LAMONT-DOHERTY GEOLOGICAL OBSERVATORY 
 OP COLUMBIA UNIVERSITY 
 Palisades, New York 10964 
 
 THE INTEGRAL SOLUTION OF THE SOUND FIELD 
 IN A MULTILAYERED LIQUID-SOLID HALF-SPACE 
 WITH NUMERICAL COMPUTATIONS FOR LOW-FREQUENCY PROPAGATION 
 
 IN THE ARCTIC OCEAN 
 
 by 
 
 Henry W, Kutschale 
 CU1-1-70 
 
 Technical Report No, 1 
 
 Contract N00014-67-A-0108-0016 with the 
 Office of Naval Research 
 
 Work done on behalf of the U.S. Naval Ordnance Laboratory, 
 
 White Oak, Silver Spring, Maryland 
 
 Reproduction of this document in whole or in part is permitted 
 for any purpose of the U.S. Government. 
 

LAMONT-DOHERTY GEOLOGICAL OBSERVATORY 
 
 OF COLUMBIA UNIVERSITY 
 Palisades, New York 10964 
 
 THE INTEGRAL SOLUTION OF THE SOUND FIELD 
 IN A MULTILAYERED LIQUID-SOLID HALF-SPACE 
 WITH NUMERICAL COMPUTATIONS FOR LOW-FREQUENCY PROPAGATION 
 
 IN THE ARCTIC OCEAN 
 
 by 
 
 Henry W. Kutschale 
 
 CU1-1-70 
 
 Technical Report No. 1 
 
 Contract N00014-67-A-0108-0016 with the 
 Office of Naval Research 
 
 Work done on behalf of the U.S. Naval Ordnance Laboratory, 
 
 White Oak, Silver Spring, Maryland 
 
 Reproduction of this document in whole or in part is permitted 
 for any purpose of the U.S. Government. 
 
 February 1970 
 
Digitized by the Internet Archive 
 in 2020 with funding from 
 Columbia University Libraries 
 
 https://archive.org/details/integralsolutionOOkuts 
 
TABLE OF CONTENTS 
 
 Pages 
 
 Abstract 1 
 
 Captions for Figures 2 
 
 Definition of Symbols 3-5 
 
 Introduction 6-9 
 
 Formal Solution 9-39 
 
 Numerical Computations 40-56 
 
 Acknowledgements 56 
 
 References 57—58 
 
 Appendix 59-64 
 
ABSTRACT 
 
 This report develops by matrix methods the integral solution 
 of the wave equation for point sources of harmonic waves in a 
 liquid layer of a multilayered liquid-solid half space in a form 
 convenient for numerical computation on a high-speed digital 
 computer. Only the case is considered of a high-speed liquid 
 bottom underlying the stack of layers. The integral over wave 
 number has singularities in the integrand and is conveniently 
 transformed into the complex plane. By a proper choice of 
 contours, complex poles are displaced to an unused sheet of the 
 two-leaved Riemann surface, and the integral solution for the 
 multilayered system reduces to a sum of normal modes plus the 
 sum of two integrals, one along the real axis and the other 
 along the imaginary axis. Both integrals are evaluated by a 
 Gaussian quadrature formula. Sample computations are presented 
 for low-frequency propagation in the Arctic Ocean sound channel. 
 These are preliminary computations and the ice layer, which 
 averages three meters in thickness, is not included in the 
 layered system. The effects of the ice layer on propagation 
 are currently under investigation. 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2. 
 
 FIGURES 
 
 Figure 1. 
 Figure 2. 
 Figure 3. 
 
 Figure 4. 
 
 Figure 5. 
 
 Figure 6. 
 
 Figure 7. 
 
 Figure 8. 
 Figure 9. 
 Figure 10. 
 
 Multilayered half-space. 
 
 Contours for integration. 
 
 Variation of sound velocity with depth for Model A. 
 
 Table 2 gives additional parameters for this model. 
 Computations for Model A. Phase - and group-velocity 
 dispersion and excitation function of pressure 
 dependent only on layering. 
 
 Computations for Model A. Variation with range of the 
 absolute value of pressure for the normal-mode contri¬ 
 bution of the sound field. Source depth 150 m. Hydro¬ 
 phone depth 50 m. Source frequency 10 Hz. Source 
 pressure amplitude 1 dyne/cm re lm. 
 
 Computations for Model A. Variation with range of 
 absolute value of pressure for the integral contribution 
 of the sound field. Source and detector same as for 
 Figure 5. 
 
 Computations for Model A. Variation with range of absolute 
 value of pressure of the total sound field. Source and 
 detector same as for Figure 5. 
 
 Same as Figure 5 but computations carried to longer range. 
 
 Same as Figure 6 but computations carried to longer range. 
 
 Same as Figure 7 but computations carried to longer range. 
 

 
 
 
 
 
 
 
 
 
 
 
3. 
 
 Km ; 
 z ; 
 
 r; 
 
 t ; 
 
 oO ; 
 C ; 
 
 L 
 
 9 
 
 Xrvn; 
 
 (fc) 
 
 Mm/ 
 
 22 ^ 
 -fc 
 
 DEFINITION OF SYMBOLS 
 
 compressional-wave velocity in the m-th layer 
 
 shear-wave velocity in the m-th layer 
 
 thickness of the m-th layer 
 
 vertical coordinate 
 
 range between source and detector 
 
 time 
 
 angular frequency 
 phase velocity 
 wave number 
 
 imaginary part of a complex number x + iy 
 velocity potential in the m-th layer 
 normal stress in the m-th layer parallel to z axis 
 pressure in the m-th liquid layer 
 
 horizontal particle displacement in the m-th layer 
 vertical particle displacement in the m-th layer 
 horizontal particle velocity in the m-th layer 
 vertical particle velocity in the m-th layer 
 density in the m-th layer 
 density at the source. 
 

