X « qlr , 



which we recognize as the Fresnel condition under which 

 the geometrical optics approximation tor X{\) itself was 

 valid in the first place. 



Thus Eq. (39) constitutes an improvement over geo- 

 metrical optics, while Eq. (35) coincides with geometri- 

 cal optics. Conversely, geometrical optics for < 1X1^) 

 is valid out to a very large range, while geometrical 

 optics for <A^> is valid only within the range x< qL^. 



It is of interest to study Eq. (39) in the limit of very 

 long range. As r- °o, the integral over ds can be ap- 

 proximately evaluated, and we find' 



{X\x) ) = [iq''p(Q)/2T,] (y + \ogAqx- |f7r) ~ / log.r , (40) 



where r = 0. 577. . . is Euler's constant, while for small 

 X, satisfying the Fresnel condition, we have the geome- 

 tric optics limit 



{X\x))^-^^\d'krp(0,K)(x-'-^^^--^ . (41) 



Between these limits Eq. (39) provides a smooth transi- 

 tion. In contrast to Eq. (40) we have from Eq. (35) the 

 result 



<|x|^>~r 



for both large and small x. 



II. INHOMOGENEOUS OCEAN 



Now let us turn to the effects of the sound channel. 

 That is, we must replace the nonf luctuating sound speed 

 C in the homogeneous case by a (specified) function of 

 position C(x). 



The wave equation for the pressure, which is our 

 starting point, now becomes altered from Eq. (10) to 

 the equation 



[V^ + 9'(x)]/)(x) = K(x)/>(x), 9(x)=(7/C(x), 



still with the same boundary condition Eq. (12). 



(42) 



We must first study the nonf luctuating part of the 

 problem, to evaluate the Green's function in the pres- 

 ence of the sound channel. This satisfies 



[v2 + 92(x)]G(x,y) = 6'(x-y). 



(43) 



Note that G is no longer a function only of x -y as it 

 was in the homogeneous case. We shall assume that 

 geometrical optics provides a good approximation to 

 the nonfluctuating sound channel problem. This means 

 that we can represent G(x, y) as a sum of contributions 

 from each ray joining x and y. To be specific, we may 

 write 



n(x,?) 



G(x,y)= E G,(x,y)^ 



(44) 



where «(x, y) is the number of rays and G, is the contri- 

 bution of the zth ray. We have, in particular, for rays 

 joining the origin and x, 



G,(x,0) = A-,(x,0)expM dsq[%(s)\\, ! = 1, . . . , «(x) , 



where ds is an element of path length along the ray, 

 Xj(s) is the rth ray joining to x, and A', is a normaliza- 

 tion factor. 



Now when the fluctuations are turned on, the signals 

 traveling on each of the rays joining the origin to the 

 point of observation x are subject to small-angle scat- 

 terings by the perturbing potential V(x). The signals 

 are thus deflected slightly from the undisturbed rays by 

 each interaction with V. The repeated action of V thus 

 produces, on each ray, a sort of random walk of the 

 signal away from the original ray. When we average 

 over an ensemble of perturbations V, the disturbed sig- 

 nals will fill up a tube surrounding the undisturbed ray. 

 Provided that these tubes around each of the original 

 rays do not overlap, the received pressure will be a 

 sum of contributions from each ray tube. (Such tubes 

 exist, of course, in the homogeneous case as well, but 

 there they never overlap. ) 



We may estimate the radius of a ray tube as follows. 

 The mean free path d between interactions of the signal 

 traveling along a given ray with the perturbing potential 

 V is of the order of hC/kC. Hence over a range x the 

 number of scatterings is w = x/d. The average deflec- 

 tion angle due to each scattering is of order \lkLy ver- 

 tically and \/kL„ horizontally, where Ly and L„ are the 

 vertical and horizontal correlation lengths of the sound 

 speed fluctuations. Since the process is a random 

 walk, the net dispacement due to n collisions (when n is 

 large) is proportional to •fyx, and hence the vertical and 

 horizontal extents of the tube are, roughly. 



Yy- 



[q) qLy\6c) ' ''" \q ) qL„\bC) 



Let us assume that the vertical extent of the tubes is 

 small enough so that the tubes remain distinct. Then 

 the pressure at x is the sum of contributions from each 

 tube 



pa) 



n(x) 

 E/'l(x); 



(46) 



(45) 



where «(x) is the number of unperturbed rays joining the 

 source to the point x. We shall be interested in ^|(x). 



We note that />, (x) is the pressure that would be re- 

 ceived at X if the source were not isotropic, but rather 

 emitted all its energy in the direction of the rth unper- 

 turbed ray. Thus /),(x) must satisfy the wave equation 

 (42) but with an anisotropic boundary condition which it- 

 self depends on x. To make this more precise, let us 

 define /), (y; x) to be the pressure at y from a source at 

 the origin which emits only within a small solid angle, ' 

 around the direction of the I'th perturbed ray joining the 

 origin to x. Thus />, (x) = />, (x; x), and furthermore 

 /), (y; x) vanishes unless y is inside the ith ray tube. 

 Then 



[v|+9'(y)]p,(y; 5) = ny)/',(y;x), ! = i,... ,«(x) . (47) 



In analogy with this definition of />, (y;x), we may also 

 define an "unperturbed" Green's function G,(y; x, 0), 

 2 = 1,..., n(x), to satisfy 



217 



