[v|+9'(y)]G,(y;5,0) = iiy*o, 

 G(x,0)=Eg,(x;x,0), 



(48) 



again with the same boundary condition. This function, 

 also, will vanish except when y is near jth unperturbed 

 ray. 



We may now directly derive the analog of Eq. (25) by 

 computing the quantity log/),(y; x)/G((y; x, 0) in pertur- 

 bation theory, and using Eq. (43). We find, setting y 

 = x, that 



/,,(x) = G,(x;x,0)exp(^^^^^-^|rf^y 



yG{x,y)V{y)G,{v;x,0)\ . (49) 



Finally, we note that the presence of the function 

 G, (y; x, 0) in Eq. (49) will restrict the integral over d^y 

 to a region surrounding the I'th ray from to x; since 

 Inside this ray tube G = G,, we can write 



p,{x) = G,(x,Q)exp ( } A 



\0(iX, U) Jithri 



yG{x,y)V{y)G{y,0)\ . (50) 



We have here replaced G,{k; x, 0) simply by G,(x, 0). 

 Equation (50) is evidently the generalization of the Rytov 

 formula (25) to the situation of an inhomogeneous back- 

 ground and many rays. The expression clearly fails if 

 the range is so large that the ray tubes overlap; other- 

 wise the validity conditions are the same as those in the 

 homogeneous background case. 



We shall now use Eq. (50) to calculate the various 

 averages of interest for the contribution of a single ray 

 tube to the pressure in the presence of the sound chan- 

 nel. We shall, for simplicity, drop the index i, though 

 we should keep in mind that when there are several un- 

 perturbed ray paths their contributions are to be added 

 to obtain the total pressure. Our interest, then, will 



be in the statistical fluctuations of the contributions of 

 a single ray, or rather a single ray tube. 



As In the homogeneous case, we define 

 -^x)=^^^J d'yG{K,y)V(y)G{y,0), 



(51) 



and we wish to compute ()^) and ( 1XI^>. We recall that 

 assuming geometrical optics to be a valid approximation 

 for the nonf luctuating background permits us to write 

 the Green's function 



K{x,y)exp[iqS(x.,y)] , 

 where 



q'S(x,y)=f ds9[x(s)] , 

 and where the normalization factor is 



(52) 



(53) 



Here i refers to directions perpendicular to the ray. 

 [An excellent approximation to this, for our purposes, 

 is to write simply A'(x, y) = (4jrlx- y I )"', as in the homo- 

 geneous ocean case; we need to be careful about devia- 

 tions from homogeneity only in the phase.] In Eq. (53) 

 the line integral is along the ray of interest joining the 

 points X and y. 



To repeat our earlier calculations requires us to in- 

 troduce the correlation function piy^, y^). In the homo- 

 geneous case, this quantity depended only on the separa- 

 tion y=yi-y2. Now, however, because of the back- 

 ground inhomogeneities, it will also depend on Y 

 = 2(yi + y2) (actually it will depend only on the mean depth 

 |(2i + Z2) because the inhomogeneities depend only on 

 depth). Thus we must now define the correlation func- 

 tion by 



4q'p{y, Y)^(V{y,)V{y^)) . 



As before, let us look first at ( I XI ^). We have 



( I X(x) I ^ ) . 4g* { d ' Y ( ^^^''^(^ y ' °y ( rf'yp(y,Y)exp{i9[S(x,Y+iy)-S(J,Y-|y) + S(Y+|y,0)-S(Y-|y,0)]}, (55) 



and we must keep in mind that we are to integrate only 

 over the ray tube surrounding the unperturbed ray of in- 

 terest. In the homogeneous background case we ex- 

 panded the exponent in powers of y, because p(y) van- 

 ished for large lyl. We may do the same here. Thus 



(|X(x)|^> = 49*J 



d'Y ^^^ 



Y)A'(y, 0)f 



K{x, 0) 



rf'yp(y, Y) 



Xexp{iq{y.Vy[S(x, Y) + S(Y, 0)]}) . (56) 



The integral on d'Y is again to be evaluated by station- 

 ary phase. The stationary phase path is evidently the 

 unperturbed ray joining the origin to the observation 

 point X. Hence we may write 



(\X(x}\') = ^^\s^ d'^As)p{^As),Y(s)) , (57) 



in complete parallel to the homogeneous case. Here 

 the line integral on ds is along the unperturbed ray, 

 ki(s) refers to the component of k perpendicular to the 

 ray at s, Y(s) is a point on the ray at s, and 



p(E, Y)sjd'yexp(-!S.y)p(y, Y) 

 Next we turn to (Jt^ ): 



<X(x)^) = 4,«J .'Y^l^f^'J d'ypilY) 



yexp{iq[S{x,Y + jy) + S(x,Y -iy) 



(58) 



218 