; Bessel function of order 0 
 
 ; Y bessel function of order 0 
 
 » O 
 
 ; K Bessel function of order 0 
 ; Hankel function of the first kind of order 0 
 
 (A'); Hankel function of the second kind of order 0 
 
 H 
 
 'Kl- k 
 
 r\ 
 
 Ov\ 
 
 k ( K, 
 
 'fyy\ 
 
 YW\. 
 
 , k > k 
 
 Piw- 1 , v< V 
 
 p^=> k > k ? 
 
 V. -K« 
 
 fwv 
 
 ' <Vv\. 
 
 
 flvv. 
 
 
 
 c <e 
 
 

P - kV\... r. 
 
 /W\ 
 
 <w\ 
 
 r/v\ 
 
 Q rwv. ^ 
 
 a C £ ?) 
 
INTRODUCTION 
 
 This report develops the integral solution of the wave 
 equation for point sources of harmonic waves in a liquid 
 layer of a multilayered liquid-solid half space in a form 
 convenient for numerical computation on a high-speed digital 
 computer. Only the case of a liquid half-space underlying 
 the stack of layers is considered. Furthermore, it is assumed 
 that the speed of sound in this last layer of infinite thick¬ 
 ness is greater than the speed of sound in liquid layers 
 immediately above it. The problem of immediate concern is 
 low-frequency propagation from harmonic sources in the central 
 Arctic Ocean at ranges up to ten times the water depth in 
 shallow and intermediate water depths. Solutions are develop¬ 
 ed for the pressure perturbations detected by a hydrophone at 
 depth or the ice vibrations detected by a seismometer on the 
 ice surface. A modification of the formulation is possible to 
 include harmonic vibration sources on or in the ice, although 
 it appears at the present time that surface sources are of 
 limited application (Greene, 1968). 
 
 The solution of the wave equation presented here, based 
 on the Thomson-Haskell matrix method (Thomson, 1950; Haskell, 
 
 1953), follows Harkrider (1964) for the solution of the wave 
 equation in an n-layered solid half space. Layer matrices of 
 the type given by Dorman (1962) for computing dispersion in 
 an n-layered liquid - solid half-space are used for the 
 liquid layers. An application of the theorem that the inverse of the 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
7. 
 
 product matrix for the layered system above the source has the same 
 form as the inverse of a layer matrix reduces the integrand 
 of the integral solution to a simple form in terms of product 
 matrices derived in the source-free, plane-wave case. The 
 integral over wave number from zero to infinity has singular¬ 
 ities in the integrand and is, therefore, conveniently trans¬ 
 formed into the complex c = k + ix plane. Performing the 
 contour integration in the c plane and noting that branch 
 line integrals corresponding to branch points generated by 
 each layer matrix except the last are zero, we are left with 
 an expression analogous to that of Sorensen (1959) and Leslie 
 and Sorensen (1961) for the two-layer liquid half-space with 
 a high-speed bottom. By the proper choice of contours, 
 complex poles are displaced to an unused sheet of the two-leaved 
 Riemann surface, and the integral solution for the multilayered 
 system reduces to a summation of normal modes plus the sum of 
 two integrals over wave number, one along the real 5 axis and 
 the other along the imaginary c axis. Both of these integrals 
 are conveniently computed by a Gaussian quadrature formula. 
 
 The physical interpretation of the final solution is 
 straightforward. The normal-mode terms correspond to waves 
 trapped in the Arctic sound channel; that is, refracted and 
 surface reflected (RSR) sounds and reflected sounds incident 
 on the bottom beyond the critical angle. The integral over 
 wave number along the real axis corresponds to waves incident 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
8. 
 
 __ r _, 
 
 
 i ' 4 
 
 2 
 
 
 2- | 
 a 
 
 
 ^3l 
 
 3 
 
 7 
 
 
 *- 3 
 
 i 
 
 \ i 
 
 
 _r _7=D 
 
 "Point Source 
 
 “ ^ x-s *- ^ 
 
 S* , 
 
 
 s-m ts 
 
 I 
 
 l 
 
 I 
 
 i 
 
 n-i 
 
 a 
 
 n.-\ 
 
 Fig. 1 
 
 t 
 
 2 
 
9 . 
 
 on the bottom at angles greater than the critical angle of total 
 reflection. Since energy leaves the guide continuously as waves 
 travel down the guide, this term is of importance only to ranges 
 of about ten times the water depth. The contribution to the sound 
 field of the integral along the imaginary axis is of importance only- 
 very near the source since the integrand decays exponentially with 
 range and wave number. 
 
 The usefulness of the normal mode terms for describing long- 
 range explosive sound transmission in the central Arctic Ocean 
 is shown by Kutschale (1969). In that work the present formulas 
 for the normal modes were extended to explosive sources and the 
 Fourier integral for each mode was evaluated by the principle of 
 stationary phase (Pekeris, 1948). Attenuation by the rough ice 
 boundaries was also included, although omitted here. 
 
 FORMAL SOLUTION 
 
 Source Free Case 
 
 Consider the n-layered interbedded liquid-solid half-space 
 shown in Figure 1 in which the last layer of infinite thickness is 
 liquid and has a higher sound velocity than liquid layers 
 immediately above it; that is, a high-speed bottom. A point 
 source of harmonic waves is located in one of the liquid layers. 
 
