ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Status Report THEORETICAL AND EXPERIMENTAL STUDIES OF THE REFLECTION OF SOUND FROM ROUGH SURFACES AND THE PROPAGATION OF SOUND IN SHALLOW WATER Julian R, Frederick William C." Meecham Irene V. Schensted James; E'. Lesch Project 1936 OFFICE OF NAVAL RESEARCH, U.S. NAVY DEPARTMENT CONTRACT N6 onr 23221, PROJECT NR 385-203 April 1956

I~~~~~~~~~lh

TABLE OF C oNm TFJTS I... ODUCTION 1 II.o A THEORETICAL INVESTIGAT!01 OF TIE RF2'IECT ION OF RADIATION FROM PERIODIC ANrD IRREGTAR SURFACES by William. C. Meecham...... O O.. OO 4 2.1. On the Use of +.`the Kirchhoff Approxima.ti..on'for the Solution of Re:flection Probloems O O. O O O O O 4 22o. A Fouriexr Transfoxnm Method for thle Treatmernt of the Pro bl em- of thez Refl.ect ion. of Radi.a-t ion from Irregu.lar S urface3 e. Q. Q O. 5 2, 3 A Var iat io:nal Method for the th.Calcul.ation of the Distr ibution. of E.enrgy Refle1.cted from. a Per iodic Surface o o * o o o. e 5 III. AN IN5V'ESTIGAT lON OF'f.:THE PROPAGATOi' OF SOTUJImID FROM A POINT SOURCE TN A LIQUID IAYER OVER CONCRETE.......... 7 3 5 O Model Progra........................... 7 3.1. Theoretical Conside:rationi.s'by, Irene V Scheinsted., n o o o o o c 8 -.,,1 o. Ca.lculation s Based on tle'ormal1 Mode Repc:c-: sentation of the Solution............... o o The Rigid Bo3ttom o....... 9 The Two Fluid Model. 1 l The Solid Elastic Bottomrl Model.o o.o o.o e. 23 Summa:ry and. Conc-lusiicn to Section 3,1,1 o,.o.31 35o1.2 The Propagat;ion o:f Pa lses.......... 3 An Approxinate Method Using Ntormal Modes, o,. 33.5 Imagfe Series Solution........O. 37.... ii

TABLE OF CONTENTS (Conrt inued) APPEND1IX to Section 3.1 The Effect of Finite Thickness of the Concrete on the Propagation of Sound in the Fluid Layer.. 4. 45?BIBLIOGRAPHY to Section 31..............47 3.2. Experimental In:rvestigation of the Propagation of Sound in a Shallow Water Model by J.oE. Lesch and JoLR. Frederick..........48 3.2.1, Equipment and Techniques............48 3.2o2. Continuous-Wave Experiments.O.O. e52 4 Kcps... o o.. o.... 52 Frequencies below 4 Kcps...........56 3o523. Pulse Experiments................ 66 Wall Reflect ions........66 Effect of Variations in Pulse Length...70 One Millisecond-Pulse Experiments....... 72 3.2.4. Study of Bottom Characteristics.........83 IV. SUIMARY AND C ONCLUS IONS.............86 4.1,1. The Acoustic Characteristics of the Tanks...86 4.1o2. The Effect of Other Factors..........87 4.103. The Propagation of Soutid in Shallow Water...91 B IBLIOGRAPHY....................... 94 1 JZ1

LIST OF TABLES Table Page I Amplitude reflection coefficients for various angles of incidence................. 46 II Calculated cut-on frequencies f for different types of bottom. cc = 12,300 ft sec, cs = 7,070 ft/sec..59 III Calculated cut-on frequencies f for different types of bottom. cc = 16,250 ft sec, Cs = 8,960 ft/sec.. 59 IV Ratios of amplitudes of direct signal to side wall reflection for pulses at 4 and 10 kcps and various depths.............. 67 iv

TLIST OF F 1GURES F IGUE TI TLE PAGE 1 The Geometry of the Model.,,..,.. 9 2 Path of Integration for the integral, rep:resentation of the soluti;on of th. propagation problem... 11 3 Calculated. acoustic pressure variation with range for'the rigid. bottom model.............. 13 4 Calculated. acoustic pressure variation with range for the rigi.dbottom model employing an extended source. Source length is 0o2 2................... 15 5 Geomet;ry for the two f luid model........ 16 6 Position of the poLes and the branch line for the integrands of the integral representation of the solution when c2 > c1................ 18 7 Positionr.ing of roots xn for the two fluid case for different values of c2............. 20 8 Calculated acoustic pressure variation with range for a two fluid m.od.el......... 21 9 The Geometrical picture for the solid elastic bottom model i............. 23 10 Positioning of roots Xn for the elastic bottom period. equat ion.., o o o 27 11 Calculated. acoustic pressure variation with range for an elastic bottom model.. cc 12,280 ft/sec s 070 ft /sec........ 28 12 Calculated. acoustic pressure variation with range for an elastic bottom. cc - 13,780 ft/sec, cs = 60900 ft/sec. 29 1.3 Calculated acoustic pressure variation with range for an elastic bottom~ c -- Wcs. c = 12,300 ft/sec... 30 v ~ s ~

LIST OF FIGtJRES (Con:tinued) F IGURES T ITLIYE PAGE 14 Calculated pulse shapes using a normal mode first order approximation for a rigid bottom. j = 59600/2t- radians/sec, = 05 ft, H 2O..... 36 15 Locationu of inages for a fluid layer bounded on the one side by a free surface and on the other by a rigid surface 37......... o. o 16 Summation: of the image and. normal mode serials... 39 17 Constructive interference of images to form modes for a rigid bottom.......... 41 18 A comparison of the integration of the time pulse by the image series and the approximate normal mode method O. O. 0 0 0..... 42 19 Calculated amplitude reflection coefficient, R, for plane waves incident at an angle @ upon the plane interface between. a fluid and an elastic solid.. 44 20 Plane wa-v incident upon a fluid layer; c2 12,800 ft/sec, c -= 4,800 ft/sec o01/02 -- 0.439 L 6 inches.. 45 21 Block diagram of apparatus....a.. 50 22 Photograph. of ta:nk.2. The electronic equipment is at the far end of the tank. The hydrophone is suspended from the trolley riding the track mounted adjacent to the catwalk........ 51 23 Acoustic pressure -variation with range. Frequency = 4 kcps, H/2 = h - dh =-. Theoretical curve is solid line. Measured. curve is dashed lineo....... 55 24 Measured sound intensity variation with range at different frequencioes. H/2 - h = d - X(4kcps) The correction for cylindrical spreading has been made... 61 25 Continuation of Fig. 24................ 62 26 Measured pressure amplitude variation with range for different pulse lengths. Frequency- 4 kcps, Hl/2=h =d"% -o o oe o o o oo o o 71 27 Calculated variation of arrival time with range for several even numbered top- and bottom-reflections and the direct path signal. Pulse length = 1 ms, frequency 4 kcps, H/2: h - d - 1.5 fto Length of vertical bars indicates spread of data..... O e. 73 ao

L:IST OF F IGUEES (Conc:luded) F IGURE TITLE PAGE 28 Ray diagram for so'rlmd paths where n -: 4 (2 top and 2 iT3ottom reflections)............ 74 29 Photograph of oscilloscope screen showing received siagal. at a ra;rnge of 1 foot, Pulse length. — 1 ms, freque:lacy - 4 kcps,: 3.7:ft, h: - d = — 084 ft.. o. 76 30 Photograph of oscilloscope screen showing received sign.al at a range of 17 ft.o Plulse length:: 1 ms, frequenc~y =4 kcps, 9:-: 3T 5 ft, h - d_ = 8o84 ft..... 76 31 Ray diagram f or souh:~:d paths where n j nD 4 (n+l)UT 5, (n'l)n D: 3 0 5.. 0 0 77 32 Measired pressiure- anpli:tude'variation with range of 1. ms, 4 kcps pulse for n - 4 ( 2 top and 2 bottom reflect ions )..... 79 33 Photograph of oscilloscope screten showing example of bottom transmitted early arri val (see arrow). Electrical pick-up at extreme left is not to be confused with the early arrival.................. 82 34ab,c:Effect of sl.igh.t vari atil ons in water depth, receiver depth, and frequency................. 88'\.J % i

I. INTRODUCTION The purpose of this report is to review the research effort as it has progressed in the period from 1 August 1954 to 31 December 1955 on contract N6 onr 23221, Project NR 385-203. During this period a significant change has occurred in the type of research employed on this project in the study of underwater sound propagation. Heretofore the main emphasis had been placed upon statistical and theoretical analyses of certain types of underwater sound propagation data and submarine contact data as collected by various naval and Navy-sponsored activities. The analyses were made to investigate the role of certain factors in the environment on underwater sound propagation and target detection. As will be evident in this report there have been two major divisions of the research conducted over the past year. The work has followed closely the outline submitted in a contract renewal proposal dated in November of 1954. The two major divisions are listed below. a. Theoretical studies of the scattering of radiation by rough surfaces. b. Theoretical and experimental model studies of the propagation of sound in shallow water. The theoretical work devoted to the rough surface effects is briefly reviewed in section 2.0 of this report. A separation of this from the theoretical approach to shallow water model studies is made for the reason that the rough surface material is not directly related to the model study. In addition, the rough surface work has in large part been accepted for publication

-2in various scientific journals. No experimental effort has been expended during this period on rough surface scattering effects by this project. In section 3.0 of this report both the theoretical and experimental phases of the model program are reviewed. For the most part the model program has dealt with situations where the water depth is equal to a few radiation wave lengths. Underwater sound propagation has been studied over the concrete tank bottom using both continuous-wave and pulsed signals. Acoustical constants of the bottom and the effects of changing frequency and water depth have been examined. More specifically section 3.1 contains the theoretical treatment of the following subjects -- a. acoustic pressure as a function of range (normal mode approach) for a harmonic source and three different bottom types, b. effect of an extended source on the field pattern, c. prediction of pulse shape as a function of range for short pulses, d. comparison of an image theory versus normal mode theory for prediction of pulse shape, e. bottom reflection coefficients for specific conditions. Section 3.2 contains a description of the model experiments and a discussion of the results obtained. The experiments were designed primarily to explore the characteristics of the model for water depths of the order of magnitude of the wave length. The section is broken into subsections treating the following subjects -- a. equipment and techniques, b. continuous-wave experiments, c. pulse experiments,

-3d. bottom characteristics 9 e. summary and conclusions.

II. A THEORETICAL INVESTIGATION OF THE REFLECTION OF RADIATION FROM PERIODIC AND IRREGULAR SURFACES W.C. Meecham The results of work on this class of problems may be collected under three main subject heads. The major part of the effort expended during the period of time covered by this report has been devoted to the consideration of the variational method (see below). These three topics are covered in detail in separate reports under the titles given. Only brief summaries of these reports are given here. 2.1. ON THE USE OF THE KIRCHHOFF APPROXIMATION FOR THE SOLUTION OF REFLECTION PROBLEMS' The methods of physical optics can be applied to reflection problems under certain circumstances, As is well known these methods are based upon the Kirchhoff approximation, which is an assumption concerning the value of the radiation field near the reflecting surface. It is assumed in the approximation that the field near every region of the surface is essentially what it would have been if the surface had been flat with a slope equal to that of the irregular surface at the point in question. Using this assumption concerning the value of the field at the bounding surface in conjunction with the Helmholtz formula it is possible to obtain an estimate of the field inregions removed from the reflecting surface. It is shown that this assumption is valid if the following two restrictions hold: i) jdM |(< (i! ii) kRm >> 1 -4

-5where ~dx is the maximum value of the slope of the reflecting surface, dx k = 2A/X where x is the radiation wave length, and Rm is the minimum radius of curvature of the reflecting surface. Furthermore it is shown that the results obtained by the use of the Kirchhoff approximation satisfy the reciprocity theorem. 2.2. A FOURIER TRANSFORM METHOD FOR THE TREATMENT OF THE PROBLEM OF THE REFLECTION OF RADIATION FROM IRREGULAR SURFACES A method is presented which can be used for the calculation of the distribution of energy reflected from irregular surfaces. The formulation is useful for the first boundary value problem and can be used in either two- or three-dimensional problems with any given incident field. The solution is reduced to quadrature with negligible error when the average square of the slope of the reflecting surface is small and when the wave length of the incident radiation is not small compared with the displacement of the surface from its average value. A numerical example is worked, the sinusoidal surface, and is compared with experiment and with a method due to Rayleigh. It is found that the Fourier transform method is preferable to previous methods, notably those which can be classified as physical optics (such as Rayleigh's) since the error in the transform method is of second order in the surface slope whereas the error in previous methods is of first order in the samequant ity. 2.3. A VARIATIONAL METHOD FOR THE CALCULATION OF THE DISTRIBUITION OF ENERGY REFLECTED FROM A PERIODIC SURFACE A variational method is presented which is used to calculate the energy appearing in the various diffracted orders set up when a plane wave is incident upon a periodic reflecting surface. Either the first or the second boundary condition can be so treated. A sample problem is worked showing

-6that if the average absolute slope of the reflecting surface is small (segments of surface with large slope may be included) and if the displacement of the surface is not large compared with the wave length of the incident radiation, the formulation gives results correct to within a few per cent. The calculation shows clearly the existence of Wood anomalies; these are discussed in some detail.

