SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION 

 METHOD IN UNDERWATER ACOUSTICS 



Frederick Tappert 



Courant Institute of Mathematical Sciences 

 New York University 



A review of the parabolic-equation method in underwater 

 acoustics is presented. Applications of the parabolic- 

 equation method discussed here include: 



• Short-range (several hundred nm) calculations of 

 transmission loss 



• Calculations of transmission loss in environments 

 with variable sound-speed profiles and bathymetry 



• Calculations of fluctuating acoustic fields in a time- 

 dependent fluctuating ocean using a model for a random 

 internal-wave field superimposed on Munk's canonical 

 profile. 



The parabolic-equation method is also used as the start- 

 ing point to derive theoretical expressions for fluctu- 

 ations of acoustic fields in random oceans. Using the 

 mathematical analogy with Schroedinger ' s wave equation, 

 two such techniques are described: the first applies the 

 wave kinetic equation approach to underwater acoustics; 

 the second applies the Pauli master equation approach to 

 the same problem. 



Theoretical and numerical studies and comparisons to 

 field data lead one to believe that the parabolic-wave 

 equation adequately describes acoustic waves propagating 

 in real oceans for frequencies between 5 and at least 

 500 Hz out to ranges of at least 10,000 nm. 



BACKGROUND 



Leontovich and Fock (1945) , two Soviet scientists, were the 

 first to approximate an elliptic reduced wave equation by a parabolic 



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