 The velocity potential in the m-th layer, which is 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
10 . 
 
 assumed to be liquid, satisfies the wave equation 
 
 * 0 
 
 In this layer 
 
 K, “ > Zjh. , = 
 
 3r 
 
 Solutions for nr and ( Jb \ 
 
 nm ^ W /Wv. 
 
 /Wv. 
 
 are given by 
 
 rt*v. 
 

11 . 
 
 and 
 
 A 
 
 
 n ^ iwt . / t \J kf- i 
 
 ^wvsL&£ -6u)p /VA (A^e 
 
 3t , 
 
 + A^e. ^ }e 
 
 
 where A and A^ are constants. 
 
 m 
 
 Placing the origin at the (m-l)-st interface we get 
 
 ^rw,-r L 1 k ^^ C^/wv'O 
 
 ( 1 ) 
 
 ^T*Aya-\~ LUJ P^ l A^ Afwx ) 
 
 for z = 0 
 
 and 
 
 (Yv\ 
 
 + O 
 
 -v- 
 
 /\ 
 
 Cr|°£i\v\ 
 
 ( A^ - A^cosT^ (2 
 
 - LVA^p^ (ArVA V ^^ )COS^Rwv 
 
 - p/vvv ( A fwvA ^ ^^ 
 
 ?-Vi 
 
 ^ . 
 
 for 
 
12 . 
 
 Substituting expressions for [A m - A^] and [A m + A^] from 
 equations (1) in equations (2) yields 
 
 
 C p^vv\ 
 
 
13 . 
 
 In general for an (\-layered liquid half-space 
 
 % • 
 
 or 
 
 o 
 
 
 « • » 
 
 ia>\, uo,; 
 
 since the vertical particle velocity and pressure are continuous 
 across each interface. For the n-th layer 
 

14 . 
 
 Applying boundary conditions that the pressure vanishes 
 at the surface and that no upgoing waves travel from 
 infinity, (P ZZ ) Q = 0, A r = 0, the relation (3) reduces to 
 
 t-O 
 
 k ' 
 
 -- e"a! 
 
 - -'V -\ 
 
 ^0 
 
 
 
 l ° _ 
 
 Consider now a solid layer between two liquid layers or at 
 
 the surface of the laminated halfspace. For this layer we may 
 write (Thomson, 1950; Dorman, 1962 ) « 
 
 
 
 ‘ A* 
 
 • 
 
 
 P- L 
 
 ( 
 
 16 U\, 
 
 
 
 ^** 1 ^ ^^ 43 , 
 
 U 1 
 
 .0 ] 
 
15 . 
 
 where the fourth element of the column vector is the tangential 
 stress which is zero at a solid-liquid boundary. The matrix 
 elements are given in the appendix and may be derived following 
 Haskell (1953). Equation (4) yields the three equations. 
 
 
 — \ 
 
 
 AO. 
 
 
 
 rvA -\ 
 
 
 A 3, 
 
 
 % 
 
 Substituting 
 
 A 
 
 \K 
 
 WVv.y 
 
 from equation (5) in equations (6) and (7) 
 
 yields 
 
 (A<vv7) a 
 
 
16 . 
 
 which is the same as the matrix relation for a liquid layer. 
 In general, then, for interbedded solids and liquids 
 

17 
 
 Point Source of Harmonic Waves in a Liquid Layer . 
 
 Divide the source layer into two layers as shown in 
 Figure 1. At z,=D the pressure is continuous. The vertical 
 particle velocity is continuous everywhere in the plane defined 
 by z =D except at the point source where the liquid above and 
 below the source moves in opposite directions. This may be 
 expressed by writing 
 
 - 3 K. 
 
 In matrix notation for the source layer 
 
 For the layers below z =D we have 
 
 and for the layers above z =D 
 
 « 
 

18. 
 
 where 
 
 = ^<v-v 
 
 5k 
 
 a <u and A s *a.a 
 
 
 Returning now to the relation for the n-th liquid layer 
 
 (11) may be written 
 
19 
 
 Harkrider (1964) has shown that for layered solids the inverse 
 of the product matrix has the same form as the inverse of the 
 layer matrices. Employing an extension of this theorem to inter- 
 bedded liquids and solids we get 
 
Hence (12) may be wrltten as 
 
 [-<1 ,T X_ l_ l^X + 3^1 1 
 
 * V( 
 
 O - + T a ^ . <u> 
 
 Adding equations (1*0 we get 
 
 0 = (J]_]_ + ^21^ ^ + ^ J i2 + ^22^ ^ and ^ We 
 
 let T = J^]_ + 22\ an< ^ ^ 
 
 + and solve for X we have 
 
 (15) 
 
21. 
 
 Carrying out the matrix multiplication J = E - " 1 A, expressions for 
 
 T and V are given by 
 
 A 
 
 T- 
 
 v\ 
 
 V=- 
 
 A \o 
 
 \ 
 
 
 \ 
 
 C OJ pr\ 
 
 A 
 
 i iK - v. a ' 
 
 ■v* 
 
 
 L oOp, 
 
 From the first equation of (13) 
 
 ^jJ " 0 - -(A s Y Kwr'i 
 
 and 
 
 (16) 
 
 from equation (15) 
 
 Hence, from the second equation of (13) and equations (16) 
 
 t ^ ve - k 1 ' 
 
 
 i 
 
22. 
 