III. AN INVESTIGATION OF THE PROPAGATION OF SOUND FROM A POINT SOURCE IN A LIQUID LAYER OVER CONCRETE 3.0. MODEL PROGRAM For the purpose of gaining a better understanding of the mechanism of sound propagation in shallow water an acoustic model program has been initiated. By this means it is desired to secure a borader knowledge of the fundamentals of some of the simpler shallow water problems. If this can be done both by experiment and by theory the plan is to turn to progressively more complex situations. It is hoped that the model may serve as an analogue computer in the investigation of phenomena too complex for adequate theoretical treatment. This section of the report describes experimental and theoretical approaches to some of the less complex problems, It contains descriptions of the use of a concrete tank model in experiments and examples of the application of normal mode theory to these problems. Shallow water is defined for the purposes of this report as the depth at which sound propagation is best treated theoretically by normal mode theory (as opposed to a ray treatment). The effort has been confined to the study of propagation in situations where the water depth is approximately twice the radiation wave length. The ranges vary from a few up to 40 or 50 wave lengths. -7

-83.1 THEORETICAL CONS IDERATIONS Irene Schensted During the course of the investigation various calculations have been made in connection with the study of the propagation of sound in a fluid layer over a concrete bottom. It is the purpose of this part of the report to present the results of these calculationso Specifically, they have been carried out to determine the acoustical pressure as a function of range for a time harmonic source using three models to represent the behavior of the bottom: (1) a fluid (water) layer over a rigid bottom, (2) a fluid layer over a semi-infinite fluid capable of transmitting compressional waves, and (3) a fluid layer over a semi-infinite solid capable of transmitting both shear and compressional waves. Comparison was made of the predictions of these three models with the experimental results with a view to determining the simplest model which might suffice to predict the results of the pertinent experiments. Although the concrete bottom of the tank was finite in extent, a series of simple calculations such as those performed in the appendix of this report indicate that the assumption of an infinite bottom is a good approximation to the actual case. The calculations are based on the normal mode theory of sound propagation. The basic work in this field is given by a classical paper by Lamb. In section 3.1.1 below are presented the results of the normal mode theory for the three models cited above. The formulas used are taken from Pekeris2 and Press and Ewing.3 The reader is referred to the work of the authors of these two references for derivations of these formulas, Calculations are also made to determine the pulse shapes to be expected as function of range for short pulses. In this study it is found convenient to use the method of images to supplement the normal mode point of view. Section 3.1oo2 deals with the problem of the pulse shape to be expected

-9under various conditions and exhibits the results of the pulse shape calculat ions 3.1.1. Calculations Based on the Normal Mode Representation of the Solution In this section are presented calculations based on the normal mode representation of the solution for the following types of bottom: (1) rigid, (2) semi-infinite fluid, and (3) a semi-infinite elastic. The Rigid Bottom - We shall assume a point source to be located at a depth d from the upper surface of a liquid layer of depth H. We shall use a cylindrical coordinate system with coordinates r,,9 and z, with an origin on the surface of the liquid vertically above the source. The positive z axis passes through the source and is oriented normal to the top and bottom surfaces as shown in Fig. 1. r r=O, z=d *rz H F ig. 1. The Geometry of the Model The surface z = 0 is to be thought of as a free surface (a surface of zero excess acoustic pressure). The bottom at z = H is for the present to be thought of as being perfectly rigid so that the z component of the velocity of the fluid at that plane is zero. All quantities of physical interest in this problem may be derived from a scalar potential function t which satisfies the wave equation 72) c2 2t where c is the phase velocity of acoustic waves in the liquido In addition it is necessary that 7 satisfy certain boundary conditions. The

-10velocity v of any point in the liquid layer is related to by the relationship v = - o. The excess acoustical pressure p is related to by the relationship p = p where p is the density of the liquid. Ot Hence since vz = O at z = H and since p = 0 at z = 0, the scalar potential 3) must satisfy the following boundary conditions: = O at z = 0, and 0 o z -=H o C z In addition W must behave like a point source in the vicinity of r = 0, z = d. For a time harmonic point source of frequency X, this means that 2-t i (t,-]r + (z-d)21/c) / (r,z,t)- as r — 0 and zo-d. In this section only a time harmonic point source of time dependence e will be considered. If one finds the response, R(L),rz), of the medium to such a time harmonic excitation, then the response to the more general excitation f(t) will simply be -.. f.-R(W -r, z)g(L-0)ei etdc 00 where g(L) is the Fourier transform of f(t) defined by 00 g() = f(t)e -itdt In section 3.1.2 the response to finite time pulses will be discussed. Accordingly for the present, set = $(r,z)eilt where B(r,z) must satisfy the equation F2, =-ko, with k =/c = 2/; x is the

-11radiation wave length in the fluid layer. Also ~ must satisfy the boundary conditions $ =o at z = 0 and - 0 at z = H as well as the condition that it behave like a point source near r = 0, z = d. The solution to the problem stated above may be expressed in the following integral form: c0 eipt | JO(kr) 2k sin Pz cos 5(d-H) dk O 0- z'd 00 / wt 2k sin Pd cos P(z-H) d( e J0(kr) d coskH. k. o d_ z _H; 2 2 where = k- k k0o k, 1i 2 2 and P -- - ko k0 ko and Jo(x) is the Bessel function of the first kind. The path of integration in -Eq. (1) makes semi-circular detours about those poles of the integrand which lie on the real axis as shown in Fig. 2. k - plane Poles of integrand.......;"',.".~+.oo Fig. 2. Path of Integration for the integral, representation of the solution of the propagation problem.

-12The integrands in the above expression are, for k arbitrary, solutions of the wave equation which satisfy the given boundary conditions at z = 0 and z = H. These solutions are integrated over the path indicated in k space with the given weighting factors so as to yield the desired point source behavior at r = 0, z = d. This technique of integrating kernels, which satisfy the wave e'quation and appropriate boundary conditions, so as to yield point source behavior is used frequently in problems of this type. The above integral representation may be transformed to a sum of residues by the use of Cauchy's integral theorem. The resulting expression is: = ei t (2ti) H 2)(knr)si n+l )Td sin+l ) 2-z (2) 0_ z 4H H 1 2 2 (n+1) where kn = 2/c2 (n+)2 (rt/H)2 (n+) t/ 4 )/c kn = -i /((n t) /H)2 -2/' 2O/c C4. (n+ ) /rlH, 2 2 H(2) is the zeroth order Hankel function of the second kind, The above representation is usually called the normal mode form of the solution each term being called a mode. Asymptotically for large r E(2)(kr) 2 -iknr + i /4 H0(2)(knr) -I~ eIknlr >>l Hence for large r the normal mode form is: - -2 ~it,2n -iknr + iT/4 )sin+2)Trd/H]si n+1 ) tz/HJ f1 ~-2Tcie I 0 O z,H For n sufficiently great it is apparent that kn will be imaginary and the corresponding modes will not propagate. The largest integer nmax which satisfies the equation (n+a) t C4H E)/c will determine the number of modes

-14th propagating in the medium; the cut-off frequency for the n mode will be 1n c In Fig. 3* the amplitude of the acoustical pressure is plotted versus range for the following special case: )/2-A = 4800/sec. c = 4800 ft/seco H = 2% z = d = H/2 These parameters were chosen so as to correspond to the conditions under which some of the experiments were performed in the model tank. For this case there are only four propagating modes corresponding to n = 0, n = 1, n = 2, n = 3. The fifth mode corresponding to n - 4 yields a value of k4 = - ti o Hence at a range of 10 feet the amplitude of this mode compared to that of any one of the undamped four would be of the order of e-103't,J10-14l The succeeding terms in the series are even smaller. It may be shown that -10 o(n)2 Tt/4 - (4n)2 -10 te-1/ 1 - -10 31t, e = == e In view of this fact one need only cons ider the f irst four modes in calculating the pattern of the acoustical pressure for these conditions. The normal mode series therefore constitutes a very practical approach to the problem of getting numerical results for large ranges for continuous-wave sources because of the series' rapid convergence. We shall see in section 3.1.2 that the image series representation of the solution of this problem is not rapidly convergent for large ranges. * The acoustic pressure units are arbitrary; in the present study only relative values were of interest,

~ w=2r 4800/sec c =4800ft/sec X= Ift H=2ft Z=d=H/2 Z C)z cr < 4 Cr m CL 0. w u; n,, )! — 2 or: R I I I I 1 1 Fig~4. Clculted cousic. pressure,= variatio'ih range f or the rigid( bottom model employing an extended sourc.e, Source length is 0o27:~ U) w v,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~J 0O 0 C~~~~~~~~~~~AG IN INT IF Fig. 4. Calculated acoustic pr-essure -varia-tion wiith range for the~ rigid bott~om model employing an extenided souarce. Source length is 8.~

Before going on to consider the two fluid cases note that source 1 it z 1 Tcd and receiver depth appear only in the form of the factor sin(n+-)- sin(n+4)-. For a linearly extended source one may therefore take into account source extension by integrating the factor sin(n )t- with respect to a weighing function representing source strength along the source lineo Since the source used in the experiment was not exactly a point source it is of interest to carry out calculations taking into account this effect. For a uniform linear source of length and of unit total strength one would obtain the solution (kr>1l): y.o ~2i eiLj09 e iknr ii // cos 5n(d- )-cos Pn(d )sin PnZ (3) H 2 knr One may account for receiver extension in a similar way. Figure 4 exhibits the effect of source extension on the pattern. The U) and H are assumed to be the same as for'the case mentioned above but the source is considered to be uniformly linearly extended, of 25 inches length. The main effect here is seen to be a change in the relative amplitudes of the peaks and valleys. The Two Fluid Model - Suppose now that the bottom is capable of transmitting compressional waves. The details of the solution are to be found in a paper by Pekeris. Here it will suffice to describe the nature of the results and the calculations based on them. Let P2 designate the density of the bottom and c2 the phase velocity of sound in the bottom. The geometry of the situation is as depicted in Fig. 5 r r z r-O, z=d H Fig. 5. Geometry for the Two Fluid Model.

-17The conditions to be satisfied by the excess acoustic pressure are as follows:;72 / 1 ___ O4z-HL 22 2 c1 ~t z > H o7'=U2 2 zH, c2 t and 0 at z = 0 (acoustical pressure is O at surface of liquid). At z = H we must require that the acoustical pressure be continuous across the interface separating the two media, and also that vz -be continuous. Pl at ='3 d az z -- H z- H+ z-> H_ z - H+ where the symbol z — H_ denotes z approaches H from the domain z < H, and the symbol z — H+ denotes z approaches H from the domain z > H. In addition we have the requirement that ~ behave like a point source in the vicinity of r = 0, z = d (see section 3olol). The solution to the time harmonic problem which satisfies the above conditions has the following integral representation: o0O = 2e t' j(kr) k sin 1iZ [lC~S P1(H-d) + ibP2sin P1(H-d) dk 1 l1cos % H + ib 2 sin 1 H j O ~~~~~~~~~for 0- z d i 2e t C k sin %31d B1cos 31(H-z) + ib32g1 (H-z) 3=2?J(r L 11coS Bl H + ib 32sin 1l H dk for di_ z z H, and oO -i2 (z-H) 7J) i c3 t b 0;rO(kr)k(sin P d) e dkz2(E-H4 2e dk z, (4) lcos 1c H + ib P2sin 51 H O where 1P= i/c - k2 k 0/cl, = -' k2 - ~o2/c12 k > 0/cl; B1 =.,/-u..) /c2.k. /C.s = i Xk2 _UD 2XC22k > uD/c1

-18The integrands in the above integral formulas satisfy the wave equation and the boundary conditions at z = 0 and z = H. The integration over the real axis in the k plane serves to give the desired point source behavior at r = 0, z = d. The path of integration over the k axis makes semi-circular detours about the poled which occur on the real axis as was shown in Fig. 2. To obtain the normal mode form of the solution one must transform these integrals to a sum of residues of poles in the complex k plane. The integrands of the integrals in Eq. (4) are not single-valued; they have a branch point at k = L/c2. Hence one obtains upon transformation, in addition to the normal mode series, a branch line integral extending along both sides of a branch cut extending from k = ()/c2 to k =- /c2-ioo as shown in Fig. 6. I /C2 /Cl IReal Roots k - plane Fig. 6. Position of the poles and the branch line for the integrands of the integral representation of the solution when c2 > clo The following is the normal mode form of the solution which is valid in the top layer (0 ~ z LH), i~t =e-2Tci Ho(2) (knr)plnH(sin 1lnd)sin! nZ T E)lnH- (s inTlnH )cosnlRnH-b2 ( s in23lnH )t anR!lH2 - H(2)(kr) k 2(sin pld) sin B! Z k k F 21 cos2 1 H + b252 sin2H1HI 0oz_ H

-19with b = Pl/P2. In Eq. (5) the Kln are the roots of the following transcendential equation (called the period equation), P1 tan P1H = b / 2 2 02 2 1 2 /2 (6) 1 - X /c2 ~l 2 2 2 2 The k are related to the 5ln by the relation: kn = /c Before going on to a further discussion of the roots of Eq. (6) we shall present the asymptotic form of the normal mode solution valid for large r. We have ( o -iknr+i E )4 9 i t) -2rti e P lnH(si'rlnd)sin PlnZ -: eJ k_ H2 [Llns-(slnlnH)cosPlnH- sin P12 cos PlnH -ikr 2ib e k2 sin (krftd) sin (k z) (klr)>12 cos >H kl) where /= 1-c - c/c2 c2 The above form is valid providing the parameters are such that cos2(//Hkl) is not near zero. It is seen from this expression that since the individual propagating modes decrease as rl/2 and the branch line contribution decreases as r2, that for r sufficiently large, one need only consider the normal mode terms. To proceed, if we make the substitution x = P1H in Eq. (6) the period equation may be written as: tan x =1/2 bE2 211 (-2 2 There are only a finite number of real roots to this equation. These real roots correspond to the undamped or propagating modes. The real roots can occur only when c2 ) c1 and in the interval LO/c2 ~ k' 0/c1l For c2 cl

-208w/2 r= 4,800/sec cI = 4,800 ft/sec PI /p2=.43 4 -8 I|Jods|-' Modes Fig. 70 Positioning of roots xn for the two fluid case for different values of c2.

