IV -35 



P. SIGNAL FLUCTUATION AND CORRELATIQN RESULTING FROM THE 

 OCEAN MICRO- STRUCTURE 



We now have at our disposal the tools required to examine the statisti- 

 cal properties of the pressure fluctuations of a sound wave propagating through an 

 inhomogeneous medium. As a first step, we consider the distribution of scattered 

 power from a relatively small region of inhomogeneities at a distance large com- 

 pared to the size of the region. The geometry of the situation has already been 

 shown in Figure IV-2 for the individual scatterer, and we need merely replace the 

 individual scatterer by a region of scatterers to make this figure applicable to the 

 present discussion. We found in our analysis of the geometry of Figure IV-2 an 

 expression for the first order scattered pressure which was: 



pi(x)--^i^lf I d?e"'^^-^ u(?) (IV-45) 



f 



4 n X 



V 

 We recall the meaning of the vector d to be the difference between the unit vector 

 directed at the observer and the unit vector in the positive xi direction, i.e. , the 

 direction of propagation of the incident sound. The magnitude of the vector d was 

 shown earlier to be d = 2 sin 9/2, where 9 is the scattering angle separating the 

 direction of the observer from the direction of propagation. 



The volume V is to be regarded as large compared to any single inhomo- 

 geneity, but small compared to the distance between the region and the observer. 

 If we consider an ensemble of scattering volumes differing from each other in their 

 micro- structure but all characteristic of the micro- structures which might be en- 

 countered in sequence over a long period of time, we would like to know the statistical 

 properties of the scattered pressure pi averaged over such an ensemble. Since the 

 average of the refractive index < v (^)> over the ensemble must be zero, it is clear 

 that the average fluctuation of the pressure must also vanish ( < pi > = 0). We would 

 like to compute the variance of the fluctuations of the pressure, which is analogous 

 to computing the distribution of scattered power: 



< D. i ">- 



(2TT)' 



r^fd?(^)fdi(^)e-^^^^^^'^-^^'^^Wi(^))u(?(^))> (IV-74) 



V 



k^ [fc 



= k- ||d?(^)d?(^)e-i'^^<i''-^'^><u^>R (?^)-?(^)) 



(2t 



<u''>E 



u "— 



VV 



artbur B.littleJnt. 



S-7001-0307 



