tion, neglecting time derivatives of * and terms of or- 

 der l/(*•o'■)^ we find 



32^ 1 a2>j/ 



Br' 



Stf)^ 



324, d'i 



w+^lkn-— 



■2/fe2— * = o, 



where k^ = OL)/ci, and we have assumed that 6c/c«l. 



The key to the parabolic equation method involves the 

 following additional physical approximations, based on 

 the structure of bc/c: 



3^* ., 3* 



gj.2 Ogy ' 



if kgl.^ •> 1, where L„ is the vertical scale of sound- 

 speed variations. This condition is equivalent to re- 

 taining only relatively forward scattering, which re- 

 sults in small changes in * over an acoustic wave- 

 length. It is valid if the objects off which the acoustic 

 waves are scattering have sizes which are much larger 

 than a wavelength; and 



32 >t 



3^* 32* 



3^ Bz^ 



which is true because the canonical profile, which has 

 100 times the sound-speed fluctuation than the internal 

 waves, affects the z coordinate only. More important- 

 ly, however, the internal-wave gradients in the verti- 

 cal are an order of magnitude greater than the horizon- 

 tal. The approximate wave equation is therefore 





37^ 



^0^* = 



3* 

 dr 



(1) 



As a result of our approximations, we have neglected 

 all azimuthal correlations. Thus we cannot study azi- 

 muthal fluctuations. We can study fluctuations that 

 can be observed in a single vertical plane, where azi- 

 muthal correlations have a small effect. 



To summarize the approximations required for this 

 parabolic equation to be valid we have the following 

 quantities not yet defined: a-'n, = largest frequency in- 

 volved in the internal wave spectrum, = 3 cycles/h; 

 LH = 'nini™ur" horizontal scale of sound-speed fluctua- 

 tions, = 1 km due to internal waves; L^ = minimum ver- 

 tical scale of sound-speed fluctuations, = 200 m due to 

 internal waves. Validity of the parabolic equation re- 

 quires: a)»aim; kgr»l; L„»iy; and^^y^^l- All 

 conditions are well satisfied in our case, where w = 100 

 Hz. 



C. Numerical realization 



We solve Eq. 1 by the "split-step-Fourier" algorithm 

 of Tappert and Hardin. '^ Given *(r, z) we find the wave 

 function at a new range from the following: 



*(r + rfr, z) = J-He'-^"'- JF[e'S'"-*(r, z)]} , 



where B = - ki,5c/c and A = {l/2ka)d^/3z^. Thus A and B 

 are operators in 2 space {A being the Fourier transform 

 of ^) and T is a fast Fourier transform operation." 

 This algorithm is fast and very stable since the total 

 acoustic energy !\'i\^dz is exactly conserved as a 

 function of range when absorption is absent. 



The FFT was used with 512 elements over a 4-km- 



deep ocean, and the range step has been chosen as 0. 5 

 km. 



D. Acoustic source and boundary conditions 



The acoustic field may be started with any function 

 of depth *(0, z). A point source at 2 = 1000 m (the depth 

 of the sound channel) with unit strength at one yard has 

 been modeled by an asymptotic matching technique 

 which prescribes the appropriate initial value.'' 



The ocean surface has been treated as a perfect 

 pressure-release boundary so that *()', 0) = 0. This is 

 accomplished through the use of a fast sine transform" 

 for the operation S . 



In order to model a completely absorbing ocean bot- 

 tom, a gradual loss of amplitude is imposed on 'I'(^) as 

 z nears the ocean bottom. The functional form of the 

 imposed loss at each step is the factor 



L{z) = exp^-adyexp- (^~^"'"') ]\ 



with Q: = 0.05/m and /3 = 0.04z„a,. 



This form effectively stops any acoustic energy from 

 penetrating below about 500 m above the bottom. Even 

 this attempt at acoustic impedance matching does cause 

 some reflection off the bottom at an extremely low in- 

 tensity level. 



III. RESULTS 



When an explosion is detonated deep in the ocean, a 

 series of sharp reports are heard at ranges up to sev- 

 eral thousand kilometers. The fact that each separate 

 sound arrives without being dispersed in time implies 

 that a geometrical-optics view of sound transmission 

 in the ocean must have a great deal of merit. Ray 

 tracing is a well-established technique for determining 

 the character of oceanic sound transmisssion. 



Figure 5 shows the ray paths where the sound speed 



10 



30 40 50 60 70 



RANGE — km 

 FIG. 5. Ray paths for the canonical sound-speed profile given 

 in Fig. 1, with a source on the sound-channel axis. 



205 