71 6 4800/sec 6 -- c i = 4800 ft/sec C 2= 13,780 tt/sec / P /P 2=.43 z 5 H =2ft Z=d=H/2 (I) cc a0 ~4 10 12 14 16 18 20 22 24.. RANGE IN X Fig.8 Calculated acoustic pressure variation with range for a two fluid model.

-22there are no real roots. The positioning of the roots of Eq. (6) is illustrated in Fig. 7 for a single value of -3O and two different values of C2. In this figure the function 1 2 is plotted against x. The function tan x is also plotted and the points at which the curves cross give the roots xn of the equation. It suffices to compute the right hand side of Eq. (6) for a single frequency u)0. For if one denotes the right hand side of Eq. (6) by f(uO xclc2) then f(O1Loaxclc2) - f(30,Xg~clc2)o By using this relationship, curves for different frequencies may be plotted with a minimum of effort. It is clear from Fig. 7 that for c2 sufficiently large compared to c1 there will be as many real roots for a fluid bottom as there are for a rigid bottom. When this happens the pressure amplitude versus range curves for the two cases will be quite similar. Figure 8 shows a plot of pressure versus range for a two fluid case. The frequency chosen was 4800 cps. The density ratio P1/P2 was taken to be 0.43, a value which corresponds to the ratio of the density of water to that of concrete. The velocity c2 was chosen to be 13,780 ft/sec, a value which corresponds to the compressional wave velocity for a typical concrete sample. The value of c1 was chosen to be 4800 ft/sec. This last figure is roughly the value of the speed of sound in water. For the case chosen the branch line integral was found at a range of 10 ft to be much smaller in magnitude than the propagating modes. The pattern in Fig. 8 is therefore due to the interference of the four propagating modes. There is a marked similarity between the pattern of Fig. 8 and that of Fig. 3 which may be attributed to the fact that the number of propagating modes is the same for the two cases and that the values of corresponding wave numbers are quite close.

-23By comparison of Eqs. (5), (6), and (3) we see that if we let b = P1/P2 approach zero in Eqs. (5) and (6) the two fluid case solution reduces to that of the rigid bottom. Hence the latter may be considered to be a special case of the former. The Solid Elastic Bottom Model - The elastic bottom model differs from the two fluid case in that the bottom is now to be thought of as being capable of transmitting shear waves. We shall in the following let cc denote the velocity of the compressional wave in concrete and let cs denote the velocity of the shear wave in concrete. It is also convenient to use the Lame constants x and //,in terms of which cc and CS may be written as: cc = Cs P2 The velocity of sound in the top fluid layer we shall denote by cl and the density of the top la'yer by plo Although it is still possible to describe the physical situation in the upper fluid layer by means of a scalar potential, in the bottom it is necessary to use both a scalar and a vector potential. We shall not enter here into the details of the solution but shall record the appropriate solution for the upper layer. The details of the derivation are to be found in reference (3). (The geometry of the situation is depicted in Fig. 9.) H t r=O, z=d z Pl~Cl 777 /77///7777 V// / / /,,//,7 /, cc p2 Fig. 9. The Geometrical Picture for the Solid Elastic Bottom Modelo

-24The integral representation of the scalar potential in the liquid layer has the form: i~~~t (i Jk Q cos P1(H-d) - P sin P1(H-d) 2e it Jo(kr)(sinlz) dk Q(cos 1H) - P sin P1H 7) < O z: d; (i2eit |J(kr) sin ld Q cos 1(H-z) - P sin li(H-z) (7) 2 ie3 in Pi Q cos P1 H-P sin P1H d2 z:HE dk where P = -LU)cks/P19 2 2_ 2 2 2 22 Q= W i k (k -P ) + 2J {c(k -s) - 2psckJ and 2 1 kB -kk k 4 k (k1= WO/c1) 2 C k kc c C = \ik2 -~ k 2 k/ k (ks= Wo/c5), -i k ks k ks. Before going on to write the normal mode form of the solution which is obtained by transforming these integrals into sums of residues of poles and branch line integrals, we note the above expression for / reduces to the corresponding expression for 0 for the two fluid case when /O 0. Hence the elastic bottom colution contains the solutions of the two previous problems as special cases o. It is seen that the integrands of Eq.o (7) have two branch points, one at k - uO/cs and one at k = t)/cc. Hence upon transformation of the

-25integrals of Eq, (7) to obtain the normal mode form of the solution, one obtains in addition to the normal mode series two branch line integrals. One of these extending from k = 0/cs to k = u)/cs-i o and the other from k = u)/cc to k = u/Cc-ioC. The following is the asymptotic form of the solution represented as normal modes. It is valid for large kr ~19 and for O z = H: = e iknrt /4 sin PlnZ sin 5lnd = eiwt 2Ti.......... Kn \H + cos 1H- k sin lH IQ iQ sin 1H + P cos 1Hj k n k=kn -ik r c 4 2 e 2i P1 cO /c k sin (c klz) sin (~c k d) 2 2 2 2 2 21 (k1r) ac cos (CcHkl) (kc(2kc - ks) e iksr sin(aklz )s in (ckld)i 8k (k -k2 )k/ /P 22 2r2 2 2 2 klr s a tcoS (asHkl)-sin (a Hk1)C +(sinsHkl) (osa Hkl)a ac'- 8 S S Sk=ks The last two terms in the above expression represent the two branch line contributions, These expressions for the branch contributions are valid providing we are removed from the zeros of the denominators. The form used for the branch line calculation is valid providing c1 4 cS cC This is true in the cases of greatest interest to us. The symbols Q and P are as defined in Eq. (8). In addition we have the following definitions, r 2 222P 2 _2 i = (0 2'k222 7 a? =P L kc(k -8s) + 2/1 jc(k -8s)

1 -c jc /C C = - Cl/c c cgs = I - C1/CS 1 4p k2 and c' = -u P c,ks The values of kn which appear in the normal mode series are the roots of the transcendental equation Q/P - tan D1H. The real roots kn of this equation occur for k1 7 kn > k > kc. The period equation may be written more explicitly as: 2 where ks C)/cs and kc = /cc as above. Making the substitution 2 c x = 31H the equation above may be written as: tan x = bE [4 (a) /cl - ~ -/ xH) [01 -2(L2/c~ -2x22/2) 2/j2) X 2 2 2 2 2 1/2 2c /C1 - /) /CC -X /H 2 Figure 10 exhibits the positioning of the roots xn of the above equation for various values of cc and cS. For cS and cc both sufficiently high there will be as many real modes as for the rigid bottom case for the same angular frequency GO and depth H. In some cases there will be one more mode than for the rigid bottom. In such a case two closely spaced elastic bottom modes will appear and correspond to a single rigid bottom mode. Figures 11, 12, and 13 are plots of the acoustic pressure versus range for an

-278 5 Modes l 41800/c 12,300ft/sec 5 Modes 4 = 10,050 ft/sec Cs= 7,070 ft/sec C 6 000 ft/sec / Modes r4 /2 3/4 -' 5r4 3w /4 /4 5/2 l r 13r/4 X — ~2 Modes -4 W/27 =4,800/sec - p1/P2=.43 c 4,800 ft /sec Fig. 10. Positioning of roots xn for the elastic bottom period equation.

6 4 W I4 03 e cC=12,280 ft/sec 8I~ \t~ ~.<$ ~Cs \ 7,070 ft/sec ro < 2 \ /=27r 4800/sec CI = 4800 ft/sec H =2 ft H/2=d=z I I 010 12 14 16 18 20 RANGE IN UNITS OF X Fig. 11. Calculated acoustic pressure variation with range for an elastic bottom model. cc = 12,280 ft/sec, cs = 7070 ft/seco.

Cs =6000 ft/sec cc =13)780 ft/sec I = 4800 ft/sec H = 2 ft H/2= z =d z cow /2yr =4800/sec Ctr a: a: a: w C, C, w a. 0 H2 Cl) 0 01 10 12 14 16 18 2 RANGE) UNITS OF X Fig. 12. Calculated acoustic pressure variation with. range for an elastic bottom. cc 13,7'80 ft/sec, c~ 6000 ft/sec. O~~~~~~~~~~~~~~~~~~

7. 0 w=2r 4800/sec X I ft H 2 ft c = 12 300 ft/sec 6.0 C — I cc f3 Cs Z = d=H/2 z Z5.0 a: cr C) 40 w U) w ar: CL3.0 0 H 0 0 RANGE IN UNITS OF X Fig. 15. Calculateld acoustic pressure'variation -with range for anLelastic bottom,, cc T c5 12 -7OO ft/sc Sy s /Sec~~~~~~~~~~~~~~~~~~~~~~;

-31elastic solid bottom, Figures 12 and 13 were plotted for values of cl, cs, and cc which might be typical of a water layer over a concrete bottom. The frequency chosen and the depth to wave length-in-water ratio was the same as for Fig. 3. For the values of cs and cc used in calculating the pressure pattern for these graphs there were found to be fewer modes than for the corresponding rigid bottom case. This was due to the relatively low speeds used for the shear wave velocity. The pattern computed is quite unlike that of Fig. 3 which exhibits the rigid bottom results for the corresponding case, i.e. for the same frequency and depth-to-wave length ratio. In Fig. 13 acoustic pressure versus range was plotted using sufficiently high bottom speeds so that five propagating modes appeared, two of which corresponded to a single rigid bottom mode. At these speeds the pattern is similar to the rigid bottom pattern plotted in Fig. 3. It should be emphasized that the high bottom speeds used in calculating the pressure pattern for this graph are not those commonly associated with concrete. Summary and Conclusion to Section 3.1.1. The normal mode forms of the solution for a point source liquid layer over 1) a rigid bottom, 2) a fluid bottom, and 3) an elastic solid bottom have been presented and the results of calculations based on these models have been shown. All three models for ranges large compared to the wave length yield a scalar potential of the form: ei~t B Cn(jIHci) e sin Gnd sin nZ (9a) where on is an amplitude factor. For the three models an are different functions of the physical parameters which describe the problem. The roots

-32kn for the three models are determined by the three period equations 1) Rigid bottom -- tan x.n - C where x =1 HD)2/c-k - H n in n 2) Two fluid -- tan xn E( ) 3) Elastic solid bottom -- Xn f (1 -2k2l/kS2 4ks n 2 2 tan xn = Eb -+ k k The real roots kn for the three cases occur for the following ranges of the parameters (assuming cl 4 cs 4.cc) 1) W3/@c k 2) t)/cl1 k > L/c2, 5) W)/cl k > /cs > ~0/cc These last conditions correspond to the fact that the non-propagating modes correspond to radiation moving at angles greater than the critical angles associated with the bottom, The fluid and elastic bottom cases are characterized by having in addition to the normal modes, branch line integral contributions which, however, vanish at large distances as 1/r2 (compared with 1/rl/2 for the propagating modes). Asymptotic formulas for the branch line integrals have been presented in Section 3.11.; from these the magnitude of the branch line contribution may be determined at large ranges.

-33301.2. The Propagation of Pulses In the preceding section the normal mode form of the solution was presented for a time harmonic pulse. We shall now briefly discuss distrubances with other types of time dependence. An Approximate Method Using Normal Modes - First consider an approximate treatment of the response to a pulse which may be represented by a Fourier transform, g( Q), which peaks at the frequency 0' Let us exiL0t press the original excitation of the medium as e f(t) where f(t) is the envelope of the pulse. The response RB(u)) of a mode to this excitation function may be written, (see Eq. (92))o ~~~C i( )t-knr) Rn() =( (') e |(s)sing nz sin gnd due (10) n when kr > 1. Assuming acn(LO), -ITk7, and Pn slowly varying functions of U) over the range -0 - 2 0 + we have: 1 ~O -~ i(Wt-knr) n(UO) - %(ujo) sin.nOzsin PnOd ~.ter | g(w )d. Expanding kn about kn0 we have to a first approximation: kn((V) - knO + (J- WO) - dL (lOa) where kno = k1r(tQ)O) and n0O = Pn(WO ) Hence i(Q ot-knor) ~c~(Xo)sinnoZSi innOd 1'R (L) - e o' ~.... - I I r — ir

S ince L -,(73 ) e duO _ ~-o "Z iIO~ -~k~Z~Jr c2dkn | r)l i(okt - k0r - |r) dknl R n( 1 )- e 8 -i ) (inO z) (s infnod)f(t - d r) The result may be expected to hold when (see Eq. (IOa)), From Eq. (11) it is$ seen that one can thi:L~ of the response for each mode as being the original excitation fIunrctiLorl f(t) reproduced in shape and travelling thitrough the medi-'m with the group velocity v group d- =k edskeJ Due to the fac-t thieat the (i1ff erenrt modes have dlifferent group velocities the originall pulse w:ll be elongated. aLnd distorted.. The group velocities associated with the various modes may be computed if one knows the period equation. We have for the t.hree models discussed in Section 35.11 the the period equations. 4- -A 4~ ~

1) Rigid Bottom 2 V V - C2 g P where vg is the group velocity and vp is the phase velocity; 2) Two Fluid a+ g P a.. C2 C2 2 eC2x tan xn Xn where HC = H2sec2xn tan xn2 n bHn and and. t2 2 2 Pcn kn - /cc 3) Elastic Solid Bottom group phase a + k + 6 + 2 2 2 2 2 2 2kn 2 where xn(l 2 4xn k3bH, 37bHP snks 8xn (1 _ 2k ) Xn8Psn bH P k 2 bE k cn sn 22 2 bH k sk k sn n s cn n c By the use of velocities computed from these formulas one may calculate the spreading of the initial pulse. Actually at ranges of interest here the results will be only qualitatively correct as we shall see from computation based on the image series form of the solution. Using the above formulas for the group velocities, calculations have beern made to show the spreading of

