TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN 

 UNDERWATER ACOUSTICS 



From a more fundamental point of view a more rigorous derivation 

 can be obtained by noting that the basic approximation is analogous 

 to factoring the elliptic equation into incoming and outgoing waves. 

 Such a factorization is, in fact, possible, resulting in a pair of 

 coupled parabolic wave equations, one for the outgoing wave and 

 one for the backscattered wave. Note that such a formulation could, 

 in principle, include a description of reverberation. All of the 

 numerical work to date, however, has been based on the outgoing-wave 

 parabolic equation. 



Since the parabolic equation is not valid near the source, an 

 asymptotic matching technique is required. Very near the source the 

 exact acoustic field is known (especially, say, for an isotropic 

 point source) , and this interior solution must be matched to a solu- 

 tion of the parabolic equation in the far field. 



One way to do this is indicated in Figure 2. Manipulation of 

 Equations (8), (9), and (10) leads to an expression (Equation (11)) 

 for the complex acoustic field at zero range which, when put into the 

 parabolic wave equation as an initial condition, simulates in the 

 far field a point source with unit pressure at a range of 1 yard. 

 Finally, boundary conditions must be specified to solve the parabolic 

 wave equation. To simulate the pressure release boundary condition 

 at the surface, an image source, 180 degrees out of phase with the 

 true source, is introduced (as shown in Figure 2). This forces the 

 pressure to be identically zero at the surface. 



The lower boundary condition is treated by extending the cal- 

 culation grid beyond the floor of the ocean, as indicated in 

 Figure 2. In this "mud" region well below the actual seafloor, an 

 outgoing wave boundary condition is needed. Rather than directly 



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