268 



THEORY OF REVERBERATION INTENSITY 



the reradiated sound from such scatterers would die 

 out in a very short time compared to the duration 

 of even a 1-msec ping. For most scatterers equation 

 (67) will be satisfied for pings of ordinary length al- 

 though it may sometimes not be satisfied with scat- 

 terers such as rocks on the bottom, for 1-msec pings. 



In the light of the discussion in Section 12.5.1, the 

 distance D in equation (67) must be interpreted as 

 the diameter of the volume within which there is ap- 

 preciable correlation between the phases of waves re- 

 flected from different points in the ocean. Scattering 

 volumes separated by so large a distance that there is 

 little correlation may be considered unrelated scat- 

 terers. In the absence of definite knowledge about the 

 scatterers, it is difficult to make the argument pre- 

 cise, but it seems unlikely that there will be apprecia- 

 ble correlation over a distance as great as a yard, 

 which is about the length of 1-msec ping. 



Along with assumption 2, Section 12.1, it has been 

 tacitly assumed, in the derivation of the theoretical 

 reverberation formulas that the outgoing ping is 

 square-topped (that is, that the intensity rises 

 abruptly to its steady-state value at the beginning 

 of the ping, remains constant until the end of the 

 ping, and then drops suddenly to zero), and that the 

 wave received from any scatterer reproduces the 

 shape of the outgoing ping. It is possible to make the 

 outgoing ping very nearly square-topped, but it is 

 apparent from equation (66) that the shape of the 

 waves returning from each scatterer is not necessarily 

 the same as the shape of the outgoing ping. However, 

 if the ping does not include too wide a frequency band 

 (that is, is not too short) and if the scattering co- 

 efficients of the various types of scatterers do not 

 vary too rapidly with frequency, then in equation 

 (66), if sound is being received from only one scat- 

 terer, B{(ji) is nearly independent of frequency, and 

 the returning scattered wave does very nearly repro- 

 duce the wave form of the outgoing ping. It will be 

 seen in Chapters 4 and 5 that even in a 1,000-c band, 

 scattering coefficients in the ocean apparently change 

 very little, so that, from equation (63), square-topped 

 1-msec pings should result in square-topped scattered 

 waves. 



We can now see the significance of the assumption, 

 made at the beginning of Section 12.5, that the in- 

 stantaneous voltage induced in the receiving circuit 

 is exactly proportional to the instantaneous pressure 

 of the sound arriving at the transducer. If this is not 

 the case, that is, if there is phase distortion or ampU- 

 tude distortion in the receiver, then the reverbera- 



tion resulting from any one scatterer will not have 

 the square-topped shape of the outgoing ping, and 

 the formulas which have been derived will be in error. 

 Thus, if measured reverberation intensities are to be 

 comparable to the theoretical formulas of Sections 

 12.2 to 12.4, it is necessary to use flat wide-band 

 systems which have little transient response to the 

 sudden changes of reverberation intensity. The use 

 of narrow-band systems with high transients will 

 usually cause the reverberation received from any 

 scatterer to last longer than the outgoing ping and 

 have a shape different from that of the ping. Since 

 this distortion will decrease with increasing ping 

 length T, deviation will result from the predicted 

 proportionaUty of R'it) on ping length t in equations 

 (24), (43), and (54). In order to derive appropriate 

 theoretical formulas for such systems it would be 

 necessary to write, using equation (66), 



1 r" 



E{t) = —7=] H'^)B{oi)C{oi)e'"'dw 



for the voltage induced across the receiver terminals 

 by each scattered wave where C(co) describes the 

 frequency response of the gear. The expression for the 

 average reverberation intensity would then involve 

 an integration over the frequency band included in 

 the ping and passed by the equipment. It may be re- 

 marked that there is an inherent dependence of re- 

 sponse on frequency in any directional transducer, 

 because the pattern functions 6(9,<^) and b'{d,4>) are 

 always functions of frequency. Thus, it may be neces- 

 sary to consider further the effect of directivity on the 

 theoretical reverberation formulas for situations in- 

 volving very short pings and highly directional 

 transducers. 



12.5.4 Neglect of Multiple Scattering 



We have assumed that the sound reaching the 

 transducer as reverberation has been scattered only 

 once. It is easy to see that the validity of this assump- 

 tion depends on the range of the received reverbera- 

 tion. For, as the ping proceeds out from the trans- 

 ducer, it loses more and more energy by scattering; 

 the scattered waves are of course no different from 

 any other sound waves and are themselves scattered. 

 Eventually, therefore, a range is reached at which the 

 ratio between the singly scattered sound returning 

 to the transducer and the multiply scattered sound 

 returning is no longer large. The value of this range 

 depends on the amount of scattering which takes 