Pulse length Pulse length Range 2 millisec 3 millisec 12.5X r 17X I 19.5 - 20X - - - 0 I 2 3 4 5 0 1 2 3 Ti me, m I liseconds Fig. 14. Calculated pulse shapes using a normal mode first order approximat ion for a rigid bottom = 9600/2T radians/see X = 0.5 ft, H = 2 o

-37the pulse for the case of a rigid bottom. It was assumed that the pulse had a square envelope. The results are exhibited in Fig. 14. At each range the fastest mode arrives first and then in succession the other modes arrive to interfere with ito The arrival time for each mode was computed using the appropriate group velocity. One can see that if the pulse length is short enough, the first mode may cut off before the arrival of the slow modes. In order to obtain the continuous wave result the pulse length must be long enough so that at the ranges of interest there is a region of the received pulse in which. all of the important modes interfere. For the parameters chosen there are at most four modes propagated. The cut-on or cut-off of the successive modes are indicated by changes in the pulse amplitude. Image Series Solution -- An image series representation of the solution lends itself more readily to the integration of the finite time pulse at moderate ranges. For the case of the rigid bottom the image series may be obtained directly as follows: H Fluid layer +, Fig. 15. Location of images for a fluid layer bounded on the upper side by a free surface and on the lower by a rigid surface. We consider a source at a depth d in a fluid layer of depth H bounded on the one side by a free surface and on the other by a rigid surface. Then the boundary conditions to be satisfied by the scalar potential 7 are

-38= 0 at z = H, = 0 at z = 0 Consider the point source solution to the Helmholtz equation, C=) ___= with k - k)/c and where RO is distance of the field point 2 2 (r,z) from the source i.e., R0 = r + (z-d) In order to satisfy the bounld.ary conditions at z = 0 and z: — H we start with two images of the source, one in the upper surface and one in the lower surface. Labelling the fields corresponding to these images by ~1 and 1 =, e -~ lwhere R1 = (r2 + (z+d) 1/2 i k where R 1 = r2 + (z-2H+d) By continuing this process of imaging one arrives at the following series which satisfies the wave equation and a.l. the boundary conditions of the problem and is therefore an exact solution to the problem: -i C + Z-2 d) - i ((/c)(r2+(z+2nH+d) 21/2 iwt2 (jl)n o + )l3~22 e / n r+ (z,-2n.H-d)2(r2 + (z+2nH+d)2) (12) For the special case of z=d=H/2 we obtain the following form,'9'lut S ( (r 21/2 (13) z~~-H/2 a 2+ (2nH)(

O (0) \ Image vector.- - Normal mode vector Range = 16 X Frequency = 4800 cps H=2ft Z= d = H /2 Fig. 1,6. Summatior of the image an.d normal mod.e serials.

The image series is especially easy to use for non-periodic time excitations. Corresponding to a source excitation of the form f(t) we havey 9jI (-fl)(t- a,r2-(z-,-2.)/ f(t- r + (z+2nH+d) ) - r + (-2n-d)21/ (r2+ (z-2nH+d)2)1/2 Each term of the image series represents rays which have undergone a definite number of reflections from the top and bottom surfaces of the liquid layer. The image series includes all such rays. It is evident that for large ranges it converges slowly. Figure 16 shows the relative rates of convergence of the image and normal mode series for the case c -= 4800 ft/sec, Wc= 4800 2t/sec, H = 2 ft, and d = z = H/2, a case which we have alreasy considered using normal mode theory. The vectors drawn in heavy lines are the image contributions. The four dotted line vectors represent the four propagating normal modes. Thus even at the moderate range considered i.t is necessary to sum many more than four terms of the image series in order to get a good approximation to the sum of the image series. With increasing range the work involved in the summation of the image series increases rapidly. Each normal mode represents the contribution of many images. This. situation may be seen from the fact that the normal mode contribution of a 1 mode to the field amplitude decreases as 7-1 whereas each image contribution decreases as r I In particular a normal mode may be thought of as representing the sum of groups of images which interfere constructively at the field point. Consider the case of a source and receiver at mid-depth where one may neglect the images ( marked x in Figuire 17 ) since these always cancel for z=d=H/2. Cne sees that the difference in phase of two rays arriving from images proceeding at an angle O will upon arrival at the

-41field point be Co Q - E. The condition for constructive interference is: Ei cos Q 4[ X - t = 2ni, n = 0, 1, 2, or 2n+l cos 0n = -H X and sin = (c/) (22 2 )1/2 sin' ) -- it - k~ (2H)2 here kn are the roots of the period equation for the rigid bottom. Thus a normal mode represents a group of constructively interfering rays from the images which rays travel at the angles ~n given by the above relationship. X + )( aI Fig. 17. Constructive interference of images to form modes for a rigid bottom. The image series, although inconvenient for computational purposes when the range is very great is, however, ideally suited to the integration of finite time pulses for moderate ranges. If one has the patience, one may achieve the exact integration of the time pulse for the rigid bottom case. This has been carried out for a pulse with a rectangular envelope and the results are displayed in Fig. 18. A comparison is made in the figure with the treatment of the finite pulse by the approximate method using normal modes

-42IMAGE SERIES Time - to t. t2 t3 t4 t5 t6 t7 t8 fg tao tl It12 Range =8 X Freq = 4800 Cycles/sec C, = 4800 Ft/sec H = 2X = 2d2z NORMAL MODE METHOD Time. mtlm I2m t3m (a) Initial pulse length =I millisec IMAGE SERIES Time - to.ti t2 1, t4 t5 t'6 17 t t9 t6o t,o t,2 NORMAL MODE METHOD Time - tom tim t2m t3m (b) Initial pulse length = 4millisec Fig. 18. A comparison of the integration of the time pulse by the image series and the approximate normal mode method.

described above. Although the normal mode treatment gives one an idea of the distortion of the original pulse it does not suffice to give the detailed picture which is obtained by summation of the image series. Here the discontinuities in pulse shape are due to the arrival of successive rays corresponding to different numbers of reflections from the top and bottom surfaces. As successive rays arrive one obtains successive approximations (see Fig. 16) to the amplitude of the scalar potential. Pekeris2 has derived an analogous ray series solution for the case of two fluids. Honda and Nakamura4 deal with a corresponding series for the elastic solid bottom. For these two cases the plane wave reflection coefficient for the bottom appears in the ray series expressions. One finds that terms corresponding to multiple reflections from the bottom are multiplied by the corresponding powers of the reflection. In Fig. 19 is plotted the reflection coefficient as a function of angle of incidence for a plane wave incident upon an elastic bottom for values of cc and cs typical of concrete. This plot can be used in connection with the work cited to obtain the approximate image representation of the field for two fluid and fluid over elastic solid problems.

1.0.8 ULi 0.6 LL cc = 12,280 ft/sec LU 0.2 o L, I, I I l l II 00 100 200 30~ 40~ 50~ ~60 700 ec es ANGLE OF INCIDENCE, Fig. 19. Calculated amplitude reflection coefficient R, for plane waves incident at an angle upon the plane interface between a fluid and an elastic solid.

APPEND IX THE EFFECT OF THE FINITE THICKNESS OF THE CONCRETE ON THE PROPAGATION OF SOUND IN' THE FLUID LAYER A plane wave treatment allows one to quickly estimate the effect of the finite thickness of the bottom layer upon the propagation of sound in the top fluid layer. The normal mode problem for many plane layers has been solved by many investigators but the results are quite cumbersome to apply. Pekeris2 for example treats the three-layer problem. Cl Pil C1) Pi Fig. 20. Plane wave incident upon a fluid layer; c2 12,800 ft/sec, c1 = 4,800 ft/sec, Pl/P2 = 0o439 and L = 6 inches. One may, however, get an estimate of the effect of bottom thickness by supposing the incident field to consist of a plane wave. One calculates the plane wave reflection coefficient for those incident angles 0n which correspond to the propagating modes for the case obtained by allowing L and observes the magnitude of the reflection coefficient for the case of a finite second layer (L 4 o ). If this is close to unity one concludes that though the modes may be shifted due to difference in phasingt their attenuation will be small. -45

-46The plane wave reflection coefficients for the situation depictedin Fig. 20 when the middle layer is a fluid, are shown in Table I.* The physical constants are those pertinent to the model experiment described in this report. TABLE I Amplitude Reflection Coefficients for Various Angles of Incidence _______ Ar 840o 99 68~0 99 510.98 290~ 97 One can see from this table that for a bottom characterized by a velocity c2 in the neighborhood of that of concrete, the solution for the six inch bottom should be much the same as far as infinite layer since little energy is transmitted into the underlying regions. * Constants used~ cl = 4800 ft/seco; c2 = 13,000 ft/seco; P1/P2 =0~43 L = 0,5 fto

BIBLIOGRAPHY 1. Lamb, Ho, "On the Propagatior, of Tremors over the Surface of an Elastic Solid.", Trans. Roy. Soc. 203A9 p.O 1 (London) (1904). 2. Pekeris, C.L., Propagation of Sound in the Ocean, Memoir 27, "Theory of Propagation of Explosive Sound in Shallow Water", Geological Society of America (1948) Baltimore, Maryland. 3. Press, F,, and Ewing, M., "Propagation of Explosive Sound in a Liquid Layer Overlying a Semi-Imfinite Elastic Solid", Geophysics 15, p. 426, (1950). 4, Honda and Nakamura, Bulletin of Science, Geophysics, University of Tohoku (1954).

-483.2. EXPERIMENTAL INVESTIGATION OF THE PROPAGATION OF SOUND IN A SHALLOW WATER MODEL J.E. Lesch and J.R. Frederick A description of the several procedures used in the experimental pursuit of information concerning the propagation of sound in shallow water as modelled in large concrete tanks is given in this section. The primary purpose of the experimental investigation thus far has been to explore the characteristics of the model. Before introducing additional complicating factors it appeared desirable to examine the simplest system available, and to investigate the extent to which the following factors might affect subsequent experiments: 1. elastic properties of the bottom 2. acoustic properties of the model 3. repeatability of the experiments. In the preliminary phases of the model study programs reported here, two aspects have been particularly emphasized. One was that of comparing the results of experiments in the tank with those calculated theoretically on the basis of various simplifying hypotheses with the hope of finding the simplest mathematical model which would be useful. Secondly the effect of slight changes of the variables on the sound field has been examined. 3.2.1. Equipment and Techniques The principal features of the apparatus and experimental procedures are described here. There is essentially nothing unique about either the equipment or techniques employed in the experiment. Possible exceptions are the use of tape recorded pulses for a signal source and one or two methods of treating the data.

-49A block diagram of the equipment is shown in Fig. 210 The variables associated with the tank in this experiment are water depth, projector and hydrophone depths, projector-hydrophone separation, sound frequency and, orientat ion of the range with respect to the sidewalls. The effect of other parameters such as s-urface roughness and different bottom types will be examined in the future. Both continuous-wave and pulsed signals are used. Except for the generation of the outgoing signal and the method of recording the received signal level, the same equipment is used for both techniques. The pulses used in the experiments were first recorded on a magnetic tape using a high quality gating circuit designed and built by another research group at the University of Michigan. This device gates a c-w voltage to provide a pulse that cuts on and off at the axis crossing of the wave; the nwumber of cycles in a pulse as well, as the pulse repetition rate are readily controlled. A continuous loop carrying one or more of these recorded pulses is used as a source of signals. Pulse lengths of from one to fourteen milliseconds are available at repetition rates of from one to ten per second. Most of the measurements have been made using a Bell Telephone Laboratories Type 1K projector and a Bell Telephone Laboratories Model 3A hydrophone. Both of these units have been obtained from the U.S.N. Underwater Sound Reference Laboratory. Some of the first experiments were performed using cylindrical barium titanate elements as projectors and hydrophoneso The former were about 1-1/2 inch in diameter and the latter 1/4 inch in diameter. Two tanks have been used. The dimensions of Tank 1 are 90 x 40 x 3i7 feet. Tank 2 shown in Fig. 22 is 95 x 53 x 3 feet deep. Gates in the tanks are provided for easy access of tractors or other vehicles used in hanidl ng various bottom materlals. A wave-making machine is available which carn produce

-50t AMPEX I SYNCHRONIZING SIGNAL TO MODEL 401-A OSCILLOSCOPE RECORDER PULSE KNIGHT AMPLIFIER MODEL 90-320 HEWLETT CW PACKARD MODEL 200CD OSCILLATOR TANK BTL TYPE BTL 3A HYDROPHONE IK PROJECTOR 8 PREAMPLIFIER BALLANTINE DUMONT HIGH PASS DECADE AMPLIFIER R-C FILTER OSCILLOSCOPE MODEL 220-A BALLANTINE ELECTRONIC VOLTMETER MODEL 300 FIGURE 21. BLOCK DIAGRAM OF APPARATUS

.GeV~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~: ii~~~~~~~~~~~~iiiiiii~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i!?~I"~lassi~~~~le~i~-,!i?*.*t i~"" ii~i~ iiiL:~l-'i-i;:ii:iiiiiiiiiiiiiiiiiiiiiiX"iiiiii! Amplifications: 1!11:~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~:HI.l~8-i": l~~~i:? 1;!iiitiiiiiiii;:i T iiiiiiii ~iiiiii!;iiiiiii::iii:gii!!jiii~i~ii iiii:iiiitiii iiiiii:iiii iiiiiiiiiiiiiii!11ii-.i::ii —::ii-i!!Uiii- i i iig i ~ii~~~~~~~~~~~~~~~~~~~~ii ~~~~~ ~ri iiii~~~~~iiiiiiii ~:ii~ iliiii~~~~~~~~~~~~~~iii~~~~'Iiii iiiiii~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~iiiiiiii!!iiiil~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~::~iii~~: -:.:.. i69~ii-__-: iiiiiZ ~ ~ ~ ~: iziiiiiiiiii ii-.ij~~ipr_,ii ii?!i!!iC~Jiij;l~* ~e~~ ~i~qii Fig. 22. Photograph of tank 2o The electronic equipment is at the far end of the tank. The hydrophone is suspended fromthe trolley riding the track mounted adjacent to the cat-walk.

