326 



VARIABILITY AND FREQUENCY CHARACTERISTICS 



be regarded as a function of certain relatively slowly- 

 varying physical parameters such as the wind force 

 and the wave height. 



The derivation of equation (2) is based on the as- 

 sumption that the number of independent interfering 

 amplitudes is large, or, in other words, that the 

 number of scatterers which combine to return sound 

 to the receiver at a particular time is large. However, 

 since this number of scatterers is proportional to the 

 volume of space illuminated by the pulse, it is ap- 

 parent that the effective number of scatterers de- 

 pends on the ping length, and cannot be large for 

 very short pulses. In fact, with pings sufficiently 

 short and directional, the received reverberation at 

 any instant will arise from at most one scatterer. 

 With such pings the average number of scatterers per 

 unit volume could be determined from the character 

 of the received reverberation, provided the dimen- 

 sions of the scatterers are small compared to the 

 average distance between them. For, under these 

 circumstances, the received reverberation would be 

 a series of widely separated echoes from individual 

 scatterers; from the spacing of these echoes the 

 average number of scatterers per unit volume could 

 easily be calculated. 



However, it has not yet been possible, and may 

 never be feasible, to reduce the ping dimensions suf- 

 ficiently to resolve all the individual scatterers re- 

 sponsible for volume reverberation. With standard 

 24-kc gear, some of the larger individual scatterers 

 are occasionally distinguishable even when pings as 

 long as 5 msec are used.^' ^ However, as a general rule 

 individual scatterers cannot be identified even with 

 pings as short as 0.1 msec. It has been pointed out ^ 

 that the Rayleigh distribution does not apply when 

 the effective number of scatterers is small, and that 

 it may be possible to determine the average number 

 of scatterers per unit volume from the disparity be- 

 tween the observed distribution function and the 

 Rayleigh form (2). The theoretical distribution de- 

 pends on the assumptions which are made concerning 

 the total number of scatterers present. The most 

 reasonable assumption is that the number of scat- 

 terers obeys a Poisson ' distribution. If so, the proba- 

 bility Pn(N) that A'' scatterers are present when the 

 average number present is n is given by 



PniN) = 



n^'e- 



N! 



(3) 



With this assumption, the expected distribution func- 

 tion of the reverberation intensity is calculated in 



reference 6, as a function of the average number of 

 scatterers in the portion of the ocean illuminated by 

 the ping at a definite instant. If this average number 

 of scatterers is 10, the deviation of the predicted dis- 

 tribution function from the Rayleigh distribution is 

 of about the same order of magnitude as the devia- 

 tions of the experimental points from the Rayleigh 

 distribution in Figure 1. However, the points plotted 

 in Figure 1 are not sufficiently numerous for a deter- 

 mination of n. It is estimated in reference 6 that 

 4,000 points, all taken with the same ping length, are 

 required to say definitely whether an observed distri- 

 bution more nearly approximates the theoretical 

 curve for 10 scatterers or the Rayleigh distribution. 

 The average number of scatterers n may also be 

 predicted, in principle, from the percentage fluctua- 

 tion in intensity. If the actual number of scatterers 

 obeys the Poisson distribution (3), reference 6 gives 

 the following relationship 



AP 



1 



= 1 + -- 



n 



(4) 



If the intensity is the resultant of a fixed number n of 

 amplitudes whose phases vary at random, it is easy 

 to show that 



!='--■ «' 



It is readily verified by direct integration that for the 

 Rayleigh distribution (2), 



AP 



1 



(6) 



in agreement with equations (4) and (5) when n is 

 infinite. A third alternative is to assume that the 

 number of scatterers obeys a Gaussian distribution; 

 this would be the case, for example, if the variability 

 in the number of scatterers were due primarily to 

 accidental variations in the length of the ping emitted 

 by the projector. With a Gaussian law for the number 

 of scatterers, still another formula would be obtained 

 for AP/(i)\ 



This discussion of reverberation fluctuation has 

 completely neglected the fluctuation due to varia- 

 bility in such factors as transmission loss, projector 

 output, and transducer orientation. With well-func- 

 tioning equipment such as setup C of Section 13.1.1, 

 variations in projector output occur so slowly that 

 their effect on the short-term fluctuation of rever- 

 beration is negligible. Variability in transmission loss 

 is known to be large and rapid; therefore, the fluctua- 

 tion in reverberation levels must be partly a result of 