 Therefore the integral solution for the surface verticle particle 
 velocity is ^ ^ i ^ k — 
 
 A 
 
 \8. 
 
 
 " J l t K- v * 
 
 e 
 0 
 
 ■ A 4," 
 
 t-==, -» 
 
 tt0 f > 2i ' ai 
 
 A 
 
 n 
 
 cfkM? 
 
 •V ^3\ 
 
 
 ri 
 
 Pa 
 
 7 
 
 aT 0 CHOV<JK 
 
 and from the formula 
 
 where 
 
23 . 
 
 It 
 
 Fig. 2 
 
24 . 
 
 Evaluation of the Integral Solutions. 
 
 The integral solutions have singularities corresponding 
 
 to zeros of 
 
 A 
 
 w 
 
 ■v- 
 
 
 c to 
 
 These are simple poles for complex wave number and, therefore, 
 the integral solutions are conveniently evaluated by contour 
 integration in the complex ^ = k + i ^ plane. The contours 
 
 are shown in Figure 2. First the transformation 
 
 
 
 is made. 
 
 In addition to simple poles at 
 
 A 
 
 n 
 
 FT 
 
 branch points occur at 
 
 i Kr * 
 
 T 
 
 - O 
 
 for the liquid layers and 
 
 T 
 
 = o = o 
 
 for each solid layer. But since the integrands are even 
 functions of 
 
 
 and 
 
 \ 
 
 kg k* 
 
 I 
 
 branch line integrals corresponding to branch cuts made to 
 these branch points cancel and only the integral corresponding 
 to the branch cut for \l\. 2 — r~\ must be 
 
 fkp 1 - o 
 
 considered. 
 
I • I i I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 1 I ' I n I ' I I I ' I ' I »!■ 
 
25 . 
 
 The Reimann surface h?s two sheets. To satisfy the vanishing 
 of UlT as C approaches infinity, we must remain on the sheet of the 
 Riemann surface where the real part of C. is greater than zero. 
 
 For the contours shown in Figure 2 complex poles of 
 
 —^==7 + - a,v - ■= O 
 
 have been displaced to an unused Reimann sheet and all poles 
 
 are on the real k—axis. See Ewincr f Jardetzkv and Press (1957 , 
 Sages 135 to 137) for the proof for a two-layer 
 
 liquid half space. Along the branch cut 
 
 «■' - f 
 
 is pure imaginary and along the real axis from 
 
 k 
 
 4, 
 
 to 
 
 infinity 
 
 c (- «■' 1 1- K ) 
 
 is real. 
 
 In the first quadrant 
 
 
 is positive and 
 
 in the fourth quadrant negative. The integral for w^ 
 
 , ,0 c ^t 
 
 (omitting the term) is now transformed to 
 
 where 
 

“\ 26 . 
 
 
 --- + \(A \ 
 
 ■V ^Jt\ 
 
 /v\>? 
 
 A 
 
 w 
 
 -V 
 
 Au \ 
 
 Cw f 
 
 *\ 
 
 V£V^ 
 
 + /x v a^o 
 
 O C GO /°° 
 
 or 
 
 (x^-(x<it 
 
 (19) 
 
 LP?JL rA " ° u 
 
 Likewise, since rl^V VJ approaches zero as r approaches 
 infinity in the fourth quadrant (18) is 
 
 L><n H- (°TJ\ + 
 
 CO } 
 
 aTitJ'ResCX;,) 
 
 OAE 
 
 or 
 
27 . 
 
 and (20) may be written as 
 
 k = - Ufc-ur)^?. 
 
 0 k 0 
 
 - aiuYKe^c^^ . 
 
 . yCO 
 
 Hence UT 0 = \ T.ci K +• ( 1 ^ k = 
 
 O *o 
 
 v Uf (.-ex Otdt 
 
 + / ^-FH'K>^ 1 X\^\kr-)kdK 
 
 O 
 
 - 3ttlY t 
 
 In the first integral 
 
28 . 
 
 3. ( ~ , 
 
 \| Vc^-v £* ^ 
 
 \ co~p *; 
 
 and therefore ^ \ v # , 
 
 w^p^A^-CV^^A^' 
 
 Likewise, in the second integral 
 
 FC^Vw? 1 ')- F( r C^v? f ') = 
 
 a * 
 M a\ 
 
 but A-^-^ ^22 “* A 2 ^21 = ^ 
 
 CO* 1 " 4 
 
 W >*V-(V *-k l )A* 
 
 Therefore, the integral solution reduces to 
 
 ,; T . / ^fvK^ lA*.V. H2c-^ht«t£ 
 
 ^*1 
 
 a wp^k^-k** (jA Si ") a ( Vlf^kr ) V dk. 
 
 Ajf - lk*-k^ A 
 
 A 
 
29 . 
 
 From the formulas Hp ctr) = 2~ K 0 C tP) *" a 
 
 » 
 
 and multiplying by 
 
 for a constant pressure 
 
 top & 
 
 source of pressure amplitude P Q at unit distance from the source 
 
 W„- 
 
 ■V 
 
 Kftr^-dg 
 
 o "^^P^L^P^A^-^-vrpAa,! 
 
 ^ aP 0 to fv'1 k,* ■- v* 7 (A s, >'/, ^tM-Oo^ikcSK 
 
 ° ^^A* - (Vs< * -kp A* ^ 
 
 - 3^<1 TRe^ 
 
 
 C to 
 
 For the normal mode contribution, — QTTCy j_ t j_, 
 
 I— \ C p s * 
 
 convenient to rev/rite the integral for W Q in the form 
 
 A 
 
 + 
 
 ^V\ 
 
 c Av> 
 
 4- 
 
 t uJ>p n -s\ 
 
 < 3 . 
 