-52waves of controlled frequency and amplitude. Plane wave fronts 20 feet or more in width can be produced. 3.2.2. Continuous-Wave Experiments 4Kcps -- The first c-w experiments were performed to determine the extent to which the tanks approximate an ideal situation, i.e. rigid bottom and no wall effects. It was decided to measure the pressure amplitude varin —. tion of the received signal as a function of range for a condition which could. be treated without unreasonable difficulty using normal mode theory. The ratio of water depth to radiation wave length (H/X) was fixed at a value of 2. Projector depth d and receiver depth h were held constant and equal to X. Under the presumed ideal conditions four normal modes should be propagated The field has been measured in both tanks under these conditions employing the 1K projector and the 3A hydrophone as the receiver. Ranges employed were in general no greater than 30X. A representative plot of the measured field is given in Fig. 23 and is compared with the theoretical field as calculated in section 3.1.1 for a rigid bottom condition. Since the theory becomes rather difficult at ranges less than 10X or so, no calculations have been made in that region. An adjustment has been made in the mean level of the calculated curve to bring it to a level comparable with the experimental curve. Several features of the experimental data should be notedo 1. In the range employed the plot compares most favorably with the rigid bottom calculation (as compared with the calculations of section 3.1.1 for the fluid and elastic bottoms). 2. The measured field shows an additional amplitude peak at the range of 25.5 feet (as compared with the rigid bottom calculation).

481 44 40A 36I\ / 1 (I)\ I I -32 - I i\ I I'\ I I ~28 z24 o I I Ld\ _I\ I \ \v =1I I\ I3 I I ~~~I Il I,,. $20 I I I I j I'-1 w I -r I ~I6I I I \ nA j I /%/ 161 1 I I 2 n~~~~~~ y: ~~~~~~~~~~~~~~II I i' Ii, I~ ~ ~ ~~~~~~~~~~~~~~~~~~~ I Ii v 8 I'III V Il II I 1 I V~~~\I I I 4 - v''I Y 4 I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~' J V~~~~~ 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 RANGE, FEET Fig. 25. Acoustic pressure variation with range. Frequency = 4 kcps, 1/2 = h = d = X. Theoretical curve is solid line, Measured curve is dashed line.

3 "Fine structure" in the amplitude increases with range with an indication that constructive interferences are occurring at range inter — vals of \/2. (Some data not shown here indicate still further deterioration of the pressure field at greater ranges.) With regard to statements 1 and 2 above it should be recalled that the calculated fields for rigid and fluid bottoms are not far different. A significant point is that both the rigid and fluid bottom calculations result in a field with four propagating modes. The elastic bottom calculation of Fig. 13 is based on certain assumed values of the compressional and shear wave velocities in the bottom, and at the frequency employed shows five propagating modes. Two of these correspond to the fourth rigid bottom mode. Although such a calculation has not yet been made, it is anticipated that a new calculation using the recently measured velocities of the bottom propagated waves will show closer agreement between the measured field and the elastic bottom calculation. The above-mentioned fine structure on the experimental curve is barely noticeable at the shorter ranges and becomes increasingly prominent at the larger ranges. The peaks appear at intervals of 0.5 - 0.75 feet. Since the standard datum interval is 0.25 feet and X 0.6 feet, one deduces that the observed fine structure is caused by constructive interference at range intervals of X/2. If one examines amplitude versus range plots of data at values of h other than X but under otherwise identical experimental conditions, one finds additional fine structure with a similar spacing of peaks. Between 7 feet (the shortest range of observation) and a range of about 25 feet the interference peak-to-trough amplitude difference is often up to one-half the total signal amplitude. Between the ranges of 25 and 40 feet the apparent x/2 spaced interference peaks are significantly reduced both in number and

55 in: relative amrplitude. As the range is increased from about1 40 to 55 foet (the largest ran-ge of observation) the interference tends to build up again. The structure just described is superimposed on -the apparent modal ampl.itude variations whose depth dependence is shown. in sectlon.l 3setol In attempting to offer a plausible explanatio:n for the presence of this anomalous interference pattern, one can quickly eliminate the possibilities of side-wall and projector end-wall reflections. It is obvious that with the side walls located 26.5 feet away any inelterference by reflections from these walls would result in interference maxima at range intervals of somewhat greater magnitude and increasing spacing as the traverse ran.ge increased. On the other hand the projector end-wall reflection will be added to the direct path signal with a phase difference that remains constant as -the traverse range is increased. Corner reflections as possible explanations can be ruled out by similar considerations. There remain. two feasible sources of interference -- the hydrophone end —wall reflection and modes of a higher order. As the traverse range is increased by an increment X in moving the hydrophone toward the hydroph.olne end. wall the path length of the end-wall reflectionl is decraeased by the same increment.~ It carn be seen then that interference peaks -with the observed spacing would arise from such variations in path lengths. Furthermore the amplitude of the interference would increase as the transverse range increases,. Amplitude-wise the problem is more difficu.lt, and. wtll be di.scussed ~irn section 3.2035 It will. suffice here to say that the ampl.itulde of the endwall:reflection observed in pulse experiments does not appear to'be sufficienttl.y large to provide the pattern discussed above. It may be, however, that wlith a coninos us-wave sficgnal the standing wave produced by this refl ecet ion builds up to sufffici ent ampli.tude.

-56That the interference is caused by the re-radiation of bottompropagated energy has been considered highly improbable. There exists no evidence for bottom waves with wave lengths of the value necessary to provide the observed spacing of the interference peaks. Ratler it is more likely that one or more modes of order number greater than four are contributing significantly to the sound field. In both the rigid bottom and fluid bottom calculation of the sound field for the conditions H/2=h=dj —>,four normal modes were propagating. Higher order modes appeared highly damped. It is possible that one or more of the latter modes is not as highly damped in the ranges used in the experiment as was presumed. Such a mechanism has the advantage of explaining also the apparent decrease in amplitude of interferences with range in the measurements at other than mid-depth. Computations and measurements are in progress to attempt to determine the effects on the sound field of various degrees of attenuation of these higher order modes and to examine their depth dependence. It has been shown? as will be discussed in a later section, that the pulse signals lose little energy per bottom reflection. This fact enhances the possibility that the higher order modes may not be as highly attenuated as was first believed. Frequencies below 4 Kcps -- In order to gain added knowledge of the acoustic properties of the bottom of the tanks a number of experiments have been carried out to measure the frequencies at w-hich the various normal modes "cut-on" or being to propagate. It was intended to make a comparison of the measured cut-on frequencies with those calculated assuming each of the various bottom conditions.

-57As shown in the preceding theoretical sections for a constant water depth and bottom condition there is a critical frequency for every normal mode below which a given mode will be attenuated. The nature and degree of dependence of these frequencies upon the depths of the layers of the bottom media and upon certain of the physical constants of these media hasve been calculated and observed heretofore with consid.erable agreement in some of the less complex situations. In a specified, constant experimental situation one might expect to be able to observe the onset (rapid decrease of attenuation) of at least a few of the modes of lower order number by observing the change in amplitude of the field with range as the frequency is increased. From theoretical considerations one would anticipate a linear relation between the logarithm of the amplitude and the range after accounting for cylindrical spreading for frequencies below cut-on of the zeroth order mode. When'this mode is fully propagating the curve should remain straight, and upon being corrected for cylindrical spreading should approach a constant average amplitude as a function of range (barring other sources of attenuation). Such change represents the increase of energy propagated by the now non-damped zeroth order mode. Cut-on of the first order mode should give a regular periodicity to the amplitude of the field at a higher mean amplitude level. The advent of each of the higher order modes results in a more complex amplitude variation at still higher mean levels. The values of the mean amplitudes with various numbers of modes propagating, the periodicities, and the frequencies at which these changes occur are related to the layer depths and physical constants of the media involved as discussed. in earlier sections. In Table II are the calculated cut-on frequencies of the various orders (n) of modes for each of three postulated bottom types. Values of

the hysical constants used are the best presently available for the aterials8 - in the concrete tank. Values of cl, b, and H are undoubtedly reliable. The values given for cc and cs are those derived by methods discussed in section 3.2.3 of this report concerning one-millisecond pulse experiments. The formulae for cut-on frequencies may be derived directly from the period equations developed in section 3.1.2. These formulae are given below. fn = n/2i For rigid bottom: c1(2n+l)it n 2H For fluid bottom: c cn+) n 2 2H C For elastic bottom: / CS r -1 C 1cc L. = t tan- _ c - +- n Explanation and values of symbols used in Tables II and IcI~ L) (angular frequency in radians per second) f (acoustic frequency in cycles per second) n (mode order number) pp (principal part) c1 = 4800 ft/sec (velocity of sound in water) cc (velocity of compresslional wave in concrete bottom) cs (velocity of shear wave in concrete bottom) b = 0~43 (ratio of water density to concrete density) H = 2o37 ft (water depth)

-59TABLE II Calculated cut-on frequencies fn for different types of bottomo cc 12,300 ft/sec, Cs = 7,070 ft/sec. Bottom type n f Rigid 0 1013 r rad/sec 507 cps 1 3040 0t 15'20) 2 5o8o0s 25440 3 7 L00C 3550 4 _9120 S 460 Fluid 0 1080i 540 1 3250t 1625 2 54ooai 2700 3 7580it 3790 4 97o40o 4870 Elastic 0 1800i 900 1 4520or 2260 2 7260ot 3630 3 10, 000i 5000 4 12 800oot 6 400 TABLE III Calculated cut-on frequencies fn for different types of bottom. cc = 16,250 ft/sec, cs = 8,960 ft/sec. Bottom type n 0f Rigid 0 10135T rad/sec 507 cps 1 3o4o0t 1520 2 508G0 2540 5 _7 _ _ l00_t... 2 Fluid 0 1064t 532 1 3192- 1596 2 55320t 2660 3 4~8~n~'47724. Elast ic 0 155~ 0 775 1 3960a 1980 2 6375y 3188 3 8780t 4390 In the experiments" conducted for the purpose of observing the amplitude behavior as a function of frequency-, conditions were set swuch that h - d = H/2 =- X for a frequency of 4 kcpso The frequency was varied thereafter and amplitude measurements were made as a function of rangeo. One set of data

-60was taken using 8 millisecond pulses and suitable band pass filters. There was essential agreement between this and the c-w data. Figures 24 and 25 show some of the c-w measurements with cylindrical spreading removed. For this experiment the voltage at the projector (1K) was adjusted at each frequency to produce identical hydrophone outputs at a range of 1 foot. This procedure obscures the observation of mean pressure amplitude increment increases as modes above the zeroth order mode cut-on. To deduce this type of information from the data one must introduce corrections for projector output variations with frequency (which are available only for a free field situation) which implies knowledge of the effect of the experimental parameters on the impedance of the projector. The impedance was not measured. However, several factors of interest are available from the data of Figs. 24 and 25. BelQw 700 cps the field is highly damped and the amplitudes in db fall off in range almost linearly. In this frequency region, which is apparently below cut-on of the zeroth order mode, there are slight incremental decreases in attenuation with each increment increase in frequency. The curves are not straight lines, but are somewhat concave upward with definite indications of a major decrease in rate of fall off beyond a range of 6.5 - 7.5 ft. The concavity and the "elbow" at the range of about 7 ft may be attributed to the higher attenuation of the higher modes. The incremental rises in amplitude with frequency are probably caused by the approach to zeroth order mode cut-on. The apparently anomalous behavior of the 400 cps curve is believed to be a result of the beginning of a deterioration from a sine wave of the projected wave form at this frequency, a phenomenon that progressively worsens at still lower frequencies.

-61-62I)O0 cps N,\ ~~~~~~ "",800cps -5 _ U_, 1;, -20 w w z -25 o z -35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 RANGE, FEET Fig. 24. Measured sound intensity variat ion with rage at differ frequerncies. H/2 =h = d =X~(4kcps). The correction for cylindrical spreading has been made. + 0 0 -j I -- z 0 -30 -3Id I 1 I I I I I I I I 2 545 67 89I 0 I 1: 41 61 81 0 1?_ 3?_ 52. 82 0~ ~ ~ ~ ~ ~~~~~~~~~AG~FE H ~ ~ ~ ~ ~ i~2oC~~.uto~o i~2.

