B-1 



APPENDIX B 



THE MULTIPLE SCATTERING OF WAVES 



The following development, due to Foldy* deals with scattering of an 

 incident wave by a random distribution of scatterers. The treatment is in terms 

 of general wave theory. The scatterers may be arbitrary in number and char- 

 acter, except that the scattered wave is assumed to be spherical and with an 

 amplitude proportional to that of the wave exciting the scatterer. The constant 

 of proportionality is assumed to be specified completely by the frequency and a 

 single parameter of the scatterer, denoted by s . 



If we have a collection of N scatterers and are given for each its 



position, £1 rjsj, and its scattering properties as specified by si S]vj, 



then we shall say that we have a particular configuration of the scatterers. The 

 ensemble of configurations in which we are interested may be described by a 



probability distribution function P(ri r^, si , . . . sjsj) so that 



P(ri , . . . , rjsj, si , . . . , Sfs^)dri . . . drj^j dsi . . . dsj^ represents the probability of 

 finding the scatterers in a configuration in which the first scatterer lies in an 

 element of volume dri about the point ri and has a scattering parameter lying 

 between Si and Si + dsi, and so on for the other scatterers. The average of a 

 physical quantity over the ensemble of configurations is called a configurational 

 average . Thus, for a function f(ri, . . .,£2^, Si, . . .s^). the configurational average 

 is 



<f> = I . . . I f(ri , . . rN, Si , . . . sjsj) P(ri , • . , rj^, si , . . . SN)dri . . drj,jdsi . . dsj^ 

 yv V (B-1) 



Subscripts will be used to indicate when the integration over one or more of the 

 scatterers is to be omitted; thus < f> j indicates that the integration over rj and 

 s^ is to be omitted. In the following we will also assume that the scatterers in an 

 ensemble are independent of each other with respect to position and scattering 

 parameter. In this case, P can be written as (1/N) n(ri, Si)n(r3, Sg). . .n(rjs^Sjsj), 

 where n(r, s)ds is the average number of scatterers per unit volume in the 

 neighborhood of the point r having scattering parameters laying between s and s + ds. 



We will tlien consider the steady state scattering of waves of a single 

 frequency w , so that the value of the scalar wave function at the point r and 

 time t can be represented asp(r)e"--^t. i^ the absence of scatterers, p(r) will 

 satisfy the wave equation v p + k p = 0, where kg = (i^/Cq) and Cq is the 

 wave velocity in the scatterer- free medium. The scatterers are assumed to behave 

 as point scatterers, scattering spherically symmetrical waves; thus in the neighbor- 

 hood of the j-th scatterer the wave function will behave like B-E(r, £;), where 



^L. L. Foldy (Ref.III-8). 



;arthur m.HittleJnf. 

 s-7001-0307 



