 
 
 Another conclusion is that increasing the damping is a very effec- 
 tive way of limiting the production of noise of all frequencies, since 
 the structure transmits sounds essentially by resonance. 
 
 The analysis which has been presented deliberately omitted some of 
 the features of the problem which would influence the results and intro- 
 duce new parameters. For instance, the fuselage of commercial airplanes 
 is usually subjected to an axial tension as well as other loads. In 
 addition the skin is curved. To account for these featiires of the prob- 
 lem one would introduce further terms in the differential equation 
 describing the plate and one could treat it in much the same way as 
 has been done here. 
 
 The general methods which have been used are adaptable in addition 
 to the study of a germane problem, the fatigue of panels which are 
 buffeted by a turbulent boundary layer. 
 
 A NOTE ON TESTING 
 
 The discussion of the various limiting solutions makes it clear 
 that for the transmission of boundary layer noise through a structxire, 
 the ratio of outside (boundary layer) noise to inside (cabin) noise is 
 in general a function of boundary layer as well as structural character- 
 istics. 
 
 This is to say, first, that an attenuation coefficient cannot be 
 defined by testing the sti-ucture alone with a standard noise source. 
 Thus accurate testing reqiiires at the outset that the model be tested 
 
26 NACA TM ll^■20 
 
 for transmission of a noise similar to boundary layer noise. The main 
 property of such a noise, as we have seen is that it must be rajidom in 
 space as well as in time, which precludes the use of one or a few 
 concentrated so\irces as noise generators. The only proper substitutes 
 for boundary layer pressiare fluctuations are forcing functions whose 
 effects on a fuselage are local.* The impact of water drops for instance 
 might be found adequate simulation. Further, similarity in testing 
 requires the matching of parameters which are ratios of plate and forcing 
 function properties. For instance if the forcing function used in the 
 test is a turbulent boundary layer, similarity parameters are: 
 
 
 *This is not true of jet noise which is generated away from the fuselage. 
 
NACA TM li4-20 
 
 27 
 
 VI . REFERENCES 
 
 1. Lord Rayleigh: The Theory of Sound. Dover Publications, 
 New York. Volume II (194$). 
 
 2. Liepmann, H. W.: Aspects of the Turbulence Problem (Part l) 
 ZAMP, Volume III (1952) pp 321-342. 
 
 3. Liepmann, H.W. : On the Application of Statistical Concepts 
 
 to the Buffeting Problem. Journal of the Aeronautical Sciences 
 19 (1952) pp 793-800. 
 
 k. Timoshenko S. and Young, D. A.: Advanced Dynamics. McGraw-Hill, 
 New York (19^*8). 
 
 5. Ribner, H. S. : Boundary -Layer - Induced Noise in the Interior of 
 Aircraft. UTIA Report No. 37 (April 1956). 
 
 6. Kraichnan, R. H. : Noise Transmission from Boundary Layer Pressure 
 Fluctuations. To be published. 
 
 7. Corcos, G. M. and Liepraann, H. V/. : On the Transmission Through 
 a Fuselage Wall of Boundary Layer Noise. Douglas Report No. 
 SM-19570. (1955) 
 
28 NACA TM li+20 
 
 APPENDIX I 
 
 THE RANDOM RADIATION OF A PLANE SURFACE: 
 
 A. FOUR LIMITING CASES 
 
 In order to determine the coupling between fuselage vibrations and 
 cabin air one has to choose a model for the correlation TJS between the 
 normal accelerations at two different points of the plate. The model 
 which was discussed and for which equation (k) was made plausible is 
 predicated upon two conditions: 
 
 A. That the observer is distant enough so that a large number 
 of plate elements vibrating independently contribute sound 
 in comparable amoimts, i.e. 
 
 Here as before, A. is the integral (length) scale for the plate 
 normaJ- accelerations and Y is the perpendicular distance between 
 the observer and the plate . 
 
 B. That the time scale of the phenomenon is large enough so that 
 the differences in phase (introduced by the unequal distance 
 from the point at Y to the various points of a plate element 
 of length ^) are unimportant, i.e. 
 
