520 HERSEY AND BACKUS [CHAP. 13 



since it is similar to echo-sounding or echo-ranging apparatus discussed else- 

 where in this book. It is expected that since the pulse of sound forms an ex- 

 panding spherical shell of constant thickness, it will generally be scattered 

 from many objects in the water at times such that their scattered waves arrive 

 simultaneously at the receiver. The receiver senses the appropriate sum of 

 these independent waves. The usual treatment of the problem is based on this 

 assumption, i.e. that at any instant scattering from many individual objects is 

 superposed. Since these individuals occupy a volume it is convenient to define 

 a scattering coefficient, m, of a unit volume, analogous to the back-scattering 

 cross-section of an individual. 



Let us now take the transducer as the center of a spherical co-ordinate 

 system (r, 6, (f)) and further let us assume that the energy emitted by the 

 source per unit solid angle per second in the direction {6, ^) is described by 

 F{r, d, (f)), where t is measured from the beginning of a pulse. The intensity of 

 volume reverberation at the receiver at time t can be shown to be related to 

 the volume scattering coefficient of the medium, m{r, 6, 0), through the follow- 

 ing integral equation, 



/W=f nr.e^)Mr.e.me.*)^y_ ,21) 



jvolume r^ r^ 



where b{d, <f>) describes the directional properties of the transducer as a receiver 

 and 



r = {cl2){t~r), (22) 



where c is the velocity of sound in sea-water in cm/sec. 

 The following assumptions are implicit in equation (21). 



(1) The sound velocity in the medium is constant and not materially affected 

 by the scatterers. 



(2) The definition of m given above has the effect of averaging the con- 

 tributions of the individual scatterers. 



(3) A volume element begins to scatter sound at the instant it is insonified, 

 and stops as soon as it is again "in the dark", i.e. there are no time lags — no 

 storage of energy in the scatterers. 



(4) Scattering cross-sections are small enough for us to neglect multiple 

 scattering as well as attenuation of the outgoing beam due to scattering. 



(5) The average reverberation intensity at the receiver is equal to the sum 

 of the average intensities due to individual scatterers. This is a "randomizing" 

 assumption to average out interference effects. 



(6) The backward-scattering coefficient, m, is independent of the direction of 

 incidence of the beam. 



In general, one must solve or invert equation (21) to determine volume 

 scattering coefficients from measured reverberation intensities. Reverberation 

 of a sinusoidal wave-train radiated from a "searchlight" or piston-type trans- 

 ducer (i.e. a directional source having one radiation lobe, the main lobe, much 

 stronger than all others) and received by the same transducer was analyzed by 



