FREQUENCY ANALYSIS OF REVERBERATION 



329 



lies of many probability coefficients cannot be com- 

 puted mathematically. However, a few such coeffi- 

 cients have been computed on the assumption of a 

 R;iyloigh distribution lor the individual intensities. 

 One of these is the joint probability P{Ii,U,a) of ob- 

 taining reverberation intensities h and h at the same 

 range on two different pings a time interval a apart.'- 

 This coefficient is a function of the average velocity 

 of the scatterers relative to the echo-ranging vessel. 

 Motion of the scatterers relative to the transducer 

 changes the ranges and relative positions of the scat- 

 terers from one ping to the next, thereby giving rise to 

 fluctuation of the measured reverberation levels. 

 Similarly, the joint probability of obtaining intensities 

 I\ and /2 at two different ranges on the same ping has 

 been computed." A general discussion of the signifi- 

 cance of these various probability coefficients is 

 given in reference 10. Other references ""'" may be 

 useful in the analysis of fluctuation and coherence. 

 It is worth noting that the measured 6- to 7-db 

 difference between the average intensity and the 

 average peak intensity in a band three ping lengths 

 long, referred to in Section 13.2, is another measure 

 of the coherence of reverberation. For example, if the 

 coherence were verj-^ poor the average peak height in 

 any finite band would be very large. For in that case 

 the reverberation intensity at any instant would be 

 very nearly independent of the intensity at any other 

 instant; crudely, the band could be divided into a 

 large number of intervals in each of which the proba- 

 bility for a given intensity would follow the simple 

 Rayleigh distribution (2) or some similar distribution. 

 Since the number of intervals would be large, in- 

 tensities much higher than average would be ex- 

 pected to occur at least once in the band three ping 

 lengths long. So far, it has not proved possible to 

 calculate theoretically, with accuracy, the average 

 peak height in a band. 



16.3 FREQUENCY ANALYSIS OF REVER- 

 BERATION FROM NARROW- 

 BAND PINGS 



The received reverberation is often used as a 

 reference frequency for the estimation of doppler 

 shift in the echo. In order to use reverberation in this 

 way, it is necessary to know the average frequency of 

 the reverberation and the average frequency band 

 width characteristic of reverberation. Such a fre- 

 quency analysis can also give valuable information 

 about the processes giving rise to reverberation. 



The theory of Fourier series tells us that any signal 

 of finite duration can be regarded as made up of 

 single-frequency components of definite amplitudes 

 and pha.ses. A so-called single-frequency ping (CW 

 ping) of the sort emitted by ordinary echo-ranging 

 gear contains not only the nominal frequency of the 

 ping, but also an infinite number of other frequencies 

 (see Section 12.5). However, the band width within 

 which frequency components of significant amplitude 

 lie is usually very narrow for ordinary ping lengths." 

 The returning reverberation is also composed of a 

 band of frequencies. With the assumptions that led to 

 the Rayleigh distribution (2), it can be shown that 

 the band width of the reverberation equals the band 

 width of the outgoing ping. For with these assump- 

 tions, the shape of the returning signal from any 

 scatterer is the same as the shape of the ping. The 

 factors which may cause the shape of the returned 

 signal from any scatterer to differ from the ping have 

 been discussed in Section 12.5; in general these 

 factors can be neglected for pings 10 msec or longer. 

 Thus the received reverberation, which is simply the 

 sum of many such signals, must apparently have the 

 same band width as the ping. Other sources of rever- 

 beration fluctuation, in addition to the fundamental 

 randomness of phase leading to the Rayleigh distri- 

 bution, can increase the band width of reverberation. 

 However, it can be shown that these sources can be 

 neglected for pings 100 msec or shorter. 



Because of the narrowness of the frequency band, 

 the experimental determination of the frequency 

 spectrum of reverberation is difficult. Besides, the 

 quantity which is measured as a function of frequency 

 and called the frequency spectrum is merely the 

 energy or intensity contained in a narrow band of 

 frequencies; the phases of the individual frequency 

 components are never measured. For many pur- 

 poses, therefore, the measured spectrum is not the 

 most useful way of describing reverberation. The 

 measured spectrum cannot, for example, give clues 

 as to those time variations of reverberation which 

 cause it to sound like a signal of wavering pitch. 



The UCDWR has, however, developed a special 

 device known as the periodmeter for the analysis of 



» The band width may be defined as the frequency band 

 containing half the energy in the ping, or as the frequency 

 band within which intensities of spectral components are 

 no more than 3 db below the intensity of the midfrequency 

 component. For a rectangular ping the band width, defined 

 in either manner, is approximately the reciprocal of the ping 

 duration in seconds; thus, it would be about 10 cycles for a 

 100-msec ping [see equation (63) of Chapter 12]. 