 X « a lie* 
 
 a-j^ is the speed of sound in the fuselage air, and is the 
 integral (time) scale for the phenomenon: 
 
 . j;^^(WO^(-;*^^ 
 
 ( ^\^ 
 
 I ot j 
 
 Then one can choose a simple model for the correlation function 'w' 
 
 where d is the delta function. The normal accelerations are assumed 
 perfectly correlated \id.thin a length 7v and not at all for distances 
 
NACA TM li+20 ^^ 
 
 greater than A • Then 
 
 
 and upon integrating 
 
 a^'. a.S^a^,^fc^, X[^:-^J^Si^z.'.r.;^^(r^ 
 
 \>i'- = Z:LiL(^\* (fi^iii 
 
 which is equation (k) . This caseYijX « Y ; l) ^ ^< Al ^^H J 
 
 corresponds to the following conditions. The passenger (or the microphone) 
 is far from the plate (in terms of ^ ) , the boundary layer is thick and 
 the airplane velocities low. One may well wonder about cases for which 
 these conditions do not apply. V/hile it appears difficult to answer such 
 a query with generality it is possible to consider other limiting cases. 
 
 For instance let us assume that condition 2 still applies but that 
 our observer is extremely close to the plate. This would correspond to 
 the following physical case: A thin fuselage skin^ a thick boundary 
 layer, a low airplane velocity and we are measuring noise by placing a 
 microphone very close to the skin and insulating it on all sides except 
 the side which faces the skin. Then 7S>7'Y • 7^< Ai,M^* • 
 
 - BOUNDARY LAyeR TA//CK/vess 
 
 fezz^^^^^^'^^^^^szzzzz^^^^,^^^ 
 
 A^/CROPHC3f/E 
 
 FIGURE ^ 
 
 Under these conditions the noise at the microphone is contributed primarily 
 from a single plate element which in the average vibrates in phase. The 
 evaluation of this contribution is particularly simple. We can witite, very 
 nearly 
 
50 
 
 MCA TM li+20 
 
 1^ 
 
 (^-^'''t-5:0lFl->-^-^^.)'-r^r 
 
 (1.2) 
 
 If, for the sake of definiteness, we assume the element circular, then 
 
 w^' f'^drri^K^-Y) 
 
 and since A ^7 Y it is permissible to write 
 
 K=A'?^f^) 
 
 The pressure intensity is therefore given as 
 
 
 (1.2) 
 
 Thus Eq. (6) applies for the very close as well as for the very far 
 field when phase effects are not important {7\«-t&a^) • 
 
 Now assume that we carry on the same experiment but that the boundary 
 layer is thin and that the velocity of the airplane is high so that the 
 exciting frequencies are high. Let us assume in addition that the skin 
 is thick, so that Y <4. X • '^ ^^ ^/^L . 
 
 FUS£LAGE BOUh/DAF.y i.Ay£R 
 
 ^aSSLAGE 
 SKIf/ 
 
 MICROPHONE 
 
 FIGURE 6 
 
MCA TM li+20 51 
 
 Now the time scale of the plate motion is short and phase effects are 
 prevalent. We define a simple time history in analogy to the space 
 description of Eq. (l.l) 
 
 
 The microphone still receives signals effectively only from one plate 
 element and all points within that element vibrate in phase but the 
 pressure pulses originating from that element do not arrive at the 
 microphone in the same time. Then: 
 
 (1.3) 
 
 A' is simply\ . Equation (1. 3) is evaluated by noticing that: 
 
 \\li)'[ii^)Vi- Mill 
 
 Here ^^ are the real roots of g( ^ ) = which are included in the interval 
 between and and b . We only have one root , namely rn = r2 • If we choose 
 to integrate, say, with respect to S2 first we get (assuming again that 
 the element is circulax 
 
 (lA) 
 
 
 (1.5) 
 
 A' - M' 
 
 and according to (l.^) the inner integral yields; 
 \ 
 
 5(^^) 
 
 y*Af. 
 
 '' I '^^ 1 V.?. L §* ) ^ 
 
52 
 
 NACA TM 114-20 
 
 so that 
 
 K 
 
 ft ai*^* 
 
 ( m^\ '-^ 
 
 = ?:.:^(|^f(,[^^-Y] 
 
 "bM^ 
 
 p ^ f.v-.'^MIr) 
 
 (1.6) 
 
 The time scale appears explicitly in the answer. For the vmbounded plate 
 however it is simply proportional to S /^oo just as the time scale for the 
 boundary layer pressure fluctuations. 
 