30 . 
 
 where the layer matrices CX / y v>< are defined in the Appendix. 
 There are, as before, simple poles of I 2 along the real axis at 
 
 for 
 
 This is the period equation for the laminated half space. 
 From the period equation we write _ 
 
 / i 00 PnC \ _ | ~ Asu \ 
 
 V L V A u 
 
 * - * 2_ k-kjj. 
 
 But again A-,A 22 - A 21 A 12 = 1, and therefore 
 
 a 
 
 U)avi)c 
 
 G-Q 
 

31 . 
 
 In summary, the expressions for vertical particle velocity 
 at the surface, vertical particle displacement at the surface, 
 and pressure in a liquid layer are: 
 
 o c A? - C<-k"> A»n 
 
 + Q 
 
 ^ HX^t + _ 
 
 LOtLO^ 
 C--Cj I 
 
+ 3TT 
 
 r t c 
 
 
 -C\ 
 
 n 
 
 C U) C? f\ P 
 
 v s„|' 
 
 n: * « 
 
 32 . 
 
 ^o-ujSL 
 
 has been used. 
 

 
 
 
 
33 . 
 
 For programming it is convenient to write: 
 
 ur 0 - -Y/^V* - cV/ 3 + cV/^ - l 
 
 and for the absolute 
 
 value of vertical particle velocity, a quantity conveniently 
 
 measured in transmission experiments, 
 
 Kol- {(.-X Y + £w s -'WsV ‘ 
 
 where 
 
 V 
 
 pn R-v* x , \ T 0 (k Ok d K 
 
 o |°s W"pXA.,f + ^ fX(Vn (_- A av 'j ^ 
 
 W=StTT 
 
 c 3 
 
 Y- 
 
 c3 A" 
 
 vOel^i 
 
 C 
 
 o TV p % Y^p' -? a * + (X<* ♦ 
 
s 
 
 
 Likewise 
 
 for vertical particle displacement 
 
 
35 . 
 
 v _ 
 
 A O 
 
 
 UJ 
 
 
 V\ ^\\ 
 
 v V* t 
 
 cVspn.A,^- \ c ^'' ^. Ti^L-Aj ^ 
 ' bc-Liimir,, n, 
 
 and for pressure 
 
36 . 
 
 Xi 
 / +*\ 
 
 ( a-p 0 Pnlk^-k"' I*\Kt-lAi>) a ^ (krXMk 
 
 
 ^ W < + ^V 
 
 V/ 4 ^- 
 
 3 -r 
 
 ? 0 w X^-fo ,\ 3 x 
 
 ^~ A '£l 
 
 ^ A M ^"X/wv (j-A^P) 
 
 3C Clrw^f^ 'X 
 
 f 1 
 
 W=Va)|. 
 
 C»C* 
 
 ? 0 wo x w \r (A^X ^x ^ (A s> \ ^ r X 
 
 cVsPv,A»ll S^x_ v X ^(-A^ j 
 
 ScLxw^n, ^ jl 
 
 « *»*>*. 
 C“Cj_ 
 
37 . 
 
 The layer matrices for the integral along the real axis 
 are programmed as 
 
 for solid layers and 
 
 (a. ^ 
 
 - w 
 
 for liguid layers. For the integral along the imaginary axis 
 the matrices are 
 
for liquid layers. Likewise, for the ndrmal mode terms the 
 layer matrices are programmed as 
 
39 . 
 
 for solid layers and 
 
 for liquid layers. 
 
40 . 
 
 NUMERICAL COMPUTATIONS 
 
 Computer programs were written in double precision Fortran 
 
 2 
 
 IV to compute the pressure in dynes/cm and the vertical particle 
 displacement in millimicrons. The programs are run on the IBM 
 360/91. For numerical results presented here, computing time 
 was under six minutes. It appears that practical limits of 
 numerical precision make the present development most useful for 
 the Arctic at frequencies below 50 Hz in water depths to 1 km. 
 
 At higher frequencies or in greater water depths the integrands 
 may be so oscillatory that it is difficult to achieve the desired 
 accuracy in the numerical integrations without excessive computing 
 time. 
 
 Computations are made in two stages. The first program, an 
 extension of Dorman's (1962) PV 7 dispersion program, computes 
 phase- and group-velocity dispersion, the excitation function depen¬ 
 dent only on layering of the medium, and the excitation function for 
 the particular source and detector depths. In the first case for 
 the m-th normal mode the excitation function is defined by 
 
 

41 . 
 
 for vertical particle velocity. In the second case the excitation 
 function is 
 
 yT/w\ a 
 
 \ T ^ \ C>A 
 
 C-* A, 1 
 
 w 
 
 
 
 
 Pn 
 
 for pressure and 
 
 
 lOslO/iH 
 
 c5 A„a 1 c ^ A 
 
 u X/w\(rA^ v A 
 
 3 c- 
 
 p 
 
 
 
 C-C 
 
 
 /wv. 
 
 for vertical particle velocity. These definitions were chosen to be 
 
 ful also at long ranges where VC Ik c) = 
 
 (% 
 
 use 
 
 e 
 
 va-kr) 
 
 The second program computes the 
 three integrals, the normal mode contribution to the sound field, 
 and writes and plots the absolute value of 
 
 4 
 
 \ 
 
 , or w Q as a 
 
42 . 
 
