SCATTERING OF SOUND 



251 



as scattered. Such changes in beam direction occur, 

 for example, when the beam is refracted by a tem- 

 perature gradient which is a function of water depth 

 only. Similarly, reflection from an infinitely smooth 

 plane surface sharply changes the direction of the 

 beam; but since all the energy theoretically remains 

 in the beam the process is not termed scattering. 



However, there are inhomogeneities of small size, 

 such as air bubbles, or small irregularities in the 

 ocean surface, which cause energy to be "detached" 

 from the main sound beam, that is, to travel in a dif- 

 ferent direction from that of the main beam. This 

 detached energy, which differs in direction from the 

 main beam and which results from local inhomogenei- 

 ties in the ocean or bounding surfaces, is called 

 scattered energy. It is apparent from this discussion 

 that the distinction between scattered energy and 

 nonscattered energy is not always too clear; for ex- 

 ample, bottom reverberation received from under- 

 water cliffs might more properly be called reflected 

 energy rather than scattered energy. This possible 

 confusion in nomenclature is of no immediate con- 

 cern. The important point is that the existence of 

 reverberation is predicted by theory from the known 

 inhomogeneity of the ocean. 



The magnitude of the reverberation reaching the 

 water near the receiver is calculable, in principle, by 

 solving some differential equation which, with ap- 

 propriate boundary conditions, takes into account 

 the inhomogeneity of the ocean. Since temperature 

 gradients and density gradients are small in the body 

 of the sea, the differential equation would be the 

 wave equation 



(1) 



where p is the sound pressure, and c is the velocity 

 of sound at the time t at the point whose coordinates 

 are (x,y,z) ; this equation was derived and its applica- 

 tion discussed in Chapters 2 and 3. The presence of 

 solid particles and air bubbles, and the nature of the 

 roughnesses in the ocean surface and bottom, are 

 described by the boundary conditions; these condi- 

 tions give the positions at each instant of all the 

 surfaces at which the density and sound velocity 

 change discontinuously and the amounts of these 

 changes. 



Because of the complexity of the ocean, neither the 

 function c(x,y,z,t) nor the boundary conditions are 

 precisely known. The bathythermograph gives the 

 broad outlines of the temperature distribution. How- 



ever, the locations and magnitudes of small local 

 temperature gradients cannot be determined with the 

 bathythermograph, although such "thermal micro- 

 structure" is known to exist.' Furthermore, the posi- 

 tions of bubbles, solid particles, waves, etc., change 

 continually and unpredictably. Even if the sound 

 velocity and the boundary conditions were specified 

 exactly, it would be an insuperable mathematical 

 problem to solve equation (1) rigorously for p. Thus, 

 theoretical formulas for reverberation cannot be de- 

 rived by solving equation (1) with boundary condi- 

 tions. 



Instead, we shall base our mathematical analysis 

 of reverberation on several simplifying assumptions. 

 Since a great deal is known about the general proper- 

 ties of solutions of equation (1), reasonable assump- 

 tions can be made about reverberation, even though 

 a complete solution of equation (1) cannot be ob- 

 tained. The principal assumptions which we shall 

 use are : 



1. Reverberation is scattered sound. 



2. Scattering from an individual scatterer begins 

 the instant sound energy begins to arrive at the 

 scatterer and ceases at the instant sound energy 

 ceases to arrive at the scatterer. 



3. Multiple scattering (rescattering of scattered 

 sound) has a negligible effect on the intensity of the 

 received reverberation. In other words, all but a 

 negligible portion of reverberation is made up of 

 sound which has been scattered only once. 



4. The intensity of the sound scattered backward 

 from a small volume element dF is directly propor- 

 tional to each of the three following quantities: the 

 volume occupied by dV, the intensity of the incident 

 sound, and a "backward scattering coefficient" desig- 

 nated by m, which depends only on the properties of 

 the ocean in the neighborhood of d V. 



5. The average reverberation intensity, which is a 

 function of the time elapsed since the emission of the 

 ping, is the sum of the average intensities received 

 from the individual scatterers in the ocean. To ex- 

 press this assumption in mathematical form, let 

 g{t)dV represent the average intensity, t seconds 

 after the emission of a ping, of the reverberation re- 

 sulting from scattering in the volume element dV 

 only. Then the average intensity of the reverbera- 

 tion received from the entire ocean, at the time in- 

 stant t, is given by 



G{t) = jg{t)dV (2) 



where the integral is taken over the entire ocean. It 



