11-12 



In a distance L the incident sound is scattered by -r-^ such slabs. The scattered 



sound arriving at the observer from the different slabs is uncor related since its 

 phase depends on the detailed constitution of each slab. Therefore, the power 

 observed by the observer is the sum of the scattered powers from each of the 

 slabs.* It is common practice in underwater acoustics to deal with the square 

 root of the ratio of scattered power to incident power; this is called the coefficient 

 of variation. Its meaning is clearly that of the ratio of a typical scattered pres- 

 sure amplitude to the amplitude of the incident wave. According to the above con- 

 siderations, the coefficient of variation will be the product of the square root of 

 the number of slabs and the scattering strength from each individual slab (i.e. , 

 powers add): 



V = -^ -J^ |v| k^R^=lv| k^R^/^L'/^ (II-6) 



P 



Thus, we see that the scattered pressure a distance L inside the inhomogeneous 

 medium will be proportional to the square root of L. 



So much for the low frequency situation. At the other extreme, when 

 the frequency of the incoming sound is high, the scattering from an individual 

 scatterer will be very directional and we must therefore distinguish two cases 

 corresponding to an observer inside the focusing range and an observer in the 

 interference range. 



Let us consider first the behavior to be expected in the interference 

 range, i.e., L> > kR^. Since we are considering the high frequency range 

 (kR > > 1), the scattering of a single inhomogeneity is confined to a cone with 



half-angle --^. Therefore, an observer located a distance L inside the medium 

 kK 



observes the scattered fields from those inhomogeneities which are located in a 

 cone of half-angle — with its apex at the observer. (See Figure II-4.) We 



again slice this cone into disks of thickness 2R. A circular disk a distance L 



L ^ 

 from the observer will have a surface area proportional to (7-5-) , The number 



''The argument that, in the case of a numJ^er of uncorrelated scattered pressures, 

 the sum of the individual powers add, may be made more appealing to the sta- 

 tistically oriented by thinking of the pressure as a stochastic variable. The scat- 

 tered pressure is a stochastic variable with zero mean; the scattered power is 

 just the variance of the variable and hence the argument that the variance of a sum 

 of such uncorrelated variables has a variance which is the sum of the individual 

 variances becomes obvious. 



:artbur H.HittlcJnf. 



S-7001-0307 



