provided that x^ - Xi » y^ - Vi, Zj 

 matrix A,/ has the form 



Therefore the 



(68) 



at an intermediate point x along the ray joining source 

 to receiver, where the 2x2 matrix A,j now is defined in 

 the vertical xz plane only. 



Au 



, /ta 



has the form 

 -tanSX 

 1 )' 



The matrix A,, 



■tan^e 



-tan 9 



it has one zero eigenvalue, associated with an eigen- 

 vector parallel to the ray, and one nonzero eigenvalue 

 j4i with eigenvector ki = {- k,ta.n9, k^) perpendicular to 

 the ray. The quantity kx,ki,j{A''^),i is just kl/A[. Since 

 A' = (l+tan^eM^., we have kl/A!, = kl/A',, = kl/A',, 

 = k^kg/Al^. Thus computing any matrix element of An 

 is sufficient to allow us to obtain the quantity we need. 

 At this point we shall drop the primes. 



There are a few regimes where Al\ may be obtained 

 without specific assumptions regarding the form of the 

 sound channel. 



First, for very deep rays, the sound channel varies 

 nearly linearly with depth, and the rays are nearly arcs 

 of circles. For this case, to first order in the radius 

 of curvature, the quantity A,, has the same value as in 

 a homogeneous ocean: 



a;\ = xiR - x)/R . 



Second, for near axis rays, the sound channel is 

 nearly a parabolic function of depth, so that the rays 

 are nearly sinusoidal. Then we find 



A]\ = (1/K) sinKxsinKiR - x)/sinKR , 



where 2ti/K is the wavelength of the sinusoidal ray. 

 Thus Ti/K^R, the range of an axis loop. Note that this 

 approximation to A'^^ becomes infinite when the receiver 

 is located at axis crossings of the ray. These points 

 are caustics for sinusoidal rays. Caustics are, in gen- 

 eral, points at which A'^^ diverges, that is, where y4„ 

 = 0. When this occurs, the matrix j4,y takes the simple 

 form 



/o 



A„=\ R/x(R-x) 



\0 0^ 



so that the passage from Eq. (60) to Eq. (62) becomes 

 altered, because the second-order term in the trans- 



verse derivative of the optical path no longer dominates. 

 To calculate {^) correctly in this region, we would 

 have to keep third-order derivatives of S(x, Y) + S(Y, 0) 

 fcf. Eqs. (59)-(61)]. As we shall see in Sec. VTO, the 

 effect of caustics on our theoretical predictions is to 

 introduce false narrow spikes in (A"^) at the caustic 

 positions. Presumably, there is in fact some unusual 

 structure at these points due to the different behavior 

 of the integral in Eq. (59), but our theory, to the level 

 to which we have carried it, cannot correctly describe 

 this structure. 



A third regime in which we can obtain A'^, without 

 knowledge of the details of the sound channel is that of 

 very long ranges, in which the rays contain a large num- 

 ber of loops. In this case, the optical path length Sj in 

 the vertical plane from the origin to a point (r, z), plus 

 the optical path length S^ from (x, z) to the receiver at 

 (i?, 0), can be written 



S= Si((*, z), (0, 0)) + S2((iJ, 0), (r, z)) 



=«iS,*'(i?r)+^Si(r, z)+w2sr(fir)+'^s2(x, z) , (69) 



where «i 2 are the number of double loops in the first 

 (second) path, S['2 '^ the optical path length of one 

 double loop, Rl'^ is the range of one double loop, and 

 ASi 2(^1 z) is the remaining path length from the end of 

 the last double loop to the point (x, z). Evidently 



X = niR{' + A^j , R- x = WaiJJ +^X2 , 



with Axi,2 «x, R- X and ASi 2 ^^ SJ'2. Thus we may ap- 

 proximately write 



S = niS{'{x/ni) + n2Sf{R- x/n^) . (70) 



Therefore 



8'S| /I l\d^S'-(Rn ,„.. 



8? I uaper.„r.ed r„ " V'l ^ ^J diR-f ' 



Since tii + n^ = R/R*', and since yii = x/R''; n2 = (R- x)/R'', 

 we have 



A„ = [R/x(R-x)]5 , 6=R*'d^S*yd(R*'f , (72) 



and therefore 



Al\ = (tan'e/d) x{R- x)/R . 



Other than in these cases, A]\ depends on the sound 

 channel, and we shall defer further discussion of it to 

 Sec. IV. 



Let us now return to Eq. (62). Once we have obtained 



(62) as 

 1 



JT.„„ 



fe„(s)*,,(s)^-'(Y(s))u=i(^^^^^ + f-) . 



(73) 



Hence, using the dispersion relations for horizontal and 

 vertical components of wave numbers, we find 



(}^) = 2qh-\B\ dxsec'eEf'^" d^(u,'-wir"F{u>,j;z)expl'-(^\ (^ 



'W 



^ A 



')] 



(74) 



Geometrical optics is valid when the exponent in this 

 equation is much less than one; this is the analogue 

 of the Fresnel condition in the channeled ocean. 



Except when the receiver is in the vicinity of a caus- 

 tic, it is in general the case that the kl term in the ex- 

 ponent, which is associated with horizontal spreading. 



220 



