TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN 

 UNDERWATER ACOUSTICS 



wave equation in the mid- 1940s in connection with tropospheric radio- 

 wave propagation problems. Since then this method has been rather 

 widely used in radio physics and ionospheric physics (Fock, 1955; 

 Malyuzhinets, 1959; Barabanenkov , et al., 1971). 



My first exposure to the parabolic-equation method was in work 

 on radar systems involving the simulation of propagation of UHF radar 

 pulses in a random ionosphere. My former colleague, Ron Hardin, and 

 I developed computer codes based on the parabolic-equation method to 

 simulate radar propagation. 



When we became involved in underwater acoustics, it was natural 

 for us to apply these same methods to the subject of low-frequency, 

 long-range acoustic propagation in oceans. These applications turned 

 out to be quite fruitful and a number of results have been presented 

 prior to this workshop (Hardin and Tappert, 1973; Tappert and 

 Judice, 1972; Tappert, 1974a; Hasegawa and Tappert, 1973, 1974). 

 Progress has been rapid (Tappert and Hardin, 1973; Tappert, 1974b; 

 Tappert and Hardin, 1974) , and other workers have continued to develop 

 and apply these methods (Spofford, 1974; McDaniel, 1974; Benthien, et 

 al., 1974). 



A key factor in the success of the parabolic-equation method is 

 the numerical technique used to obtain the solution. The parabolic- 

 wave equation is solved directly by the finite-difference split-up 

 Fourier algorithm which makes use of Fast Fourier Transforms to 

 achieve accuracy, efficiency, and unconditional stability. This 

 yields a full-wave (all diffraction effects included) , fully coupled 

 (all mode-coupling effects included) solution for the acoustic field 

 at all depths and ranges. Realistic ocean environments with depth- 

 and range-dependent sound speed and volume loss, and layered 



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