TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN 

 UNDERWATER ACOUSTICS 



applying such a boundary condition, its equivalent is simulated by 

 preventing waves that reach so far below the floor of the ocean from 

 scattering back into the ocean. This is achieved by introducing a 

 strong volume attenuation which absorbs all the acoustic waves that 

 reach this subbottom layer, colloquially called mud. This is a 

 numerical, artificial absorption introduced solely to remove reflected 

 waves. There is, of course, a corresponding image mud in the upper 

 ocean. 



While substantial analyses have been performed on the validity 

 of this approximation, it is still a rather open subject and there 

 have not yet been developed necessary and sufficient conditions for 

 its validity. 



The best way, of course, is to compare it to field data, and 

 this has been done in a number of cases by myself, and Spofford (1974) 

 who also compared it with ray and normal-mode results. Such com- 

 parisons are not conclusive, however, nor are they a replacement for 

 precise analytical estimates for the conditions under which the 

 parabolic approximation is valid. 



One such analytical approach is to begin with the geometrical 

 acoustics approximation to the parabolic wave equation (Figure 3) . 

 The exact ray Equation (12) is shown for a two-dimensional stratified 

 ocean, where z is the ray depth as a function of range r, and 6 is 

 the vertical angle of the ray (rather than the azimuthal angle, as 

 earlier) . In the parabolic approximation, the corresponding ray 



equation is given in Equation (13) and is the same except for the 



2 



factor 1/ (n cos 6) . However, they both have as a first integral 



Snell's invariant as expressed in Equation (14). 



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