 Finally we may consider a physical case for which phase effects are 
 important ajid for which the microphone has been placed a large distajice 
 away from the plate . i.e.: t^ 4^ >» /jj^ ) "Ts C<, ^ 
 
 BOt/A/OA/iY 4.AYER 
 
 J^aS£i.AS£ SM/A/ 
 
 MlCROPH0A/£ 
 
 FIGURE 7 
 
NACA TM li+20 35 
 
 Now the contribution from each sub -element of vibrating plate is still 
 in the average independent from that of the next one. However, there are 
 in addition cancellations from within one element just as in the previous 
 case. This will happen if the boundary layer is thin, the airplane 
 velocity is high if&i small) and the observer is far from the fuselage wall. 
 
 In order to evaluate this limiting case we first specify the time 
 behavior of the correlation function: we write 
 
 
 so that 
 
 A A 
 
 and we integrate first with respect to P~ . Using the same techniques as 
 in the previous example , we get : 
 
 ft o '••I 
 
 Now we assiarae that 
 
 Tjf (0,8.-6.) =/^-J^f^S,(6.-6v)jXLo) 
 
 Integrating with respect to Og, 0, , O, successively: 
 
5^ MCA TM 11+20 
 
 For a circular plate of radius R, this would give 
 
 Pr; Zl i:^\^ CO) X(o)a; f ii da 
 
 S(tr--'"(-M 
 
 In general, and defining 
 
 K^.vo=( i^ii 
 
 a function of the plate geometry and of the distance Y only, we have 
 
 g^~.(^^)\^Xl,(,.V) 
 
 (1.8) 
 
NACA TM 11+20 55 
 
 APPENDIX I 
 B. THE NOISE GENERATED BY SKIN RIPPLES OF FIXED VELOCITY: 
 
 If the turbulence pattern is frozen, as discussed in section IV-l 
 ripples will travel through the (infinite) skin at a fixed convective 
 velocity. Then the correlation function 'UT must be vrritten differently: 
 
 where \)t is the speed of propagation of the ripple (turbulence 
 convective speed) and therefore 
 
 or 
 
 
 where now 
 
 
 The inner integral is of the form 
 
 \tis]'\^ls)\H 
 
 which can be written 
 
 
 as in part A. The expression 
 
 ^/^)- [['='.' ^''O-l^''"^-^'^] 
 
 (1.9) 
 
36 NACA TM 11+20 
 
 has either one or two real roots depending as ''\,<^ or r'yj>\ respectively. 
 For M 4 V J "tlie only real root is 
 
 Pi *» r-j, 
 
 ajid thus the inner integral yields 
 
 I 
 
 ;li + HiXi 
 
 so that : 
 
 
 n.. (ii+rtpc/) 
 
 (l.lOa) 
 
 notice that equation (I. 10a) above tends to equation {k) for low 
 convective speeds. 
 
 For ^i>| ) (l'9) has, in addition to the root X, « ?C), , another 
 
 root given by 
 
 Mr- 1 
 
 It is easy to show that this root exists for all values of x-j*. In order to 
 simplify the integration let us assume slightly supersonic conditions; i.e. 
 let us write 
 
 where 
 
 £«.» 
 
 Then 
 
 and 2 
 
 t 
 
 «l 
 
 * -!1* [r,*xl) 
 
NACA TM 1^4-20 
 
 57 
 
 and the inner integral 
 I 
 
 ti f Mgx-v 
 
 1%« '»L 
 
 so that for the supersonic case : 
 
 M.(r.^ M.x.'J 
 
 (UJilV, 
 
 
 (I. 10b) 
 
 A result which is save for a constant coefficient the same as I. 10a. 
 
Q MCA TM 1420 
 
 APPENDIX I 
 C. THE GENERATION OF NOISE BY PLATE DEFLECTIONS OF ZERO SCALE 
 
 If the space average of the correlation function is zero and if 
 the plate vltrations are isotropic in x' and z' one can define a new 
 length scale as a moment arm: 
 
 3-U^'' '*'') ( ^« is a fixed point) 
 
 alternatively one can define a modified integral scale 
 
 'to 
 
 
 Here 
 
 where c is a constant. 
 
