II- 2 



functional results from a detailed model of the scatterer.* The total field must 

 therefore satisfy the functional equation: 



cp = cp. + L (cpj (II- 1) 



inc I. J 



It is usually not possible to find the exact solution to this equation, i.e., to ob- 

 tain that field cp which results when a given exciting field cPjnc ^^ scattered by a 

 known inhomogeneity characterized by L. Instead, a sequence of approximate 

 solutions is obtained by a simple iterative scheme. If the effect of inhomogene- 

 ities were negligible, it would be adequate to take '^{^c ^^ ^^^ total field. Sup- 

 pose therefore that we choose cpv°) - cp^j^^, as the initial approximation. We may 

 then hopefully improve this approximation by substituting in (I-l) and obtain a 

 first order approximation 



(^) = cp + L {cp 

 inc I- ii 



We may repeat this procedure any number of times, obtaining, in general, a k*-" 

 order approximation which is: 



cp(k) = (1 + L + L^ + ...+L^) {cp. 



The meaning of the different terms in this expression is the following. The first 

 order correction to the initial approximation (choosing the exciting field as the 

 total field) is the direct effect of every portion of the scatterer in scattering the 

 incident field. However, this first order scattered wave gets scattered repeatedly, 

 thus causing successively higher orders of scattering. In all cases with which we 

 are concerned, the first order scattering approximation is an adequate descrip- 

 tion of scattering from a single inhomogeneity, since we are only interested in 

 the far field. 



""It might appear more natural to regard the scattered field as a functional of the 

 incident field rather than of the total field, e.g. , cpg^. = T {"Pji^cJ " ^ practice, 

 however, one usually finds the functional L more readily than the functional T. 

 Furthermore, knowledge of T would not permit as easy a formulation of the 

 problem of simultaneous scattering from many scatterers as is obtained above 

 by the use of the functional L. 



artbur a.littleJnc. 



S-7001-0307 



