IV-10 



The relevant vectors are those shown in Figure IV- 1, and r is again the vector 



connecting the scattering element and the point of observation. We now obtain 



the correction to the complex phase by substituting the above expression for W 



in (IV- 33): 



2v2. r ikr p (?) 



*i (x) =-^ d?- L'(?)^-^ (IV-36) 



4n J - r - Pq (^) 



V 



First of all, we observe that for small values of '^i, (IV-36) and (IV-12) do indeed 

 correspond according to the correspondence principle given in (IV-20). Second, we 

 note that the integral to be evaluated either in the small perturbation theory in 

 (IV-12) or in the "'smooth" perturbation theory leading to (IV-36) is essentially the 

 same. WTiat differs is the interpretation of the integral. 



In the small perturbation theory, the integral essentially represents the 

 first order correction of the pressure as a result of the inhomogeneity . In the 

 m-ore uniformly valid approximation, the integral corresponds, loosely speaking, 

 to the correction to the complex phase resulting from the inhomogeneity. Depending 

 on the situation, we shall avail ourselves of either interpretation. In much of the 

 later work of this chapter, however, we shall be concerned with integrals of this 

 type and therefore introduce the symbol pi to connote the integral: 



^ 2 /• ikr 



p, (x) - £ p (-)(x) = ip (x) ti (x) = - ^ d? - 'J(?) p (?) (IV- 37) 



— — o— — 4Tri— r — o — 



B. SCATTERING FROM AN ISOLATED INHOMOGENEITY 



The micro- structure of the ocean consists of a multitude of weak inhomo- 

 geneities of every size and shape. The statistical description of the inhomogeneities 

 encountered in the actual ocean medium will be studied in the next section. In the 

 present section we shall confine ourselves to the diffraction of a plane wave by a 

 single inhomogeneity (of the refractive index) in an otherwise entirely homogeneous 

 ocean. In particular, we wish to determine the far field of the scattered wave and to 

 gain some insight into the dependence of the far field on the characteristics of the 

 scattering inhomogeneities and the frequency of the incoming plane wave. 



To simplify the geometry of the problem, we shall locate the scattering inhomo- 

 geneity so as to include the origin of the coordinate system. The incoming plane wave 

 of unit amplitude is taken to propagate in the positive xi direction (see Figure IV-2). 



arthur B.Hittle.Ilnc. 



S-7001-0307 