Between 600 and 1000 cps it appears that the zeroth order mode becomes fullyv propagating. The data show rapid decreases of attenuation as a function of frequency with the major jump occurring between 700 and 800 cps. From these data it would appear that the mode cuts on relatively slowly with increasing frequency and that indeed it does not become fully propagating until a frequency of 900 or 1000 cps is attained. This is to be compared with predicted cut-on frequencies of 50'7, 540 and 900 cps for rigid, fluid, and elastic bottoms, respectively using tank experimental values for velocities in the bottom. Substitution of Shell Exploration and Production Company values for bottom velocities in wet concrete yields analogous cut-on frequencies of 507, 532, and 775 cps. In Fig. 25 amplitude data for the frequencies of 1800 and 2800 cps are shown. It is fairly evident that at both frequencies the spacing of maxima and minima of the amplitude is uniform with smaller spacings at the higher frequency. This trend is further exemplified by a still shorter spacing at 3300 cps as revealed in the corresponding 8 ms pulse data. Ideally the trend is predictable, and the exact spacing is also predictable if the proper longitudinal and shear wave velocities and the density of the bottom are known. The problem is to compute the difference in range between two points in the periodic variation of the field which are one cycle apart. If one chooses a frequency at which two normal modes should be propagating, the real part of the field of each of the modes can be represented by the form A0sin(kor-wt) and Alsin(klr-ut) respectively (for modes of order numbers 0 and 1). Here A is the amplitude, r the distance from the source, ) the angular frequency, t the time, and ko and kl the k numbers associated with each mode (see section 3.1,1)o Constructive interference between modes will occur where the two modes are in phase or at range r1 where

kor1l klrl 2ni n = integer and at range r2 where kr2 - klr2 =2(n+l)Tc From the above equations one finds the desired spacing 2it r2 rl Ar k2 (1) 0 1 The frequency chosen for comparison was 2800 cps. At this frequency if one assumes an elastic bottom and uses the Shell Exploration and. Production Company bottom velocity values, two propagating modes are predicted. The spacing between maxima produced by these two modes can be computed as above. The two-mode spacing has been calculated for this condition. For the calculation of the spacing considering an elastic bottom an iteration process must be employed to operate on the period equation and thus arrive at values for the k numbers involved. The period equation as derived in section 3.1.1 is given below. x 2 x2 (c - h J-2- 3 tan xb- 2 _ - X bH C, H2 Ai X The meanings of the above symbols are: 1. = 4.,800 ft/sec (velocity of sound in water) Cs - 8,960 ft/sec (velocity of shear wave in concrete) c = -16,250 ft/sec (velocity of compressional wave in concrete) H - 2e37 ft (water depth) b - 0@435 (ratio of density of water to that of concrete) - 2i x frequency = 2 (2800)

-65222_ - tan x = 0.982x n - 3.86 (.268 2k) (-.5 k k2 - 1.17 n where kn = k (n/2) (t/) and k2 = 2/c 2 By iteration the k's are obtained. ko = 2.20 for x = 6.94 radians k1 = 3.15 for x = 4.42 radians From Eq. (1) Ar =6.6 ft 0,1 From the measurements of amplitude versus range in the tank at 2800 cps one obtains a spacing of 3 feet. Obviously this is not in agreement with the value computed above in the elastic-bottom two-mode calculation. The reason for this behavior is unknown. A combination of a low amplitude for one of the two propagating modes at mid-depth and a beginning of the third mode cut-on (predicted at 3188 cps) might possibly result in such a spacing. Both experimental and theoretical effort on the possible contributions of modal amplitude depth dependence, branch line integrals, and highly damped modes at this frequency are high in priority for future work. Another factor to be noted in the figures is the increasing "fine structure" of the amplitudes with increasing range. This is similar to the interference noted previously at 4000 cps.

-663.2.3. Pulse Experiments Considerable effort was expended in experiments using short pulse signals both at 10 kcps and 4 kcps for conditions where h = d = H/2 - X. The possible value of the short pulse experiments, it was believed, was twofold. First it was expected that data could be obtained at least over a portion of the range free from wall-reflection interference. Secondly, one should,be able to get a direct measure of the amplitude of the wall reflection relative to the direct field. With the closest walls to the range line in the two tanks being 20 feet and 26.5 feet respectively, one expects no interference with the direct path signal from reflections from these walls using a 1 ms pulse over the entire range used in the experimental work. With a 4 ms pulse this range of non-interference of side-wall reflections with the direct pulse would be limited to about 30 feet in the narrower tank. If one considers elongation or distortion of the received signal due to image effect or mode group velocity differences then this non-interference range becomes somewhat shorter. Wall Reflections - Experiments designed to examine the amplitudes of side-wall reflections have been carried. out at 4 and 10 kcps in various depths of water. at each depth where both frequencies were employed all experimental conditions were identical except for pulse lengths. The projector and receiver hydrophone were at mid-depth in all cases. Ideally, it would have been desirable to use 1 ms pulses in all cases. However at 10 kcps only 2 ms taperecorded pulses were available. Table IV lists the water depths used, frequencies, pulse lengths, and ratios at ranges of 20 feet of the amplitude of the direct water path signal to that of the side-wall reflection wherever the latter was observed.

-67TABLE I-V Ratios of amplitudes of direct signal to side-wall reflection for pulses at 4 and 10 kcps and various depths 4 kcps 10 kcps Water depth Pulse length Amplitude ratio Pulse length Amplitude ratio at range=20 ft at range=20 flt Direct Reflected Direct. R ef lec ted 0.95 ft 1 ms no reflection observed 2 ms 6 1 2.36 1 It Y 2.86 1 3.00 1 35.70 1 3 1 2 111 It should be noted that H 0o95 ft 2 at 10 kcps aznd H = 2.36 ft = 2X at 4 kcps. Under these conditions and the assumption of a rigid bottom it has been calculated that 4 normal modes should propagate. Also when the direct path range is 20 ft the path length for the side-wall reflection is 44.6 ft. It is seen above that there are no discernable side-wall reflections at 4 kcps except at the deepest water depth. Here at the range of 20 ft the wall-reflected signal is superimposed upon the higher order top- ard bottom,reflected pulses which are of comparable amplitude. Acknowledging the difficulty of amplitude measurement imposed by the presence of top-, and bottom,reflected pulses superimposed on the side-wall reflections it appears that the ratio of the direct path signal amplitude to that of the wall reflect io i is of the order of 3:1 at this deepest depth. On the other hand it is to be noted that for 10 kcps there is clear evidence for the wall reflections at both the shallow and deep dept'hs estimated ratios of direct to wall-reflected signal amplitudes are (61 and 1:1 respectively at a 20 foot range.

Over the traverse ranges employed the path length of the side-wall reflections in the tank of greatest width (where most of the c-w data was obtained) varied between 50 and 75 feet. The path length for the hydrophone end-wall reflection varies between 75 and 130 ft. Some- time was spent in pulse experiments searching out the end-wall reflection. Positive identification was made by correlating times of arrival of the reflected pulse with changes of range of the hydrophone. Since all the effort at the time these measurements were made was in pulse work, this reflection was deemed a negligible influence on the field for two reasons. Its amplitude was very small, and its arrival time was such that the reflection did not interfere with the received signal over most of the ranges being worked. The later appearance of interference peaks in c-w experiments at range increments of approximately X/2 indicates that at the long traverse ranges where one is working relatively close to the end wall and the reflected signal amplitude becomes more nearly comparable to that ofthe direct signal, such interferences may be the result. In order to give a plausible interpretation to these observations, several other facts believed to be pertinent are described in the following paragraphs. To date almost all ranges employed in both the c-w and pulse experiments have been of the order of 20 - 35 ft. It has been noted in section 3.2.2 "Continuous-Wave Experiments?" that essentially good agreement between the c-w pressure amplitude measurements and the rigid bottom calculation has been obtained. These c-w measurements and those made in the mode cut-on experiments both show an increase in "structure" at the longer ranges measured. In a subsequent section 3.2.2 "One Millisecond Pulse Experiments" there is a discussion of the separation in time of source image reflected pulses and of the measurements of their individual amplitudes as a function

69of range and incid.ent angle It will. suffice at this point. fto say that the measurements provide no observable loss on reflection at the bottom. Ia additlion there is a small peak in the amplituudes of each of these source image reflections at an incident angle of approxima-tely 23~. The data further suggest that this phenomenon is followed by a slighlt decrease in amplitude and a subsequent rise as the incident angles increase. These effects are quite small and are observed as slight perturbat.ions on the reflected pulse ampllitude range cu.rves -Which seem otherwise to cojnform within experimental error to the spherical spreading lawo At these ranges anld availablJe water depths, _i is certainly questionable as to whether measurements are being made in the far field.~ That is, the data takera may be indicat.ive of intermediate ranges wherein certain pertinent phenomena are occurring. These phenomena include the possibility of a spherical wave-front. which would influence phase considerations and reflection coefficients also the change in inc id.en-t angle for the various source image rays from 0 to angles beyond those critical for cut-off of compressionalI and shear wa-ve propagation in the bottom does not ordina.7arily occur in the far field. Any vali d hypothesis regarding the nature of the propagation. in the taiks must satisfactorily reconcile the observations mentioned above. These are s-ummarized as follows -- T1. Wall. eflections decrease (with respect to direct signal) with decrease in water depth. and witbh increase in frequency at a given range. 2. Over the limited range considered the measured field agrees moderately with the rigid bottom calculation. 3. The source -image amplitude range depende:nce indicates a high bottom reflectionl coefficient at all angles of inc idenceo 4o The soure image amplitudes possess a small peak at an angle of incid.ence of 23~, an.d then a slight fall off anld. subsequent recovery as the incident angle is increased.

-7oEffect of Variations in Pulse Length - In section 3.1o2 calculations are made to show the time dependence of the pulse shapes for the normal mode approxi.mation and for the summation of the image series with a rigid bottom assumption. For a 3 ms pulse (H/2 = X = d. - b. - | ft) it can be seen that in only a fraction of the received signal are the four normal modes in time coincidence because of the different group velocities of the modes. That is, one would be compelled to be extremely careful in observing the received signal. in order to measure the amplitude corresponding to the vector addition of four modes. Experience with this experimental arrangement in. the model has demonstrated that one must go to pulse lengths on the order of 6 to 8 ms in order to obtain a field which corresponds with the c-w field at the ranges used here; the pulse length must be longer for larger ranges. Figure 26 shows the results of range runs (H/2 - = d - h = 1 ft) using 4, 6, and 8 ms pulses. Only the 8 ms pulse lerngth provides a field which corresponds closely with the c-w field and,r with the exception agai.n of the additional measured peak at 26 feet range, corresponds with the rigid bottom calculat ions. The shape of 1 ms pulses in the tanks is qualitatively similar to that predicted. in section 311o2 and shown in Fig.o 18 Experiments with various values of H and. w'ith several values of h and d have been performed. The records from these clearly demonstrate the arrivals of the pulses reflected from the water surface and. the bottom and extendling to very high order reflectionso Greater values of H serve to separate clearly the individual reflected. pulses. A more complete discussion of the 1 ms pulse data is given later in this section. No attempt has been made thus far to achieve detailed agreement between experimental pulse shape and that predicted. It is hoped that in the future the pulse shape may be studied in more detail. To date the prime

28 - - Pulse length=4ms -- -- Pulse length=6ms Pulse length=8ms 24 z \ LLJ~~~~~~~~~~~~~~~~ \ c 20 a.6 \ "' \ o~~~~~~ I I 2 \! /~ //_qH 3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - ~~~~~~~~'-/'J /i a 12 ~~~ ~ ~~~~~~~~~~~I I / -_~\,J I -' \ LI-i I % I CO (' / U; 4 V 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 RANGE, FEET Fig. 26. Measured pressure amplitude variation with range for different pulse lengths. Frequency = 4 kcps, H/2 = h = d = X.

purpose for varying the pulse length has been to attempt to observe the field in the absence of wall reflections, although this is not to minimize the value of other information gained in connection with the pulse experiments. One Millisecond-Pulse Experiments - It had been planned at the outset of-. the tank work to investigate the behavior of very short pulse lengths. It was believed that it might be particularly interesting if the cycles in the pulse could be readily discernable in order to examine interferences in detail. The image-like appearance of the received signal in the instances of the shorter pulse-length experiments indicated that a considerable amount of information might be obtained through the use of even shorter pulse lengths. A number of features which one might wish to examine by these means are: a. top and bottom reflections, b wall reflections, c. normal mode build-up mechanism, d. bottom transmitted energy. Various experiments have been run using one ms pulses. The main parameter varied was H, the water depth. Other factors that were varied were h and d, (receiver and projector depths), and the position of the range. The experiments were run in both of the two available tanks. The first tank used allows a maximum water depth eight inches greater than the second and possesses more cracks in the bottom. The significance of the cracks is clarified in later paragraphs "Study of Bottom Characteristics". A loop of magnetic tape was made on which several one-millisecond 4 kcps pulses were recorded, The transmitted signals were examined and their shape at very short ranges was quite similar to that of the pulses recorded on the magnetic tape. The x-axis of the oscilloscope was expanded so that the four cycles of 4 kcps pulse were clearly separated. Figure 27 is a representative plot of data taken when H = 3 ft and h = d = 1.5 fto The range was along the center-line of the tank parallel

TIME OF ARRIVAL, ARBITRARY UNITS N) N N 4,0 (D a)N 0 OD) N )0 t-'~j~. 0 P o - ~ -(D Ot _ C H (DCD Q PDH ~ - ~ DII C) Y Ht' ~ I c Q (B H-' ct 0 Pi1 Pi —-,d 4C &(D p( c0 G + I1 00...... Dc+ c+ co H0 <:Fd0 0 r~ 1 P~3 y P3 5 c- (D O" 1D I 0 > ro.... PI ": c m Pi~"' H-FJ II CD oC ~ l(D OD FoI N N ~-I- r m (PH' NN ct _-, ci-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~N II(D' F "d ~ r~ N 1 c3 0 N HC+c * tr (!-o c co~ 0 +0 ND F N [O) II (D o t('D O~ 0 II ~~~D 0 ~ O':I' F.-' t09

to the longer (side) walls and midway between the shorter (end) walls. The curves are the calculated times of arrival (TA) of the various even-numbered top and' bottom reflections at the ranges showno. For example, n = 4 is the case where the reflected signaal had experienced two top and two bottom reflections. (See Fig. 28.) It is apparent from the small signal size that the.L80 phase shift occurring at the surface reflection and, incidentally, the lack of significant phase shifting at the bottom, eliminated odd-numbered arrival.~ This means, of courseg that:effectively the odd-numbered reflections of equal path length are phased out. water surface projector receiver V\/~~~ L \/;bottom Fig. 28. Ray diagram for sound paths where n - 4 (2 top and 2 bottom reflections). Most of the spread in the data (length of the vertical bars) can be attributed to the method of reading the pulses. Times were measured on the oscilloscope photographs from the sweep start to the peaks of the highest amplitude positive half-cycle and highest amplitude negative half-cycle in each pulse at each range. This procedure results in at least a half-cycle to a whole-cycle spread in the data. Very slight varilations in the start of the oscilloscope sweep also contributed slight errors, The curves on the plot represent the TA s of the pulse "centers" minus a time corresponding

