328 



VARIABILITY AND FREQUENCY CHARACTERISTICS 



middle of the blob is plotted on the vertical axis. 

 The ping length r is given in an upper corner of each 

 plot and is marked along the time axis for comparison 

 with the blob width of the reverberation. It will be 

 observed that the diagram of each blob consists of a 

 peak with wings trailing off to either side. It appears 

 from the figure that the duration of a reverberation 

 blob is very close to the ping length, but that the blob 

 is peaked in shape and not rectangular like the en- 

 velope of the ping. 



The coherence of the reverberation can be de- 

 scribed mathematically by the correlation between 

 the reverberation amplitudes at different times. This 

 correlation coefficient p is defined by 



_ Uih) - J«l)][/(<2) - 7«2)] 



(7) 



If p is very nearly 1, then a high value of the intensity 

 at time h is likely to be associated on the same record 

 with a high value of the intensity at time k. If p is 

 near zero, a given value of the intensity at time h 

 gives no information about the intensity on that 

 record at time i2. If p is very nearly — 1, a high value 

 at ti implies a likely low value of the intensity at k- 



It can be shown theoretically that p depends, in 

 general, only on the difference in time | <i — <2 1, pro- 

 vided the average intensities at ii and fe are not too 

 different. '" If the outgoing ping is square-topped, and 

 if the average intensities at the times <i and 4 are 

 equal, and if, furthermore, the individual intensities 

 follow the Rayleigh distribution (2), then it can be 

 shown that the correlation coefficient (7) reduces to 



( a> T 



(8) 



where a = | <i — <2 1 and r is the ping length." In 

 other words, reverberation levels at times close to the 

 center of a blob will be high, but levels at times more 

 than a ping length away from the center will bear no 

 relation to the intensity at the center of the blob. The 

 value of p from the relation (8) is plotted as a func- 

 tion of a/r in Figure 3. Ev-idently the result (8) for 

 the correlation coefficient explains the dependence of 

 the blob width on ping length noted in Figure 2. 



The precise functional dependence of p on a in 

 relation (8) depends on the assumption of a rectangu- 

 lar ping and also depends in part on the assumption of 

 a Rayleigh distribution for the individual intensities. 

 If, for example, reverberation resulted from echoes 



returned from widely spaced single scatterers, rather 

 than from a dense population of scatterers, each re- 

 verberation blob would reproduce the shape of the 

 original ping and p(a) would be unity for a < t. 

 However, the result that p is zero when a ^ t should 

 be quite independent of the assumed distribution 

 function for the reverberation intensities. For, when 

 a ^ T the reverberation at time h arises from scatter- 

 ing in a volume of space which does not overlap the 

 volume causing the reverberation at time U. Thus, if 

 the reverberation levels from two nonoverlapping 

 volumes are independent of each other — • and this is 

 a reasonable assumption — p will be zero whenever 

 a ^ T. In other words, no matter what the distribu- 

 tion of the reverberation intensities, a decrease of 

 blob width with decreasing ping length would be ex- 

 pected. A corollary to this discussion is that the ob- 

 served dependence of blob width on ping length does 

 not lend any appreciable support to the assumptions 

 which led to the Rayleigh distribution. 



a. 



ID Z 



S < 



U, K 



^ 2 



o ^ 



o <x 



1.0 1.6 



RATIO OF TIME INTERVAL cC TO 

 PING LENGTH 



Figure 3. Theoretical curve for self-correlation of re- 

 verberation intensity. 



It is possible to determine all sorts of probabihty 

 coefficients from the observed reverberation records; 

 these coefficients can then be compared with compu- 

 tations based on various assumptions regarding the 

 sources of reverberation. However, the labor re- 

 quired to analyze the observed records is so great 

 that this method of investigating reverberation has 

 not been considered practical. To illustrate, only one 

 very crude attempt has been made to quantitatively 

 compare the functional dependence predicted by re- 

 lation (8) with experiment; the agreement cannot 

 be said to have been better than quahtative. Another 

 difficulty in this approach is that the theoretical val- 



