IV-3 



By collecting terms of the same order in e in (IV-7), we find a hierarchy of 

 equations for the successively higher- order terms in the power series for the 

 pressure: 



V3 p(0) . 1 4p<Z) 



c 

 o 



5p av-8a) 



o o 



The first of these equations has as its solution the pressure distribution in the 

 homogeneous medium. The equation for the first order correction is also a wave 

 equation, but with a nonzero inhomogeneous term. This term shows how, to 

 first order, the inhomogeneities of the medium interact with the unperturbed 

 pressure distribution and thus act as sources of secondary waves. 



In all the problems of interest, we shall be concerned with a pressure 

 distribution that has harmonic time dependence: 



-iuut 

 p(x, t) = p(x)e (IV-9) 



Also, we shall usually be concerned with either a plane or spherical incoming wave: 



p (x) - A e (plane wave) 



C ) ^^ 



p (x) = A - (spherical wave) 



(IV- 10) 



(Note that x = | x j and k = | k | 



uu 

 c 







In either case, the equation governing the first order correction term to the pres- 

 sure satisfies a Helmholtz equation with a distributed source function:* 



v^ p(') + k^ p(') = - 2^k= p(o) (IV-11) 



''Since the temporal variations in the index of refraction occur very slowly compared 

 to the time required for the acoustic wave to pass by a typical inhomogeneity, we 

 may regard [i (x, t) as constant during the passage time. 



arthur H.littleJnir. 



S-7001-0307 



