TAPPERT: SELECTED APPLICATIONS OF THE PARABOLIC-EQUATION METHOD IN 

 UNDERWATER ACOUSTICS 



asymptotic expansions involving Pearcey functions for the cusp, and 

 Airy functions for the smooth caustics. In the parabolic-equation 

 method, such features are treated routinely. 



Figures 7 and 8 correspond to the same geometry for frequencies 

 of 100 and 2 00 Hertz, respectively. As the frequency increases, the 

 diffraction effects are reduced and the caustics are more pronounced. 

 At 200 Hertz, the second cusp, at twice the distance of the first, 

 is clearly present. It must be re-emphasized that these contours 

 are not rays. The ray-like patterns correspond to interference be- 

 tween up- and down-going rays near the surface, and between pairs 

 of rays associated with smooth caustics. 



The preceding three figures correspond to a bottom with high 

 volume attenuation so that essentially no energy is returned when 

 it enters the bottom. Figure 9 is for the same case as Figure 8 

 (200 Hertz) but with a low-loss bottom (simulated by a strong posi- 

 tive sound-speed gradient and no volume attenuation) . Here the 

 bottom-reflected paths are spectrally reflected and interfere with 

 the RSR paths distorting the field contours even around the cusps. 



The second example corresponds to the slightly more complicated 

 environment of a bilinear profile. The ray trajectories for a source 

 in the thermocline segment of the profile are shown in Figure 10, 

 compliments of Richard Holford of Bell Labs. Note the formation 

 of smooth and cusped caustics, RR caustics which effectively surface 

 reflect, and the intersections of caustics. The correct ray treat- 

 ments for these cases are extremely complex. 



Figures 11 through 14 illustrate the field contours generated 

 by the parabolic-equation method (again using a high- loss bottom) 

 for frequencies of 50, 100, 200, and 400 Hertz. At the lower 



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