-75to one-half the pulse length. The length of the vertical bars represents the spread of the measured time of arrivals; the bars are arbitrarily spaced on the graph at 16 times the measuring range intervals. With the velocity of sound in water constant, the change in path length with increasing image order number is a function of the water depth and range. For a given pulse length and over the ranges here considered there exists a minimum depth at which the received signals of consecutive order images just fail to overlap each other in time. As the depth is increased beyond this minimum the signals separate further in time. In a deeper water experiment the separation of the pulses is more readily discerned. This fact results in clearer observation of reflected pulse arrival times and amplitudes. With the projector and receiver not at mid-depth the path lengths for the odd-order reflections are no longer equal. In an experiment where H = 3.7 ft and h = d = 0.84 ft one can note the effect just mentioned. The phase shift between top and bottom reflected odd-order images is no longer 1800 due to the path length differences, and signals of significant amplitudes are introduced. (See below.) Figures 29 and 30 show photographs of the oscilloscope screen taken at hydrophone ranges of 1 foot and 17 feet, respectively. At one foot there are few reflected pulses. The wall reflection appears toward the right end of the sweep. At a range of 17 feet several even-numbered reflections can be seen. Gain increases between two and 17 feet have been made as the direct pulse decreased in amplitude in order to observe the detail in the reflected pulses. In comparing a plot of the measured times of arrival with the calculated times it is seen that when the projector and hydrophone are not at mid-depth the odd-numbered reflections are not cancelled by phase shifts.

-76Fig. 29. Photograph of oscilloscope screen showing received signal at a range of i foot. Pulse length = 1 ms, frequency = 4 kcps, H= 3.7 ft, h = d = 0.84 ft. Fig. 30. Photograph of oscilloscope screen showing received signal at a range of 17 ft. Pulse length = 1 ms, frequency = 4 kcps, H = 3.7 ft) h = d = 0.84 ft.

-77The calculation, shows a group of families of cllrves, each family consisting of three closely spaced curves,, Tihe trar:smiss:. paths which yield these curves are showr1 i'~.t Fi g i 31Fg.571. IiL a g'iverL fam'ily, the ce.ter curve is for an even-n.'u.mbered. reflec.t i'.o time of arrival.:, then irmledAiately below it is the curve for (n-1 )Do The subscript D in-dicates that the.. ef lected. pulse left the projector in. a dcwr,T:.ward. direktioxn.o Similarly, im.rlediately above the n curve is the curve for (n.+L )T where the subscript U in.dicates the reflected pulse-.Lft the pro',ec'tor i.z an,'; upward direct+io:an The comparatively small. diffe re-es ain path length betwee:r. these t'hree r(efalected pulses account for the grouping i-.nto a "fami. ". Th. sp(read. of the plotted data shows the broade. iir.g.r-freslt xg fom ~.':ro this The shapes of the calculated and measured curves arco id.os-.;.ti.cal. except:for a slight error due to th;e nonlinearity of theI-N osciDl.oscope sw'eepo water surf ac e projector' r" ~ce ~ver 7/7770 - 7roe.]0 /e r'Fig. 31., Ray d.iagram for sou.:rd. paths wF-here nT — T n -- 49 (:rntl)TST'5, (5 3(

Referring again to Fig. 30 the appearance of reflected pulses at the longer range suggests an irnvestigation of the amplitudes of the reflected pulses as a function of range. The pressure amplitude of each of the evennumbered reflections was measured, corrected for spherical spreading, and plotted as a function of range. Such a plot for n = 4 is given in Fig. 32. Here one might expect, as predicted in section 3.1.2 of this report, to find an ind.lcation of the range at which the critical anrgle of reflecti.on for n 4 is exceeded. The rise in pressure amplitude giving evidence of a peak at a range of 6 feet is interpreted as an indication of such a cut-on, As is shown in the figure the critical angle calculated for this cut-on is 2'20. This may be inrterpreted as the critlal angle for the compressional wave in the concrete, Likewise predicted by section. 3.1.2 of this report is a cut-on or critical angle for the shear wave propagated in the bottom. The experimental evidence, though very tenuous, suggests that the critical angle for shear wave is approximately 43Qo Similar plots for higher orders lend further credence to this interpretation. The mean measured compressional wave critical angle for six different orders of even-numbered reflections is 23527' with a maximum deviation from the mean of 10241". In the same range of n's only two vague suggestions of shear wave critical angle can be foud giving angles of 42033' and 420551. These critical angles yield compressional and shear wave velocities in the bottom of 12,280 ft/sec and 7,070 ft/sec, respectively. The characteristics of the bottom are discussed later in the report. These values of velocities, it should be noted, are reasonable for concrete. The probability for the data to yield such. consistent values of the compressional wave velocity by chance. is computed to be less than 1:370. This ratio was calculated as follows, STever different even-ordered reflected pulse amplitude vs range curves were measureabl.- Po = 2 through n = 14). Two

z::15 Q Angle of incidence=22005' Angle of incidence= 42055' cO 10 _J W.~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~\ a-: 5 U) Cn UwCx 2 4 6eI0 12 14 16 18 20 a. RANGE, FEET Fig. 32. Measured pressure amplitude variation with range of 1 ms, 4 kcps pulse for n = 4 (2 top and 2 bottom reflections).

-80of the seven did. not meet the acceptance criterion which was that after correction of the amplitudes for spherical spreading, the highest amplitude peak on each plot must fall within the total spread in angle of 2,50 centered on the critical angle. The average range of incident angles over which the amplitudes of the various orders of reflection were measured was about 150. One of the two rejected measurements gave a clear peak at 24~31' but a higher one existed at 28~. The second rejection gave peaks at 18~ and 260 but at a range where the reflected pulse and direct water path signal interfered. The above data are not consistent with the velocity information from the Shell Exploration and Production Company. The latter results give compressional wave velocities of 15,350 ft/sec and 16,250 ft/sec for dry and wet concrete and shear wave velocities of 9,160 ft/sec and 8,960 ft/sec respectively for dry and wet samples, Using the same value for the velocity of sound in water (4800 ft/sec) and'the Shell wet concrete velocity values, the calculated values of compressional and shear wave critical angles are 17027' and 28~33' The reason for the lack of agreement is not clear. Weinstein5 has observed experimental variations in. the value of the reflection coefficient from that predicted by plane wave theory when the angle of incidence is such that the change of phase upon reflection varies greatly with a small change in the -angle of incidence. in these regior.s especially one carnnot use a divergent beam several degrees wide to approximate at a plane wave. Weinstein demonstrates a shift of the reflection peak at the critical angle for the compressional wave and a major.'minimumn in the reflection amplitude in the region near the critical angle for shear. He uses a thick aluminum plate as a semi-infinite reflector. These observations are qualitatively similar to some of the reflection data taken in the tanks, Further experimental effort is indicated. It is possible that theoretical work now in progress will determine the correction factor needed to convert reflection coefficients obtained with

spherical -waves to the plane wave coefficientso Another possibility is that the introduction of a damping factor d.ue to shear wavave attenuatio.. Imay clear up the inconsistency. Attention is now turned to the consideratl.on of bottompropagated energyo Exhaustive search for evidence of this energy at the wosrki: g frequencies (4 and 10 kcps) have been made in. tank:number on.e. The neot result was that there was only slight evideince for some tow~-leve-1 bottom-transmitted energy. It was concluded that a better approach woul.d. probably be to invest.gate this phernomeneon by means of an. impulse source. However, since the signal at these frequencies was so weak that it was believed. of litt.le signif icance to the amplitude measurements, and t:he impulse-type of experiment was deferred. It Ls evident that joints and. cracks in the bottomr of the tank will strongly influence bottom propagated energy. These effects will be discussed in a later sectio'n. After the experiment was moved to the second. tank. anrd more ru.ns were made with one millisecond pulses, an.y existing doubt about the presence of bottom propagated enrergy was removed.. The photograph showr._. i.n. Fig.o 3 clearly shows a low amplitude early arri.vaalo It should be poainted out; that the data from which this photograpJh. -was selected were.lnot taken for the express pu-rpOSe of investigating bottom-transmitted enze.lfrgy. It is fe.Lt that these experiments can be improved in the future. The above-mentioned data give an apparent bottom-propagated wavre which preceeds the direct water-propagated wave at a range rou.gh:ly determined to be between 5~5 and. 6,5 feet. The uncertainties of the experimernt do not allow one to compute exactly from the data the veloci.ty of t;h.s bottom wavet Calculations reveal, however, that the measured range spread given above is close encugh to the calcu.lzated. range of the first arrival preceding the water propagated wave so that a presumed. shear or Rayleigh wave velocit;y of the order of 7000 - 9000Q ft/sec could account for the observatioino At somewhat

-82Fig. 33. Photograph of oscilloscope screen showing example of bottom transmitted early arrival (see arrow). Electrical pick-up at extreme left is not to be confused with the early arrival.

-83greater ranges there is evidence for an early arrival travelling with the approximate velocity of the compressional wave in concrete in addition to the one described. P.N.S. O'Brien6 has with a similar experimental arrangement employing exponentional pulse forms obtained quantitative data regarding the amplitudes of the so-called "head" wave. He worked with a water-over-concrete model and observed close to the interface direct to compressional and direct to shear wave amplitude ratios of the order of 1:100 and 1:10 respectively. One may deduce from these ratios a compressional to shear wave amplitude ratio of the order of 1:10. Since in our experiments the shear wave signal (if it is shear) is very low in amplitude, one might expect that the compressional wave signal is down in the background noise. It is hoped that the questions raised here can be resolved eventually. Further effort in this problem of bottom-propagated energy is planned. A number of runs employing a 10 kcps one-millisecond signal were made. These have proved difficult to interpret for the reason that a "spool effect" was obtained on the transmitted pulse. This phenomenon makes it difficult to identify reflected pulses and obviates amplitude measurements. 3.2.4. Study of Bottom Characteristics It is well recognized and has been further emphasized in the theoretical treatment of the model program that in shallow water sound propagation, especially at frequencies below 5 to 10 kcps, one must evaluate the degree to which the elastic properties of the bottom influence the sound field. These properties influence such questions as the amount of energy lost in the bottom, the energy re-radiated by the bottom, the velocities with which the energy is propagated in the bottom, and the critical angles of reflection at the bottom for water-propagated waves. For these reasons several independent measurements were made to determine the values of the pertinent elastic constants of the concrete bottom of the tanks employed in the experiments.

The concrete agglomerate contains pebbles ranging in size from that just visible to the naked eye up to those with a (-coss-sectio:n of about one inch, The d.ensity, as determined. by weighing a large representative bottom sample, is 2.3 grams/cc. The thickness of the concrete varies between five and. seven inches. It has been laid in. sections each about 30 feet square. The sections are separated by one-half inch. of an expansion joint material composed of asphal.t- impregniated paper. The bottom of the first tank used had been. further broken i.n many places; many cracks are clearly visible. The bottom of the second'tank exhibits no such crack.s The methods that have been used in an attempt to determine the elastic properties of the bottom are briefly described below. a. Dynamic method - Prismatice bars of concrete sawed from the bottom were vibrated in various longitudinal and torsional modes using a concrete testring apparatus in the Mich!.igan state EHighway Departmer.t Laboratory. No information was gained. du.e primarily to the difficulty in exciting suitable modes in the bars of the small dimensions availabl.e b. Static compression method - A pr..smatic bar of concrete was compressed and the resultant strain measured.~ No conclus ive information was obtained since strains were localized. by the inlomogeneiti es and yl.elded an excessive spread. in the datao co Inpulse method - A test bar, of the concrete was sent to the Shell Exploration an.d Develom.ent Company for a determination of t'he velocities of propagatior and attenuation of acoustical pulses. This was done in response to an offer by Dr. A. Ginzbarg of that company to make the meas-irements with an apparatus which they use for sl.ml.lardeterminat4ions. The method consists of coring the concrete sample, measuri.ng transit times and pulse amplitudes in the core, and then cu-tt lu.ng the sample into two unrequal. parts and repeatting the experiments

These pieces are then saturated with water after being evacuated, and the measurements repeated. The method gives three points on plots of time versus length and amplitude versus length for both dry and wet samples. The frequency of observed pulses was approximately 10 - 15 kcps. The velocity data from both dry and wet samples gave consistent values from sample to sample. The values of velocity as obtained from graphs of travel time versus sample length are shown below. Dry~. Wet cc (compressional velocity) 15,350 ft/sec 16,250 ft/sec cs (shear velocity 9,160 ft/sec 8,960 )ft/sec Shear wave attenuation was also measured by observing the dif — ference in amplitude between once and twice reflected shear pulses. Corrections were made on the basis that both attenuation and reflection coefficients affect the observed amplitude. The actual value for the shear attenuation in the wet samples seemed to be about 1/3 db/cm (within a factor of two). Such a value is close to the upper limit of attenuation and, in general, one must expect that the attenuation is actually somewhat less. d. Reflected pulse method -This technique has been adequately described above in the discussion concerning the one-millisecond pulse experiments. As was explained there a discrepancy exists between the values obtained in this manner and those obtained by method c. e. Early arrival method - Now that definite observation of early arrivals (see "One-millisecond pulse experiments" above) has been made, it should be possible with refined experiments to get a reliable check on the velocities obtained by the reflected pulse method previously described. In the second tank one can work within a single concrete block with no cracks in the bottom over the working range.