 Then, one can idealize the correlation function as 
 
 provided 
 
 It follows that 
 
 and integrating with respect to | 
 
 - CM ^ I ^^^' S?-4!' 9['^y>'0 
 
 Now, iinless a or b = 
 b 
 
 \JNn^]h-lii^^'b)^% 
 
MCA TM lit-20 39 
 
 and thus 
 
 
 If there is a (time) microscale 
 and 
 
 F^^-S'L^^^->['-^^-'-^-M 
 
 ?^^ 
 
 
 Here the plate has been assumed circular and R is its diameter. 
 
1^0 
 
 NA.CA TM II4-2O 
 
 APPENDIX II 
 
 THE SIMPLIFICATION OF THE PLATE RESPONSE INTEGRAL 
 
 (Equation I6) 
 
 We consider the approximation equation 
 
 m^s-^m]''' 
 
 kcLk 
 
 4Xf 
 
 •Slice 
 
 Sr^' 
 
 W 
 
 The right-hand side is clearly iinbounded as the damping constant /3 - 
 since its value is explicitly proportional to l//3 (see for instance 
 Eq..(l8)). On the other hand the difference between the left and the 
 right-hand integrals is finite for ^ = 0. To show this we write 
 
 I = 
 
 :Kk' 
 
 §[ioo 
 
 Then the left-hand side becomes for /9 = 
 
 S\i 
 
 00 
 
 •Ma 
 
 (r-i)' 
 
 and the right-hand side becomes for 
 
 ^ = 
 
 (III) 
 
 (16) 
 
 cflico 
 
 poo 
 
 Jo 
 
 df 
 
 {$-'}' 
 
 Now 
 
 (112) 
 
 (^^-0^% I a-i/"" a-o^'^a-D cr-o 
 
 (113) 
 
 so that the difference D between expressions (ill) and (II2) is 
 
 or 
 
 
 r 
 
 Schn 
 
 re + 
 
 I 
 
 I 
 
 (^1/ ^ (^.1; (?-') 
 
 (11^) 
 
NACA TM 114-20 ^^ 
 
 This expression is finite and of course independent of /? so that we can 
 conclude that the left-hand integral of (l6) is unbounded for /3 = 0. 
 Further, it is clear that D is a regular 1 unction of /3 so that the ratio 
 of the left-hand side to the right-hand side of Bq.. (l6) can be made ar- 
 bitrarily close to unity, by choosing arbitrarily small ^ . If a correc- 
 tion is desired a numerical check indicates that Eq. ijlh') gives a good 
 approximation to the error made even with moderately large damping. 
 
k2 
 
 KACA TM 1420 
 
 APPENDIX III 
 
 THE EVALUATION OF THE INTEGRAL SCALE 
 
 Our starting point Is Eq,. (ll). In terms of non-dlmenslonal variables 
 It becomes 
 
 We now simplify the denominator by writing successively 
 
 (nil) 
 
 Ki«0 
 
 (III2) 
 
 Then we define 
 
 ch <!. 
 
 A > >r'K.'' 
 
 Eq\iatlon (III2) can now be written 
 
 and If we replace Kq by its value at resonance,* namely 
 
 * The justification for that step Is identical to that advanced in 
 Appendix II. 
 
NACA TM 11^-20 
 
 45 
 
 We can write 
 
 
 reo 
 
 »o 
 
 
 
 K.tfO 
 
 
 d/i 
 
 (III3) 
 
 If we compare (III3) to (l3) we get immediately 
 
 
 
 (IIIM 
 
 NACA - Langley Field, Va. 
 
• ii a> 
 ■*-• ■ — I rt 
 
 ■a 3 
 
 C M 
 
 3 
 
 
 § i s ^ s 
 
 aS§oo.so^gd 
 
 rt ■::; h 
 
 en 
 
 Q 
 
 Z 
 
 o 
 
 n 
 
 IS 
 
 ■^1 
 
 o H 
 O P 
 
 >-£S 
 
 o og 
 CO 2 E 
 
 •* > 5 
 
 ^ -o o 
 
 < rt Z 
 ZZ O 
 
 O ' 
 
 S " - 
 P-2S 
 
 <a . 
 
