TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN 

 UNDERWATER ACOUSTICS 



The split-step Fourier algorithm is expressed in (21) as the 

 solution at a new range, r + Ar, in terms of the solution at range r 

 operated on by a product of three factors. A is a differential 

 operator which, when carried in an exponent, is difficult to evaluate 

 by direct methods. But in Fourier space the operator A is simply a 

 multiplication and therefore this operator acting on a function of 

 depth can be quickly and accurately evaluated by first Fourier- 

 transforming the function of depth, then doing a multiplication by a 

 precomputed and stored phase function, and finally inverting the 

 Fourier transform (22) . 



One can prove by the Trotter product theorem of functional 

 analysis that in the limit as Ar goes to zero, the iterated version 

 of this does converge in norm (that is, in the space of discrete 

 functions or functions defined on a discrete grid) to the solution 

 of the parabolic-wave equation. 



Some of the features of this algorithm are listed in Figure 4. 

 The advantages of this method (listed in Figure 5) are that, without 

 any extra effort or computation, it can treat range-dependent 

 velocity profiles, range- and depth-dependent volume losses, and 

 variable bathymetry (that is, the deptn of the ocean can be an 

 arbitrary function of range) . It is easy to solve numerically 

 by marching in range. 



The disadvantages (also Figure 5) are that for very large 

 angles with respect to horizontal, which sometimes occur with steep 

 slopes, there are inaccuracies. (Recently, methods have been de- 

 veloped to reduce these inaccuracies.) Discontinuities require special 

 treatment (essentially smoothing) , but this can be done in a way that 

 is consistent with the physics and mathematics of low-frequency 



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