K = 0.5 cycles/km 



as a function of time: 



MODE 



1 



MODE 

 3 



MODE 

 10 



6C 

 C 



FIG. 2. Sound-speed profiles due to internal-wave modes. 

 Realistic internal -wave spectra have significant intensities for 

 horizontal wavenumber k less than about 0. 5 cycles/km. Note 

 that the major internal-wave contributions to sound-speed 

 fluctuations occur at depths less than 1 km. 



where the summation sign means integration over the 

 continuous variables k^ and k^. We normalize W(j, k, z) 

 so that 



r 



-'o 



N\z)W^(j. k, z)dz = \. 



where z„^ is the depth of the ocean. 



The numerical difficulty in projecting this three-di- 

 mensional field onto the two-dimensional vertical plane 

 used in the acoustic propagation code has caused us to 

 consider a simplified version of the internal wavefield 

 where internal waves propagate only in (or opposite to) 

 the direction that the sound waves propagate. In addi- 

 tion we combine real and imaginary parts to reduce 

 fluctuations in the overall energy in the internal waves 



6c,„=-tJ- (ReA +ImA) 



and 



J. ft 

 where r is the horizontal range. 



The A(j, k) are complex Gaussian random variables. 

 From a synopsis of diverse oceanographic measure- 

 ments, Garrett and Munk^ have proposed the following 

 model: 



{A(j,k)) = Q 



{AU, k)A*(j', k')) = 6jj,5„. X|32^(j)B(j, k), 



where 



H(J/)=6/(77j)^ 



B(j,fe) = (2A)fe,fcV(fe' + fe?f, 



kj:^(TT/B){Wt/N„)j. 



The spectrum is normalized so that 



J2f H(j)B(j,k)dk = l andf B(j,k)dk = l. 

 From the above equations it can be shown that 



r{(^i)-''- 



But Garrett and Munk^ have shown that 



\2\ 



0)=' 



Hence ^^=>'^B/3 where y is a measure of the fractional 

 sound-speed fluctuations due to internal waves. From 

 Ref. 4 we have y = 4. 8x10"*. 



It is useful to point out a few properties of the dis- 

 persion relation and the spectrum. The frequency 

 (j}(i, k) varies between the inertial frequency (1 cycle/ 

 day) and A^p (3 cycles/h). Frequency increases with in- 

 creasing values of k and decreasing values of i. 



For a fixed mode number j, the function B(i, k) gives 

 the relative contribution from each value of k. The 

 peak of the k distribution is at fej. The function H(i) 

 gives the overall contribution from each mode number 

 j: the gravest mode (j = l) has the largest contribution, 

 with other modes decreasing as \/j^. The relative in- 

 tensities of the various modes are shown in Fig. 3. 



Note that the magnitude of the sound speed fluctua- 

 tion due to internal waves is bc/c ~ 10'*, a factor of 100 

 below the magnitude of the deterministic structure. 

 Also the spatial behavior of the sound-speed variations 

 due to internal waves is of the order of a few hundred 

 meters vertically and several kilometers horizontally. 

 Figure 4 shows some typical sound-speed profiles due 

 to internal waves. 



C. Final expression for sound-speed structure 



c(r, = Ccp(z)+5Ciw 



203 



