IV -7 



For small perturbations, we may therefore identify the correction term of (IV-6) 

 with the corresponding term of (IV- 19), which yields: 



e p (^) = ip 1^1 (IV-20) 



When the perturbations are indeed small, the approximation theory about to be 

 developed should go over into the small perturbation theory according to the 

 relation (IV-20). 



From (IV- 18) we conclude that: 



\lti = 1^ - i^ = (S - S ) - i ^n 4- (IV-21) 



O O A 



O 



We may, therefore, identify the changes in phase and logarithm, i.e., amplitude 

 due to the inhomogeneities of the medium: 



AS - (S - S ) = Re i|;i (IV-22a) 



AtnA = ^n— = Imiii (IV- 22b) 



o 



The wave in the inhomogeneous medium must satisfy the wave equation (IV- 1), and 

 if we introduce harmonic time dependence (IV- 9) and the definition of the index of 

 refraction (IV- 2), we obtain the wave equation for the pressure: 



(V^ + n^k^ ) p = (IV- 23) 



The complex phase function must, therefore, satisfy an equation which is obtained 

 by substituting (IV- 17a) in (IV- 23). After a little manipulation, we find that i|; is 

 governed by: 



i V^ lit - (V i|;)2 + n^k2 = (IV-24) 



Since p^ is the solution of the wave equation with an index of refraction of unity, 

 the complex phase function i|(q must satisfy equation (IV-24) with n = 1: 



i v2 ^ - (V il( )2 + k^ = (IV-25) 



o o 



If we substitute (IV-18) in (IV-24) and make use of (IV-25), we obtain an equation for 

 the correction to the complex phase function i|ii: 



iV2^^ - (V ,1,1)2 - 2(V,|, -Vi^i) + (n^ - i)k^ = (IV-26) 



o 



:artliur B.llittleJnf. 



S-7001-0307 