IV. SUMMARY AND CONCLUSIONS The results of the experimental work may be classified in two ways. One set of results provides information about the acoustic characteristics of the water tanks. The other deals with the more general problem, namely the characteristics of the propagation of sound in shallow wat er o 4.1.1. The Acoustic Characteristics of the Tanks The'Bottom - Various standard experimental techniques and one rather unique one have been employed to determine values for the physical constants of the concrete bottom of the tanks. These have been described in Section 3.2.4. From the experiments two sets of values for the velocities of propagation of the compressional and shear waves associated with the concrete have been obtained. Although the two sets of values differ in a ratio of 4~3, both lie in a range of velocities which are reasonable for concrete. Remembering the difficulty of ascertaining the true values for the physical constants of the concrete, a reasonable similarity has been found between the measured and calculated sound fields over the ranges employed in the tanks. It has been shown that the assumption of a perfectly rigid bottom is a fair approximation to the existing situation. In addition the calculation including the effect of a compressional wave in the bottom (fluid bottom) bears unmistakeable resemblance to the -86

-87measured sound field. The basic reason for the similarity between the rigid and fluid bottom calculations lies in the fact that in the situation for which calculations were made, four normal modes are propagating in either case. It appears that agreement between the calculation and the measured field will be improved by more accurate determination of the physical constants. In the short pulse experiments where the amplitude-range dependence of the source images was measured, it was found that over the path lengths and number of associated bottom reflections considered, little attenuation was apparent. This implies low energy loss per bottom reflection. It is necessary to postulate only a slight loss per reflection to account for considerable attenuation of the normal modes at long ranges. No evidence-has been found to lead the authors to believe that the bottom cannot be considered a semi-infinite half-space at least at the frequencies of interest. In the Appendix to Section 3.1 a calculation indicates that the above conclusion is probably correct.:Experimentally no reflected pulses attributable to reflections from the interface between the bottom of the concrete and the ground or from the ground below have been observed. 4.1.2. The Effect of Other Factors In the course of the experiments several factors, other than the bottom and its effect upon the acoustic characteristics of the tanks, were investigated. For example, the measured sensitivity of the sound field to variations in water depth, receiver depth, and frequency is shown in Fig.o 34a, b, c and the AH effect, Ad effect and Af effect, respectively. The plots of pulse data were selected to demonstrate the

-8828 26 (a)AH Effect 4 —-- H= 10-1/8 Inches 242 - H= 10-9/16 Inches 22 20 18 16 8. I I I l 26 o (b) A d Effect r 24# \i I -- -- d = 5 -1/4 Inches I \ // V - 18() Effect I'I / \ /. 6 /II \/ u> II I I6 V w6 4) 18go 34a3 b., Ge~(c)Af Effect 8 6

-89order of magnitude of the shift in the sound field with small changes in individual variables, They serve merely to point out the care with which the parameters must be controlled in order to obtain repeatable results. The degree to which the bottom approximates a level surface has been examined along the range under the trolley track. The steepest slope found amounted to 0.004 ft/ft over a 10 ft range. This is a slope of 14 minutes. Over a 40 ft range interval the slope was more like 0.0015 ft/ft or 5 minutes. Variations of depth and bottom slope of this order of magnitude are not believed significant over the range and with the water depths employed in these experiments. Water temperature has been. measured each day near the surface, at mid-depth and near the bottom. The total spread between the top and bottom is generally less than 20C. This has been considered a minor effect insofar as the gradient is concerned. To determine the sound velocity for use in computations an average temperature, which is near the mid-depth temperature, is used. The velocity of sound at the average temperature is taken from British Admiralty Tables of the velocity of sound in fresh water. (7) The problem of wall reflections has been examined. It seems clear that when conditions are such that more than three or four modes are propagating, wall reflections must begin to contribute significantly to the observed signal. For example this occurs with deeper water or higher frequency, or both. By shortening the pulse it is possible to observe in a region wherein the wall reflections do not arrive in time to influence the measurement. One approach to the solution of the problem of working with higher numbers of propagating modes is to change the scale factor. This can be done by lowering the water depth and raising the frequency to the limit imposed by the fact that the bottom is not perfectly level, To observe the sound fields at ranges longer than 40 or 50 X such a change is necessary anyway.

-90It does not appear that wall reflections offer a suitable explanation for the high amplitude interference signal superimposed on the modal field at range intervals of approximately X/2. The possibility that the interference is the result of re-radiated bottom-propagated energy has been posed. It is considered more likely that the normal modes of order number higher than four (presumed heretofore to be highly damped) are the source of this phenomenon. Particular attention is being given the attenuation of these modes in order to determine their contribution to the sound field. That the departures of the calculations from the measurements are probably related to the bottom constants is further emphasized by the results of experiments utilizing different bottom types. One set of measurements was taken with a fiber glass bottom. Asphalt covered fiber glass 7/8" thick was placed over the concrete along the range. The sound field was measured with H/2 = h = d = X. The periodic nature of the pressure amplitude vs range plot indicated a field with two propagating modes. Assuming the water layer was bounded by pressure release surfaces, a calculation of the spacing of the modal interference peaks was made. The calculated spacing was just three per cent different than the measured spacing. The fact that good agreement was obtained serves to indicate that the assumption of a pressure release bottom was correct and that no spurious effects were encountered from the bottom or from wall reflections. It may be inferred then that effort must be placed on the problem of arriving at consistent and reliable values for the velocities and energies of the waves propagated in the concrete bottom. Another experiment was carried out utilizing a bottom of sintered clay blocks laid over the concrete. The effect upon the sound field was that which would be anticipated with a bottom which is "lossy". In one millisecond pulse runs the amplitudes of the top-and bottom-reflected pulses were observed to fall rapidly with range.

_91= In summarizing the acoustic properties of the tanks it can be stated that they offer a definite promise of value in their employment as shallow water models. The degree of versatility of the model, particularly in investigating the sound propagat;ion mechanisms acting in. the near and intermediate fields, has been suggested in this report and is reviewed below in Section 4.1.5. It is believed that work in progress will solve the problem of the inconsistency in the values of the elastic constants of the concrete bottom. Once this has been done the more serious application of the model to shallow water sound propagation problems will begin. The restrictions imposed by wall reflections and/or concrete-propagated energy are not believed to be serious. The extent to which ranges can be lengthened probably in the final analysis will be determined by the bottom roughness o. 4.1.3. The Propagation of Sound in Shallow Water In the investigation of the acoustic properties of the tanks and the probable value of the tanks as models, several interesting factors related to propagation have been explored. The following summary of these factors serves to exemplify the different types of problems to which the model can be applied, First the convenience of the model for making rapid checks on theoretical results and on c-ertain calculations is evident. Wi thin the capabilities of the model one is able to set up an experlmelt and very often in the period of a day or two have a set of pertinent data. Su.ch a procedure enables both the theoretical. and experimental phases of the program to procede with a minimum of extraneous difficulty. At the present time this procedure is being utilized inz the in-^vestigation of the role of presumably damped modes and branch-linse integral contributions in the near and intermediate sound field.

-92It has been shown that the model offers a good facility for the determination of the frequencies at which the normal modes cease to be highly damped in a prescribed situation. With little additional complication of the experiment one can also expect to observe changes in the energy in the sound field associated with mode cut-ons. This type of experiment includes the capability for the examination of the excitation function as a function of frequency. Thus both frequency and amplitude studies of the Airy phase can be made. While no effort has been expended in this type of investigation, attention has been given to the possibility of exploring certain other modal mechanisms which are of considerable interest. These are the examination by means of pulses of the build-up of modes at longer ranges and the examination of the decay or the conversion of higher modes to lower modes due to scattering using a c-w signal at long range. Scattering caused by bottom irregularities (using larger scale factors) and by a roughened water surface are possible mechanisms to employ in the latter problem. An investigation of the effect of the shear wave attenuating in the bottom would also be of interest. In the former problem it can be shown that as the path lengths of the various source image signals approach that of the direct signal at long ranges, the pulses will merge in time to form modes. As these modes are formed one should be able to observe their group velocities and amplitudes. Some effects of varying pulse length upon the form of the received signal have been observed. Qualitatively these observations are in agreement with those computed using the approximate normal mode method and the image series summation method. It is believed that a rigorous experimental and theoretical program concerned with these c-w pulses could possibly result in the development of practical techniques applicable to the investigation of propagation mechanisms in the ocean.

-93In the matter of determination of reflection coefficients of the bottom it is proposed that the technique employed in Section 3.2.3 of this report be considered as a practical shallow water technique. Here c-w pulses were used, and the pressure amplitude-range dependence of the various orders of the received pulses was observed. The critical angles for cut-off of the compressional and shear bottom propagated waves (and so the velocities) were determined by observing peaks in the amplitude curves. The multiple reflections provide a convenient means of measuring mode attenuation. Simultaneous measurements at a few select frequencies could reveal a great deal regarding the bottom characteristics and transmission loss. Two preliminary experiments which were performed employing bottom types different from the concrete bottom have been described above. These represent lhe attempts made thus far to change the bottom material. The possibility of introducing various bottom materials and configurations in the model gives rise to a number of feasible experiments. One of the advantages offered by the model in this type of experiment is the possibility of extending the pressure amplitude measurements into the bottom materials. Thus direct measurements of velocity, amplitude, and attenuation in the bottom materials could be made. In summary it is believed that the model discussed in this report can be employed to investigate some of the probable mechanisms which may account for the attenuation of the sound propagating in shallow water. The more probable mechanisms include (1) depth variations; (2) attenuations produced by the non-plane upper surface; (3) attenuations due to scattering at the lower surface; and (4) attenuations produced by dissipation in the bottom.

(Continued from page 47) BIB1LIOGRAPHY 5. Marvin S. Weinstein, J. Acoust. Soc. Am., 24, 284 (1952). 6. Peter N.S. O'Brien, "Model Seismology - The Critical Refraction of Elastic Waves", Geophysics 20, 227 (April 1955)o 7. Mathews, D.Jo,'"Tables of the Velocity of Sound in Pure Water and Sea Water for Use in Echo-Sounding and Sound Ranging" 2nd Ed., Hydrographic Dept. British Admiralty, 1939 p. 282. -94

UNCLASS IF IED Distribution list for Technical Reports University of Michigan Contract N6onr 232(21), NR 385-203 Office of Naval Research Commanding O:ff icer and lDirector Acoustics Branch. (Code 411) U. S. N Underwater Sound Lab Washington 25, D.C. (2 copies) New London, Cormo (1 copy) Office of Naval Research Conimanding Officer and Director UnTdersea Warfare Branch (Code 466) U.oSo.No Electronics Laboratory Washington 25, D.C. (1 copy) San Diego 52, Califo (1 copy) Director C ommander Naval Research Laboratory Naval Air Development Center Technical ]Information Officer 7ohnsvlle, Pa, (1 copy) Washington 25, D.C (6 copies) Chief, Commanding O.ffic er Bureau of Ships (Code 847) Office of Nava:l Research'Branch. Office'Navy Department 1030 E. Green. Street Washington 25, DoC o (1 copy) Pasadena 1, Calif. (1 copy) Superint endent, Commanding Officer U oS. Navy Postgraduate School Office of Naval Research Branch Office Physics Department 10th Floor Monterey., Cal i.f. The Jbioh Crerar Library Bldg. Attn: Prof. L.E. Kinsler (1 copy) 86 E Randolph Street Chicago 1, Ill, (1 copy) Harvard UTniversity Ac oust ics Laboratory Commanding Off icer PDivision of Applied. Science Office of Naval Research Branch Office Cambridge 38, Mass. (1 copy).16] Broadway Ne-w York 135 N.Y. (1 copy) Cathclic U:nriversity of America Departmen.t of Physics Commanding Off icer Washington [-7, D o.C. Office of Naval Research Attn~ Dr. K. Herzfeld (1 copy) Navy No. 1009 Fleet Post Office New Yorkl, New York (1 copy) Brown University Department of Physics Office of Technical Services Providence 12, Rhode Island Department of Commerce AttnL Pro:f AO. Williams (1 copy) Washington, D.C. (1 copy) Mass. Institute of Technology Armed Services Tech. IOnfor. Agency Acoustics Laboratory Document Service Center Cambridge 39, Mass. (1 copy).nott Bu ldi.ng Dayton 2, Ohio (5 qopies) Director, Marine Physical Laboratory Director Un.iversity of California Naval 6Ordnance Laboratory U.So Navy Electronics Laboratory White Oak, Mdo San Diego 52, California (1 copy) Attnm Sound Division (1 copy). Hudson Laboratories Colutrb ia Univers ity 1.45 Palisades Street Dobbs Ferry, N.Yo (1 copy)

Distribution List (Conti nuned) Iamonrt Geological ObservatoryColumbia University Torre Cliffs Palisades, N.Y. (1 copy) Woods Hole Oceanographic Inst itution Woods:Fole, Massachusetts (1 copy-) Attn: L B. o ers ey UoS. Un.dersea Warfare Committee Navy Department Washington 25, D C.

UNIVERSITY OF MICHIGAN 3 9015 02826 6081