TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN 

 UNDERWATER ACOUSTICS 



acoustic propagation. Also, the neglect of azimuthal coupling can be 

 remedied at the cost of greater computer running times. 



In summary, the technique treats a wave equation in an especially 

 useful way by applying effective and very rapid computational methods 

 to generate its solution. The basic program is easy to write because 

 the algorithm is so simple and stable. For production runs on a daily 

 basis, a highly optimized version of this program is needed, such as 

 that developed at AESD by using machine language programming and 

 sophisticated Fast Fourier Transform techniques. The AESD version 

 has achieved enormous increases in speed over earlier versions. 



APPLICATIONS 



The following examples indicate the application of the parabolic- 

 equation method to several problems in underwater acoustics. These 

 are displayed in terms of iso-loss contours in range and depth from 

 the effective "source" which may actually be the receiver. The top 

 figure of each pair is the basic field contoured in 5-dB intervals. 

 The lower figure represents a range-averaged field with only the 80- 

 and 90-dB contours shown as light and heavy, respectively. The shaded 

 regions are either less than 80-dB loss if inside the 80-dB contour, 

 or greater than 90-dB loss when bordered by the 90-dB contour. 



The first example corresponds to a simple pressure-gradient, 

 or linear, profile in water 16,000 feet deep over a high- loss bottom. 

 The effective "source" depth is 8,000 feet. Figure 6 illustrates the 

 field contours for a frequency of 50 Hz. Point C is the location of 

 a cusped caustic, the two smooth branches of which are migrating 

 toward the surface and bottom with increasing range. In ray- tracing 

 programs, the fields in these regions must be found by using uniform 



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