TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN 

 UNDERWATER ACOUSTICS 



Using the quasi-static approximation in which the frequencies 

 of the fluctuations are small compared to both the carrier frequency 

 and the transit time of the acoustic waves over a horizontal correla- 

 tion length L , the parabolic wave equation for a random ocean with 

 H 



time-dependent fluctuations varying in both range and depth is ex- 

 pressed in (25) . 



The solution of this equation gives the pressure as a function 

 of range and depth and time (26) . It is a function of three vari- 

 ables, represented in the form of a complex envelope and a carrier 

 wave; ij; is simply the complex-demodulated envelope which would be 

 measured experimentally. Hence, ij; is a quantity that can be compared 

 directly to experimental measurements of acoustic fluctuations in 

 the ocean. Typical quantities of interest are correlations of the 

 pressure at different ranges, depths, and times. This approach has 

 been carried out numerically, and the results of that calculation 

 are presented in subsequent workshop papers dealing with both theory and 

 comparisons with experimental results. 



Two additional theories are being developed in connection with 



this problem of wave propagation in random oceans. Using the 



analogy with the Schroedinger equation, following Pauli, a Pauli 



master equation can be derived using normal modes (Figure 23) 



(Agarwal, 1973) . The envelope ^ is represented as a sum of normal 



modes (28) with random coefficients a . A density matrix (29) is 



n 



defined as the correlation between normal-mode amplitudes, and 

 coupling coefficients (30) between normal modes are used to represent 

 the effects of the randomly fluctuating component of the sound speed. 



Transition probabilities (31) are developed and finally a master 

 equation (32) involving only the diagonal elements of the density 



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