acoustic signals. Their model cannot be complete as 

 it fails for rays whose turning point occurs within the 

 thin layer. We consider a full spectrum (excluding 

 tides) and treat the acoustic-internal-wave interaction 

 within the full volume of the ocean. 



The remainder of the paper is organized as follows: 

 Section I describes the sound-speed field derived from 

 the internal-wave spectrum, and its computer realiza- 

 tion. Section II describes the acoustic propagation 

 method (which depends on the parabolic equation ap- 

 proximation) and its computer realization. Section III 

 presents our quantitative results. Section IV is a sum- 

 mary and conclusion. The Appendix describes our 

 method of vertical beamforming. 



I. OCEAN SOUND-SPEED STRUCTURE 



A. Deterministic profile 



On the scale of the depth of the ocean (4 to 5 km) the 

 sound speed as a function of depth z is determined by 

 the gross behavior of the density, temperature, and 

 salinity. We use the profile derived by Munk'" whose 

 input is an exponentially decreasing density gradient. 

 The resulting "canonical" profile is 



<rcp(2) = fi{l+«[e"'-{l-'7)]}, 



where t] = 2(z- z^/B. Note that fcp(^) has a minimum 

 at z^ , that the width of the minimum is B, and the de- 

 viation of the sound speed from the minimum value Cj 

 is of the order e. 



Figure 1 shows Ccp(2) for the typical (though not uni- 

 versal) values we have chosen for the parameters: ^^ 

 = 1000 m, B = 1000 m, € = 0.57- 10"^ and ri= 1500 m/ 

 sec. 



1 52 1,54 1.56 l.i 



SOUND SPEED — km/s 



FIG, 1. Deterministic sound-speed profile as a function of 

 ocean depth (Canonical Profile). The value of c at the mini- 

 mum U^ = 1000 m) is c(2^) = 1500 m/sec. 



One must point out that realistic ocean profiles in 

 most cases have significantly different behavior from 

 this general form. For example, within a few hundred 

 meters of the surface a mixed layer usually results in 

 a lowered sound-speed gradient. However, we ignore 

 these details in our present treatment. 



B. Internal waves 



The density gradient in the ocean leads to the pos- 

 sibility of waves traversing the volume of the ocean 

 just as the density discontinuity at the surface leads to 

 the possibility of surface waves. The density gradient 

 is usually presented in the form 



JV(z) 



7:i£ Spy 



VPo 32 / 



where A'(^) is called the local stability (Brunt-Vaisala) 

 frequency. 



Following Garrett and Munk, ' we assume a stratified 

 ocean with 



where iVo = 3 cycles/h. 



Let (((r. /) be the vertical component of fluid velocity 

 at position r and time t. It can be shown" that w sat- 

 isfies the equation 



^(V2„■)+^'2(^)V2,»■ = 0. 

 The eigenmodes of this equation can be found by taking 



where k = (b\jrk\)^''^ is the horizontal wavenumber. 

 Substituting and modifying our equation to account for 

 the rotation of the earth, we find 



3'W 



32 = 





^^w=o. 



202 



where a)j=inertial frequency = (2 cycles/day) sin(lati- 

 tude). We will use 0); = 1 cycle/day. Boundary condi- 

 tions are "Wiz) = at surface and bottom (assumed 

 flat)." 



A particular mode, characterized by mode number j 

 and horizontal wave number k, will have a vertical ve- 

 locity profile given by W( ;, k, ^) and a definite frequen- 

 cy a)( 7, k). Because every fluid element moves with the 

 same frequency, the vertical displacement J of a fluid 

 element from its equilibrium position for a single mode 

 will also be proportional to W( j, fe, z). The sound-speed 

 fluctuation he is related to the displacement X. by^ 



6c =f, -W^(z)J a:iv2(2)W( j, *, z) . 



Several examples of the sound-speed profiles due to 

 particular modes are shown in Fig. 2. 



The sound -speed fluctuations caused by a full internal 

 wavefield may be represented as a linear superposition 

 of eigenmodes, leading to 



|J.»1.*2 ' 