 TABLE 1. 
 
 Model Parameters for Sorensen's (1959) Computations: Source 
 
 2 
 
 Pressure Amplitude re lm,.3048 dyne/cm ; Source Depth, 15.2 m; 
 Hydrophone Depth, 15.2m; Range, 3.048 m 
 
 Layer Thickness, m Compressional 
 
 Velocity, m/sec 
 
 Shear 
 
 Velocity, m/sec 
 
 Density 
 gm/cm 0 
 
 18.3 
 
 1463.0 
 
 0 
 
 1.03 
 
 Infinite 
 
 1609.3 
 
 0 
 
 1.24 
 

43 . 
 
 TABLE 2. 
 
 Integral Real Axis, Real Part 
 
 F requency, 
 Hz 
 
 Sorensen’s (1959) Value, 
 dynes/cm 2 
 
 Our Value, 
 dynes/cm 2 
 
 10 
 
 .004713 
 
 .004685 
 
 20 
 
 .030170 
 
 .030153 
 
 40 
 
 .040278 
 
 .040485 
 
 80 
 
 .023193 
 
 .023121 
 
 160 
 
 - .021839 
 
 - .022201 
 
 320 
 
 - .074395 
 
 - .074011 
 
44 
 
 TABLE 3. 
 
 Integral Real Axis, Imaginary Part 
 
 Frequency, 
 
 Hz 
 
 Sorensen 1 s Value, 
 
 dynes/cm^ 
 
 Our Value, 
 
 dynes/cm^ 
 
 10 
 
 - .002567 
 
 - .002551 
 
 20 
 
 - .022260 
 
 - .022240 
 
 40 
 
 - .061960 
 
 - .062274 
 
 80 
 
 - .050986 
 
 - .050641 
 
 160 
 
 - .047979 
 
 - .047750 
 
 320 
 
 .006299 
 
 .008316 
 

45 . 
 
 TABLE 4. 
 
 Integral Imaginary Axis 
 
 Frequency, 
 
 Hz 
 
 Sorensen’s Value, 
 dynes/cm 2 
 
 Our Value 
 
 dynes/cm 2 
 
 10 
 
 .095080 
 
 .095146 
 
 20 
 
 .081938 
 
 .082076 
 
 40 
 
 .041887 
 
 .041759 
 
 80 
 
 .030795 
 
 .030937 
 
 160 
 
 .030214 
 
 .030356 
 
 320 
 
 .010593 
 
 .011043 
 
DEPTH (M) 
 
 H6. 
 
 SOUND VELOCITY (M/SEC) 
 
 Fig. 3 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
TABLE 5. 
 Model A 
 
 Layer 
 
 Thickness, m 
 
 Compressional 
 Velocity, m/sec 
 
 Shear Velocity, 
 m/sec 
 
 Density, 
 
 gm/cm3 
 
 50 
 
 1431). 0 
 
 0 
 
 1.03 
 
 50 
 
 1437.0 
 
 0 
 
 1.03 
 
 50 
 
 1440.0 
 
 0 
 
 1.03 
 
 50 
 
 1443.0 
 
 0 
 
 1.03 
 
 50 
 
 1446.0 
 
 0 
 
 1.03 
 
 50 
 
 1450.0 
 
 0 
 
 1.03 
 
 50 
 
 1453.2 
 
 0 
 
 1.03 
 
 Infinite 
 
 1600.0 
 
 0 
 
 1.20 
 

 
 
48 . 
 
 NOllJNnj NOl LV1I0V3 
 
 CD CM 
 
 (03S/IM ) A1I0013A 
 
 bO 
 
 •H 
 
 fa 
 
 FREQUENCY (HZ) 
 
RANGE (M) 
 
 49 . 
 
 o — 
 
 Ol 
 
 o - 
 
 o 
 
 o 
 
 o 
 
 o 
 
 Fig. 
 
 .0001 
 
 • • • . 
 .0055 
 
 PRESSURE (DYNES/CM 2 ) 
 
 .0J09 
 
 5 
 

 
RANGE ( M) 
 
 50 . 
 
 o - 
 
 m 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 Fig. 
 
 6 
 

RANGE(M) 
 
 o 
 
 o“ 
 
 o 
 
 .0064 
 
 PRESSURE (DYNES/CM 2 ) 
 
 .0127 
 
 Fig. 
 
 7 
 
52 . 
 
 Fig. 8 
 

2000 RANGE (M) 
 
 53 . 
 
 o 
 
 o*“ 
 
 o 
 
 oi 
 
 o — 
 
 o 
 
 o 
 
 -U 
 
 o * 
 
 o 
 
 o 
 
 .00027 
 
 PRESSURE (DYNES/CM 1 ) 
 
 .00054 
 
 Fig. 
 
 9 
 
RANGE ( M ) 
 
 Fig. 10 
 
 5 ^. 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
55 . 
 
 function of range. In this program provision is also made to 
 compute the three integrals as a function of frequency or detector 
 depth at a specified range. The integrals are evaluated by 
 Gaussian quadrature formulas (see, for example, Hildebrand, 1956). 
 Care must be taken to employ a sufficient number of Gaussian points 
 to obtain the desired accuracy of the integrals. A thirty-two 
 point Gaussian quadrature formula was found to give sufficient 
 accuracy for the computations presented here out to a range of 
 1500 m, but at longer ranges, higher frequencies, or greater water 
 depths a ninety-six point formula is used. Abscissas and weight 
 factors for the Gaussian integration are given by Davis and 
 Rabinowitz (1956, 1958). The integration along the imaginary 
 axis extends to infinity, but in practice a finite upper limit is 
 chosen which gives sufficient accuracy. This may be done because 
 the integrand is an exponential function of -kr for large k or r. 
 
