IV -2 



Note that we are using the vector notation x = (xi , Xg , X3 ) to indicate 

 position coordinates of a point. We shall also have occasion to denote the length 

 of the vector x by I x | - x (see Appendix C - Notation). The variation in the 

 index of refraction is supposed to be extremely small: 



I u| << 1 (IV-3) 



Typical values of the variations in the index of refraction for the ocean are of the 

 order of magnitude of 10' or less. 



The problems of practical interest are of two kinds: 



(a) An incoming plane, spherical, or cylindrical wave hits a region of 

 inhomogeneities and is scattered. 



(b) A source of acoustic waves is situated inside an inhomogeneous 

 region, and the waves emitted by the source are distorted by the inhomogeneities. 



For sufficiently small regions of space, these problems can be attacked 

 by a formal perturbation theory using the magnitude of the variations in the index 

 of refraction as the parameter of smallness. We may write the index of refraction 

 as unity plus a very small multiple of a function of space and time: 



n(x, t) = 1 + u (x, t) = 1 + e |i (x, t) (IV-4) 



where 



e << 1 , 1 n| ~1 (IV-5) 



We can now consider a family of problems depending on the parameter e . For 

 z - 0, the solution to the problem is just the corresponding solution for the homo- 

 geneous medium. If the pressure distribution is an analytic function of e near e =0, 

 we may expand the pressure in a formal power series in the parameter of smallness: 



p = p(°^ + ep(') + e^p('^ ... (IV-6) 



In this case, the wave equation (IV- 1) becomes: 



'o 



v^ (p^ '+ep^ ' + ..) = (1 + e ^)2 -^ _ (pv-^ + epv ' + . .) (iv-7) 



artbur Sl.ltittlcHnir. 



S-7001-0307 



