BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 



223 



distance, as shown in Figure 23. One might at first 

 suppose that refraction of this sort would be similar 

 to refraction in the water alone, and that the re- 

 ceived pulse would be a replica of the pressure wave 

 emitted by the source, with an intensity which could 

 be calculated by ray theory. It is easily shown, how- 

 ever, that this is not the case. We shall first show that 

 when the bottom is acoustically uniform, so that rays 

 in the bottom are straight lines, the intensity pre- 

 dicted by ray theory for a ray such as EABH in 

 Figure 23 is zero. Figure 25 shows a ray having 

 inclination 6 in the water, dx in the bottom, together 

 with a neighboring ray. By Snell's law of refraction 

 we have 



cosfli 



Ci 



cos 



(9) 



where c and Ci are the velocities of sound in water and 

 bottom respectively. Now the energy which leaves 

 the source E in an interval dB of inclinations and in a 

 fixed narrow interval of azimuth is partly reflected 

 and partly transmitted, and the transmitted part is 

 distributed over the region between the rays AP 

 and A'P' in Figure 25. If R{e) is the reflection 



Figure 25. Spreading of adjacent sound rays on enter- 

 ing the bottom. 



coefficient of the bottom at the angle B, this trans- 

 mitted energy is proportional to [1 — R{B)']dB. If the 

 range r = AP \s large compared to EA, the distance 

 between P and P' will be approximately 



ds « rdQi 



c sm S . 

 r-- dB 



(10) 



Ci sm 6i 



by equation (9). By introducing another factor r to 

 allow for azimuthal spreading, the energy received 

 at P per unit area is then proportional to 



Cl-fi(W^_l.^,_^(,^^.c,sin^. 

 rds r^ c sin B 



(11) 



As the ray AP approaches the horizontal, sin Bi ap- 

 proaches zero, and according to equation (11) the 

 intensity at P must do likewise. This conclusion is 

 made even stronger by the fact that, according to 

 Section 2.6.2, R{B) approaches unity as B approaches 

 the angle for total reflection. Thus, ray theory cannot 

 account for the sound received via a path like EABH 

 in Figure 23, when the bottom is uniform. 



The argument just given to show the inapplicabil- 

 ity of ray theory to arrivals of the type shown in 

 Figure 23 would of course not be strictly correct if 

 there were a gradual increase of the velocity of sound 

 with depth in the bottom, a situation which is quite 

 common, especially for soft bottoms. It will be in- 

 structive to consider briefly the sound ray paths for 

 this case, since the limitations of ray theory can be 

 most clearly seen by studying this case where it is 

 partially applicable. 



/ \ / '\ / V **'" 



1 i,; / %' 





-nw 





WATER 



:^ 



RECEIVER 



Figure 26. Ray paths in and over a bottom giving 

 weak upward refraction. 



Figure 26A shows a family of rays connecting a 

 source and hydrophone, both of which are lying on a 

 bottom characterized by weak upward refraction. 

 According to ray acoustics the first signal to reach 

 the receiver H will arrive via the path I^. This will be 

 followed almost immediately by arrivals along other 

 paths, such as Ig, which likewise lie in the bottom 

 but which involve one or more reflections at the inter- 

 face between bottom and water. Some time later an- 

 other group of arrivals will be received, each of which 

 comes along a path involving one reflection from the 

 surface of the water. One path of this type is shown 