 The programs were checked by computations for the same two-layer 
 model of Table 1 used by Sorensen (1959). Tables 2, 3 and 4 show 
 that the two sets of numerical results are in very close agreement. 
 
 As an illustration of the progrms for the Arctic, computations 
 were made for the layered model A of Figure 3 , which closely follows 
 the observed variation of sound velocity with depth. Additional 
 parameters for the model are given in Table 5. This is a prelim¬ 
 inary model and the ice sheet, which averages about 3-m in thick¬ 
 ness in the central Arctic Ocean, was not included. The effects of 
 the ice sheet on propagation are currently under investigation. 
 Figure A shows phase - and group-velocity dispersion and the 
 excitation function for pressure. At a frequency of 10 Hz only 
 two normal modes are excited. The range dependence of the absolute 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
56 
 
 value of pressure for the normal mode contribution, the integral 
 contribution, and the total pressure are shown in Figures 5 through 
 10 . 
 
 ACKNOWLEDGMENTS 
 
 Dr. J. Dorman of the Lamont-Doherty Geological Observatory 
 kindly supplied a copy of his PV-7 dispersion program. Computing 
 facilities were provided by the Columbia University Computing 
 Center. This work was supported by the U. S. Naval Ordnance 
 
 Laboratory and the Office of Naval Research under contract 
 N00014-67-A-0108-0016• 
 
57 . 
 
 REFERENCES 
 
 Davis, P. and P. Rabinowitz, Abscissas and weights for 
 
 Gaussian quadratures of high order, J. Res . NBS , 56 , 35-37, 
 1956. 
 
 Davis, P. and P. Rabinowitz, Additional abscissas and weights 
 
 for Gaussian quadratures of high order: Values for n = 64, 
 80, and 96, J. Res. NBS , 60, 613-614, 1958. 
 
 Dorman, J., Period equation for waves of Rayleigh type on a 
 
 layered, liquid-solid half-space. Bull . Seismol . Soc . Am., 
 52, 389-397, 1962. 
 
 Ewing, W. M., W. S. Jardetzky and F. Press, Elastic Waves in 
 Layered Media, McGraw-Hill, New York, 1957. 
 
 Greene, C. R., Arctic operation of seismic transducers. Sea 
 
 Operations Department, AC Electronics - Defense Research 
 Laboratories of General Motors Corporation, AC-DRL TR 
 68-53 , Santa Barbara, California, 1968. 
 
 Harkrider, D. G., Surface waves in multilayered elastic media 
 
 I. Rayl&igh and Love waves from buried sources in a multi¬ 
 layered elastic half-space. Bull . Seismol . Soc . Am., 54 , 
 627-679, 1964. 
 
 Haskell, N. A., The dispersion of surface waves on multilayered 
 media. Bull . Seismol . Soc . Am., 43 ., 17-34, 1953. 
 
 Hildebrand, F. B., Introduction to Numerical Analysis, McGraw- 
 Hill, New York, 1956. 
 
 Kutschale, H., Arctic hydroacoustics, Arctic , 22 , 246-264, 1969. 
 
 Leslie, C. B., and N. R. Sorensen, Integral solution of the 
 
 shallow water sound field, J. Acoust . Soc . Am., 33.> 323-329, 
 1961 . 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
58 . 
 
 Pekeris, C. L., Theory of propagation of explosive sound in 
 shallow water in. Propagation of sound in the ocean, 
 
 Geol . Soc . Am. Memior, 27, 1-117, 19^8. 
 
 Sorensen, N. R., Integral solution of the shallow water sound 
 
 field, U. S. Naval Ordnance Laboratory, NAVORD Report 6656 , 
 White Oak, Maryland, 1959. 
 
 Thomson, W. T., Transmission of elastic waves through a strati¬ 
 fied solid medium, J. Appl . Phys ., 21 , 89-93, 1950. 
 

 
 
 
 
 
 
 
 
 
 
 
 
59 . 
 
 APPENDIX 
 
 Matrix elements for normal modes (see Haskell, 1953, and 
 Dorman, 1962). 
 
 Solid layers 
 
 =(5^,,=cos,^ 
 
 ^ A, - ' L \ vv S m\ * 
 
 c^'JL 
 
 V C0 ^ 
 
 (A-Ojs * ^ l ^ C t< + 
 
MM. w • 
 
 VWk.-'X - co^q^ 
 
 (£W) S ,,- c(V,c 
 
 . ^ V . s " 2' 3 *"1 
 
 ^Ar- ^^'VNT^-Cr^r^-if)S1«Q 
 
 Liquid layers 
 
 VS^-fWv'X^ fw^^^ COSV^wv 
 
 ^ smT^ 
 
 P*wO _ 
 
 (A*vx\. = 
 
 ' 'o^/vv\ 
 
 Matrix elements for integrals along real axis: 
 
 Solid layers 
 
 laA,=^\,- co^- C) • 
 
 CO^Q^ 
 
 ^ Q * T \ 
 

61 . 
 
 
 Av\ 
 
 
 -V 
 
 
 
 
% 
 
 62 . 
 
