VOLUME REVERBERATION 



253 



12.2 



VOLUME REVERBERATION 



Volume reverberation is defined as sound scattered 

 back to the transducer by scattering centers in the 

 volume of the sea. Let a transducer be located at 

 in deep water far from both the sea surface and sea 

 bottom. This transducer sends out a pulse of sound, 

 or ping, of duration t. Because the ping is of finite 

 duration, a large part of the sound energy at a time 

 t/2 seconds after midsignal will be contained between 

 two closed surfaces Si, S2, portions of which are 

 shown in Figure L If the sound velocity c in the 



Figure 1. 

 t/2. 



Portion of ocean scattering sound at time 



ocean is everywhere the same, these surfaces are 

 spheres centered at 0, with radii ct/2 — ct/2 and 

 ct/2 + ct/2. With refraction, however, the surfaces 

 may be far from spherical. The reason for saying that 

 a "large part" instead of "all" the energy in the ocean 

 lies in the volume between Si and S2 is that the very 

 existence of reverberation shows that some sound, 

 scattered earlier out of the ping, does not lie between 

 these two surfaces at the time t/2. However, accord- 

 ing to assumption 3 in Section 12. 1, the amount of the 

 previously scattered soimd which is rescattered back 

 to the receiver is neghgible. Therefore, because of 

 assumption 2, the only scattering taking place at 

 time t/2, which is important in producing reverbera- 

 tion, occurs at those scatterers located within the 

 volume S1S2 defined by the transmission laws of ray 

 acoustics. 



Now consider the sound scattered at time t/2 by 

 the volume S1S2. Obviously, all this soimd will not 

 return to the receiver at the same instant since the 

 sound scattered near Si travels a shorter distance 

 than does sound scattered near S^. It is shown in 

 Section 12.5.5 that it can be assumed as a conse- 

 quence of Fermat's principle,^ that the average travel 

 time of sound along the path from the transducer 

 to any point X in S1S2 equals the average travel 

 time from X back to 0. Using this result, we can 

 readily delimit the region where the sound returning 

 to the transducer at the time t is scattered backward. 

 If T is the signal duration, and if all times are meas- 

 ured from the middle of the signal, the signal emis- 

 sion starts at time — t/2 and ends at time t/2. The 

 sound emitted first, at time — t/2, and returning as 

 reverberation at time t, is scattered backward at 

 some definite time which we shall call t'. The corre- 

 sponding travel time Ti out to the scatterers must be 

 t' -\- t/2; the travel time back to the receiver must 

 have an equal value because of our assumptions; and 

 the sum of these two travel times and the time of 

 emission —t/2 must equal t, the time at which the 

 reverberation is received. Thus, 



<'■+-;) 





and therefore 



^^ = 2- + i- 



(5) 



Similarly, the sound emitted last, at time t/2 and re- 

 turning as reverberation at the time t, must be 

 scattered at a time t" given by 



t" = '- + ''-■ 

 * 2 + 4' 



and the corresponding travel time T2 is t" — -> or 



^^ = 2-4- 



(6) 



Thus, all the scatterers effective in producing the 

 reverberation at time t must lie between a pair of 

 surfaces out to which the one-way travel times are, 

 respectively, t/2 — t/4: and t/2 + t/\. These sur- 

 faces are indicated in Figure 2 by the labels S'l and S2. 

 If there is no refraction, these surfaces S'l and S'2 are 

 spheres with radii c{t/2 — t/4) and c{t/2 -f- t/4) 

 respectively; the volume SiS'2 is thus a spherical shell 

 with average radius ct/2 and thickness ct/2. In 

 general, even with refraction present, the volume 

 S'iS'2 is about half the volume S\S2', that is, the 



