II-3 



Consider now an ensemble of very many scatterers- -distinct scat- 

 terers such as air bubbles or indistinct scatterers such as thermal patches. 

 Suppose that the i^" scatterer is characterized by a scattering functional Lj. 

 This scattering functional describes how the scattered field due to the ith scat- 

 ter may be derived from the total field in the vicinity of the i'^^ scatterer. A 

 field cpinc incident on this ensemble of scatterers causes a total field cp which 

 has a scattered component consisting of the sum of the individual scattered 

 fields: 



cp = E L. {cpj 

 sc 1 1 < J 



The total field, therefore, satisfies the functional equation 



cp = cp. + (E L ) {cp} (II-2) 



inc i 1 ^ J 



Suppose we proceeded to solve this equation by the same successive approxima- 

 tion scheme introduced for the single scatterer. Choosing the incident field 

 again as the zeroth order approximation, we obtain a first approximation of the 

 total field 



)(i) = cp + (S L) (cp "l 

 inc i i "^ inc J 



The first order correction term is therefore just the sum of the first order scat- 

 tered fields from the individual scatterers. This approximation is called, for 

 self-evident reasons, the single scattering approximation. If we proceed to the 

 second order approximation, we find a second order correction term whose 

 physical meaning is that of the sum of all ways of scattering the incident field 

 twice: 



cp(2) = cp. +(EL.){cp}. +(.EL.L.) {cp. } 

 mc i 1 L Jmc i,j i J *• mcJ 



In other words, in addition to the direct scattering from each individual scatterer, 

 the second order approximation also takes into account scattering from all pairs 

 of scatterers. The next higher approximation would contain an additional term 

 from all triplets of scatterers, etc. Usually the single scattering approximation 

 suffices. Sometimes we can demonstrate the precise conditions under which the 



artbur B.littlcJnt. 



S-7001-0307 



