TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN 

 UNDERWATER ACOUSTICS 



matrix is obtained. These elements are simply the squared amplitudes 

 of the normal modes . 



Everything in Equation (32) is known and can be numerically 

 obtained using computers. Comparisons are then possible between 

 these normal-mode results and either numerical experiments (using 

 the parabolic equation) or field data. 



Another approach, outlined in Figure 24, is based on the wave- 

 kinetic equation, basically applying transport theory to acoustic 

 propagation of random oceans. The Wigner phase-space distribution 

 function f defined in (33) is introduced where f is quadratic in the 

 complex demodulated signal ^ and hence depends on depth z and range 

 r as well as vertical angle 9. The ensemble average F (36) satis- 

 fies the integro-dif ferential equation (37). This equation, which 

 describes the evolution of the ensemble average Wigner distribution 

 function (Tappert and Besieris, 1971; Besieris and Tappert, 1973), 

 is essentially the covariance of the pressure. 



Again, everything in this equation is known in terms of the 

 fluctuations. It has the form of a classical radiation transport 

 equation and numerical techniques may be used to solve it. The 

 virtue of this approach is that it leads directly to the ensemble- 

 average acoustic field (and hence mean intensities) not just at one 

 point but at two points. 



Figure 25 shows a simple example of this method, applying a 

 diffusion approximation. The correlation function of pressure at 

 two depths is obtained (41) as an exponential, and (42) gives the 

 coherence length in depth as a function of range. This is a definite 

 prediction of the theory that can be compared to either numerical 

 experiments (for example, using the parabolic equation) or field data. 



190 