 
 
 
 
 Liquid layers 
 
 COS”V^ 
 
 l a Oi , ^ SlVVf^ 
 
 ''■' '*\ iv. 5 -M* 
 
 Matrix elements for integral along imaginary 
 Solid layers 
 
 Tm - 
 
 «<& a, - - a (|-^ 
 
 ^ \\ 
 
 J vfi -v » “3^ 
 
 
 axis: 
 
 CO^Q^ 
 
 (£W)^= [ a 
 
 0 . t. 
 
 (£- t + vNU^a- - 
 
 V~ 
 
 '(k&F* 3 SMQ^ 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 ** 
 
 l^^ai ~^'VV\\ ~C 
 
 
 COS?^ 4. 
 
 * 
 
 Aw 
 

64 . 
 
 
 
 Liquid layers 
 
 T “ K t k X +T^ 
 
 'YA ''WV'IX 1 
 
 Cos?^ 
 
 l^\ x = ^{vg^vV^ 
 
 ^Vv\ 
 
 L p/w\ lO ^ \ V\ 
 
Corrections 
 
 Page 7, line 3, should read as follows: 
 in terms of elements of 
 
 Page 13, line 4, should read as follows: 
 
 Page 16, line 8, should read as follows: 
 
 ■&n-, ^ 
 
 
 ! 
 
 &<T\ 
 
 
 
 ■ b^a^-'-b, 
 
 o 
 
 Page 19, line 1, should read as follows: 
 
 A ~ Ac,^ ^n-\^n-a ' ’ ' '°n-\o 
 
 Page 22, line 7, should read as follows: 
 
 at depth # at the bottom of the layer is 
 
 Page 22, line 10, should read as follows: 
 
 <Xt> - \ • • • Q t 
 
 Page 28, line 2, should read as follows: but from the relation 
 
 A/\ '-I ) ^ V ' ~ ^'3. Aq, “ l 
 

 
 
 
 
 
\ 
 
 Page 41, second line, should re< as follows: 
 
 3 \ u) 
 
 v 
 
 p 
 
 
 c~ /a Au 2 
 
 A 
 
 \\ 
 
 •V 
 
 
 TwIrO 
 
 P*. 
 
 to-CO, 
 
 C^c 
 
 AY\ 
 
 /V/\ 
 


 DOCUMENT CONTROL DATA - R & D 
 
 (Security classi fication of title, body of abstract and indexing annotation must be entered when the overall report is classified) 
 
 1 ORIGINATING ACTIVITY ( Corporate author) 
 
 Lamont-Doherty Geological Observatory 
 of Columbia University 
 
 2fl. REPORT SECURITY CLASSIFICATION 
 
 Unclassified 
 
 2b. GROUP 
 
 Arctic 
 
 "I’lie 1 integral Solution of the Sound Field in a Multilayered Liquid-Solid 
 Half-Space with Numerical Computations for Low-Frequency Propagation 
 in the Arctic Ocean. 
 
 4 DESCRIPTIVE NO T ES (Type of report and inclusive dates) 
 
 Technical Report 
 
 5 authoRIS) (First name, middle initial, last name) 
 
 Henry W. Kutschale 
 
 6- REPOR T DATE 
 
 February 1970 
 
 la. TOTAL NO. OF PAGES 
 
 64 
 
 lb. NO. OF REFS 
 
 13 
 
 0a. CONTRACT OR GRANT NO. 
 
 N00014-67-A-0108-0016 
 
 6. PROJECT NO. 
 
 NR 307-320/1-6-69 (415) 
 
 9a. ORIGINATOR’S REPORT NUMBER(S) 
 
 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned 
 this report) 
 
 10 DISTRIBUTION STATEMENT 
 
 Reproduction of this document in whole or in part is permitted for 
 any purpose of the U.S. Government. 
 
 11 SUPPLEMENTARY NOTES 
 
 12- SPONSO RING Ml LI T AR Y ACTIVITY 
 
 U.S. Naval Ordnance Laboratory, 
 
 White Oak, Silver Spring, Maryland 
 Office of Naval Research,Washington, 
 
 13 ABSTRACT 
 
 D.C. 
 
 This report develops by matrix methods the integral solution 
 of the wave equation for point sources of harmonic waves in a liquid 
 layer of a multilayered liquid-solid half space in a form convenient 
 for numerical computation on a high-speed digital computer. Only the 
 case is considered of a high-speed liquid bottom underlying the stack 
 of layers. The integral over wave number has singularities in the 
 integrand and is conveniently transformed into the complex plane. By 
 a proper choice of contours, complex poles are displaced to an unused 
 sheet of the two-leaved Riemann surface, and the integral solution 
 for the multilayered system reduces to a sum of normal modes plus the 
 sum of two integrals, one along the real axis and the other along the 
 imaginary axis. Both integrals are evaluated by a Gaussian quadrature 
 formula. Sample computations are presented for low-frequency propaga¬ 
 tion in the Arctic Ocean sound channel. These are preliminary compu¬ 
 tations and the ice layer, which averages three meters in thickness, 
 is not included in the layered system. The effects of the ice layer on 
 propagation are currently under investigation. 
 
 DD , F °?„1473 
 
 Security Classification 
 
Security Classification 
 
 1 4 
 
 KEY WO R D S 
 
 LINK A . 
 
 . LINK 8 
 
 LINK C 
 
 ROLE 
 
 W T 
 
 ROLE 
 
 W T 
 
 ROLE 
 
 W T 
 
 Hydroacoustics 
 
 Sound field 
 
 Layered media 
 
 Matrix 
 
 Integral solution 
 
 Digital computer 
 
 Numerical computations 
 
 
 
 
 
 
 
 Security Classification 
 
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