 W CO 
 
 9 S S 
 
 m rt '" 
 
 O o " 
 5^ • (I) o 
 
 w ^ 
 
 < U o 
 
 S-fcJ Ih 
 
 
 M-l 
 
 ' >. > T3 
 
 ) -r t« 2 
 
 T* TO ra e 
 
 s» ?< 
 
 4-> 
 
 q 
 
 
 X 
 
 XJ 
 
 
 
 H 
 
 u 
 
 o 
 
 0^ 
 
 
 3 
 rr 
 
 CO 
 
 < 
 
 o 
 
 (1) 
 
 H 
 
 
 •o 
 
 to 
 
 CO 
 
 CO 
 
 a; 
 o 
 
 
 (0 
 
 XI 
 
 
 o -a 
 
 2 3 
 
 ■3.2 
 
 Q.S 
 
 c o 
 
 - ^ " 
 
 C CO 3 
 
 — CD C 
 3 bD— ' 
 XI rt (0 
 
 « t'. ^ 
 
 c:S.S 
 o ?* 
 
 S ;,.2 
 
 ^ C > I* 
 
 2 ° CD c« 
 
 "1 2 Si c 
 
 2 § " § 
 
 
 — g i: 3 
 
 O iJ't ^ 
 
 — c< C 'c 
 
 ■a c; S o 
 
 a 5i u c3 
 
 3 C » 5 
 
 m 5 <u 
 
 rt «£ m 
 
 CO (fl "O a> 
 
 CO 
 
 . a> CO N 
 
 g^.2 ■? 
 
 o gi: 
 
 O T3 
 
 .5 •" o 
 g so a, 
 
 <D OJ O tV 
 
 03 eit a 
 
 ^ be V 
 
 T3 J3 
 
 h 
 
 cs a J2 
 
 V 
 
 — „ nl 1- 
 
 S£ 
 
 CD QJ O ^ 
 
 i; w »Zh XI 
 
 
 S B 
 
 
 0) o < 3 ■ 
 
 >o<uzuj?zqS 
 
 en m U 
 
 u UJ CO 
 •13 fH P 
 
 1- ^ M 
 
 <oa 
 
 , ffi CO 
 
 oP2 
 
 — H " 
 
 Qj g CO 
 
 l§° 
 
 S g W 
 
 S-z w 
 
 ^ o H ^ ' 
 
 U - •** ' 
 
 < c« Z K 
 
 z z o< 
 
 < ■■3' 
 
 3 ~^ 
 
 o 
 Q . 
 
 CO 
 
 -in 
 I o> 
 
 a '^ 
 S ^ 
 
 c ^ 
 
 Ci3 _< 
 
 aZ 
 
 " c 
 
 li 
 
 T3 " 
 
 So 
 
 T3 <" 
 c *-• 
 
 o 5i 
 ■a E 
 
 c< 3; 
 
 Cfl rt 
 
 2 en 
 
 §•5 
 
 2§' 
 
 o 
 
 u .2 
 
 o o ■3 
 
 •3.2 
 
 ° 2 
 
 " a 
 
 ig s 
 
 t 2^ 
 
 ^ c^ =: 
 
 t^ X! 
 Q ■" Oj 
 
 >. > 73 
 
 -r «'i 
 
 i. ■- -S 
 
 •a- § . 
 S c o ■ 
 3- CO 
 
 *- .^ CJ 
 
 = » -3 • 
 
 cu ~ 
 
 3 bO — 
 
 ■? J3 CO 
 
 « '!^ ^ 
 
 c -S s 
 
 r: o 2 
 
 3 11) 
 
 Sg£ 
 
 t. .2 -a 
 cd .n c 
 
 0) cd C4 
 
 □■ 2 
 
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UNIVERSITY OF FLORIDA 
 
 DOCUMENTS DEFV\RTMENT 
 
 1 20 MARSTON SCIENCE UBRARY 
 
 P.O. BOX 117011 
 
 GAINESVILLE. FL 32611-7011 USA 
 
 T. «. 1420 
 
 •.,.;;.•, .^\0. LAB. 
 1 CE.NTER 
 
 BUlLUlNii 704-W 
 KOoS.^OUMT, MtN!JE33TA 
 
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