B-4 



Consider now the operator v + k^ and note that (V^ + kg ) E(r, r') = 4 tt 6 (r-r'). 

 Applying this operator to both sides of (B-9) gives 



v^ < p(r) > + k^ < p(r) > = - 4 n G(r) < p(r) > (B-10) 



With k^(r) = k^ + 4TTG(r), we see that v^ < p(r) > + k^(r) <p(r)> = 0, and so <p(r)> 

 satisfies the wave equation in a "continuous medium" in which the velocity of 

 propagation depends upon the scattering coefficients and density of the scatterers 

 and is, in general, a function of position. This is an important result, which gives 

 the same characterization for propagation as obtained by considering the complex 

 compressibility of a bubbly medium. 



The problem of finding < p(r) > has thus been transformed to solving a 

 boundary value problem for the wave equation, where the boundary conditions 

 depend on G(r). If G(r) is everywhere continuous and approaches a constant value 

 or zero at infinity, then the boundary conditions are that < p(r) > - Pq (£) be every- 

 where continuous, have a continuous gradient, and at infinity, represent outward 

 traveling waves. It should be noted that, in principle, it is possible to solve the 

 integral equation (B-9) directly by using the Liouville- Neumann method of successive 

 approximation. Repeatedly substituting for < p(r) > we have 



<p(r)> = p^(r) +y^ p^(r), where Pj^(r) = rg(r')p^_^(r') E(r, r')dr' (B-11) 



m=l V 



This also gives the desired solution, if the series converges uniformly. 



In order to interpret these results, it is useful to consider the case of a 

 single scatterer (the following discussion does not rely on any approximations). In 

 this case the wave function becomes 



p(r) = p^(r) + giP^(ri) E(r,ri) (B-12) 



which is the sum of the incident wave and the spherical scattered wave. Consider 

 a plane incident wave 



p (r) = Ae^4'- • (B-13) 



o — 



Then 



P^c^L) = gi Po(li ) E (r , ri ) (B - 1 4) 



represents the scattered wave . 



Arthur Sl.ltittle.Hnt. 



S-7001-0307 



