266 



THEORY OF REVERBERATION INTENSITY 



lected, and we can conclude that the rms reverbera- 

 tion intensity, averaged over a number of pings, will 

 equal the sum of the average intensities received from 

 the individual scatterers in the ocean. 



We may expect the phases to vary in a random way 

 because of the properties of equation (1). This equa- 

 tion implies directly that sound propagates through 

 the ocean as a wave with a definite velocity, and that 

 the phase of the returning wave depends on the 

 travel time of the wave from the transducer to the 

 scatterer and back. At 24 kc a phase shift of 2-k, 

 amounting to a shift of a complete cycle, results from 

 a relative displacement between two scatterers of 

 about an inch, or a difference in travel time of about 

 40 /[isec. Such phase changes or changes in ray 

 path from ping to ping could result from thermal 

 fluctuations, from the rise and fall of the transducer 

 in the ocean, from wave motion, and from drift of the 

 scatterers and the projecting ship. A relative dis- 

 placement of one inch in five seconds (the approxi- 

 mate time between pings) corresponds to a relative 

 drift of only 60 ft per hr. 



It is worth stressing that the phase shifts discussed 

 in the preceding paragraph are relative phase shifts 

 between waves from different scattering points in the 

 ocean. At any instant sound is being received from 

 many different points on any one scatterer; but if the 

 scatterer is a rigid sphere, for example, the phases of 

 the waves arising at different points on the spherical 

 surface always bear a definite relation to each other. 

 These waves from the different points on the spherical 

 scatterer will always combine to give the same result 

 in equation (62), irrespective of relative displacement 

 between the scatterer and the transducer. Thus, the 

 likelihood that the double sum in equation (62) will 

 average to zero over a number of pings is connected 

 with what might be called the "correlation" between 

 conditions at various points in the ocean. If the ocean 

 were rigid, so that the relative positions and orienta- 

 tions of the scatterers never changed, knowledge of 

 the phase of a returning wave from one point in the 

 ocean would completely determine the phases of re- 

 turning waves from all other points. In this event the 

 assumption that the double sum in equation (62) 

 averages to zero would be more difficult to maintain. 

 However, since the ocean is not rigid and the posi- 

 tions and orientations of the scatterers change with 

 time, knowledge of the phase of a wave returning 

 from one point determines the phases only of those 

 waves from the immediate neighborhood of the par- 

 ticular point. 



The averaging to zero of the double sum in equa- 

 tion (62) is made even more probable by our averag- 

 ing procedure, which focuses attention on a definite 

 instant relative to the midtime of the emitted signal. 

 As the transducer drifts or otherwise changes its posi- 

 tion in the ocean, the reverberation received at a 

 definite instant comes from different points of space 

 on different pings. In many cases, the phases of the 

 scattered waves returning from these two portions of 

 sfjace will be almost completely uncorrelated; in such 

 cases, random phase relations on successive pings are 

 even more likely. 



In the absence of definite knowledge about the 

 scatterers responsible for reverberation, it is not 

 possible to make this argument about equation (62) 

 more precise. However, it is apparent from this dis- 

 cussion that in all types of reverberation there are a 

 number of mechanisms that can cause random varia- 

 tions in the phases of the individual returning scat- 

 tered waves. When these phases are truly random 

 the assumption involved in averaging the individual 

 scattered intensities, to get the average reverberation 

 intensity, is justified. 



12.5.2 Definition of Backward 

 Scattering Coefficient 



The backward scattering coefficient was defined for 

 a volume small enough so that its relevant properties 

 do not change too sharply with changes of position 

 inside the volume, and large enough that it contains 

 a reasonable number of scatterers. Mt&r an explicit 

 assumption that the scattering by such a volume is 

 proportional to the volume, the scattering coefficient 

 of V was defined by formula (4). 



It is easy to see that this assumption should be 

 valid if the double sum in equation (62) vanishes. If 

 this term vanishes, the total reverberation intensity 

 will be just the sum of the average intensities of the 

 waves from the individual scatterers; therefore, it 

 should be proportional, on the average, to the size 

 of the scattering volume. 



12.5.3 ( Duration of Scattering by a 

 Scatterer 



In Section 12.1, it was assumed that scattering 

 from an individual scatterer begins the instant sound 

 energy begins to arrive at the scatterer and ceases the 

 instant sound energy ceases to arrive. 



In considering this assumption, we must recognize 



