1 9 



Leontovich and Fock (1946) have shown that if the term 8 U/3r can be neglected, a 

 marching solution to the resulting Parabolic Equation can be written in the following form 



U(r+Ar,z) = exp I -i Ar (K2-kQ2)/2l 



jexp [(i Ar/2 k^) a^/az^l J U (r,z) 



op 



Using results that they had developed for electromagnetic propagation problems associated 

 with the SAFEGUARD ABM program, Tappert (1977) was able to show that the exponen- 

 tial operator acting on U could be calculated using a discrete Fourier transform F. 



I j U(r,z) = F-Wexpr-iArs2/(2kQ)l FU(r,z)| , 

 ^ -'op 



where s is an index of F, and F~ is the inverse transform. Because there are fast forms of 

 the discrete Fourier transform available, the P.E. model has proved to be a reasonably 

 efficient method for calculating the sound field for non-bottom hmited cases. 



In bottom hmited cases the P.E. method runs into serious difficulties. The sound 

 pressure is continuous across the interface between the water and the sediment. However, 

 the density is discontinuous since the sediment density usually is at 20 to 100 percent 

 greater than that of water. Since U - p '- p, where p is the density and p is the pressure, 

 U is discontinuous. This leads to the well known Gibb's phenomenon in which oscillations 

 are generated by taking the Fourier transform of a discontinuous function (Figure 27). To 

 avoid Gibb's phenomenon it is necessary to replace an accurate representation of the bottom 

 sediments by one in which K*^ and p have minimum variation with z but which also is con- 

 sistent with the value of bottom loss that is calculated for the reaHstic sediment model, 

 Figure 28. The filled circles in Figure 29 represent the correct values of |R|. We want to 

 generate an equivalent sediment model that has smooth K and also has the bottom loss 

 shown in Figure 29. 



This is done with a simple algorithm as shown in Figure 30. The symbol x repre- 

 9 9 



sents either the real part of K~, the imaginary part of K or the density. A change Ax is 



made and the least mean square error of the difference, E, between the desired curve and 



the calculated bottom loss curve is calculated. If E is reduced by a change in +Ax or -Ax 



then ±Ax is increased. If the calculated E is larger, then x and E remain the same but Ax 



is reduced for the next iteration. 



The final results are shown in Figure 3 1 where there is good agreement between the 



the bottom loss for the equivalent sediment mode (the line) and the desired bottom loss 



(the filled circles). 



SUMMARY 



We have developed a general purpose plane wave reflection model that can account 

 for both liquid and/or solid layers. Earlier models which we developed, such as the sohd 

 model and the hnear gradient model, contributed and laid the basis for our most recent 



58 



