II-8 



However, the velocity of sound c is now a space - time dependent quantity which 

 differs slightly from its average value Cq 



c 



— = (1 + V (x,t) ) 

 c — 



Here v is the fluctuating part of the index of refraction, Cq/c, and is much less 

 than unity. The usual approximation scheme for weak inhomogeneities is to re- 

 gard V as very small, and to ignore all effects which are quadratic or higher 

 order in v. 



The time fluctuations of the index of refraction are very slow compared 

 to the passage time of sound waves . From the point of view of propagating sound 

 waves, therefore, the medium has a spatial structure which is stationary. In the 

 course of time this micro- structure changes slowly, causing corresponding slow 

 changes in the amplitude and phase of the transmitted sound. In a theoretical 

 analysis, therefore, we may confine ourselves to the problem of sound propagation 

 through a stationary medium whose micro- structure is any one of the many de- 

 tailed micro- structures which occur in sequence in the course of time. The ex- 

 pected values to be measured in an experiment are then obtained by taking an aver- 

 age over the ensemble of possible micro- structures. 



The sound field consists again of an incident and a scattered wave 



cp = cp + cp 



inc sc 



The scattered wave is now a functional of the detailed micro- structure v; in fact, 

 the functional must be linear if we keep only first order terms: 



cp = cp I v(x)[ 

 sc sc ^ - J 



We desire to determine the mean square fluctuations of the amplitude and phase 

 of cp. These correspond to the variances of the slow fluctuations of amplitude 

 and phase to be expected in the received signal due to slow changes in the structure 

 of the medium. We therefore wish to evaluate averages over the ensemble of pos- 

 sible micro- structures which are of the form: 



cp 

 sc 



<|cp (v} h N ,, , , (II-3) 



sc V J / ensemble of V s 



Arthur m.IlittleJnt. 



S-7001-0307 



