TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN 

 UNDERWATER ACOUSTICS 



Therefore, the rays which can be obtained from the parabolic 

 wave equation are the same as the rays which can be obtained from the 

 elliptic wave equation except for a shift in the range scale. This 

 constant factor (or value of Snell's invariant) can be absorbed into 

 the range for any one ray, but not for all rays. The approach used 

 by AESD is to effectively make this change in range scale for those 

 rays which tend to dominate the acoustic field. In this way, errors 

 in the parabolic approximation can actually be reduced. 



Because the parabolic equation is a wave equation, in the case 

 of a stratified ocean the solution can also be expanded in terms of 

 normal modes, as shown in (15) and (15) for the elliptic and parabolic 

 solutions, respectively. The parabolic equation also has a continuous 

 part to its spectrum, and the summation implicitly implies an integra- 

 tion over the continuous part of the spectrum as well. 



Both the normal mode eigenvalues, y , and eigenf unctions, d) , 



a a 



are the same for the elliptic and parabolic equations. However, the 



phase velocities (as reflected in the exponential factors in (15) and 



(16)) are different. By expanding the square root in (15) and 



retaining only the leading term in y , the parabolic phase velocity 



a 



is obtained. The error, therefore, for a single mode can be estimated 

 by carrying the expansion to the quadratic term, as is shown in (17). 



At 100 Hertz for a typical mode in the sound channel, a phase error 



4 

 of 2iT would be accumulated at a range of 10 nm. Hence, if no change 



is made in the range scale, as mentioned earlier, a range error of 



about 5 percent accumulates for a ray near 20 degrees. 



In Figure 4, the parabolic wave equation is rewritten in terms 

 of a differential operator. A, and a multiplication operator, B, 

 leading to (20) . Note that both n and a are variable quantities 

 and A and B do not commute. 



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