FIG. 6. Comparison of cal- 

 culated (dotted lines) and 

 "experimental" (solid lines) 

 rms phase fluctuations at 

 100 Hz as function of range 

 for six rays with inclinations 

 on the axis ranging between 

 ±9°. Lines connect calcu- 

 lated values, with no attempt 

 at interpolation. 



rms phase is nearly a step function of range, reflecting 

 the fact that the major contribution to the integral in 

 Eq. (96) comes when the rays cross an apex, and that 

 there is little contribution while the ray is deep. The 

 near axis rays (±1°), on the other hand, vary much 

 more smoothly with range (nearly like fl"^). 



B. Intensity fluctuations 



Table III shows the rms intensity fluctuations for the 

 same six rays, at various ranges. Since the numerical 

 experiment makes use of a vertical beam former rather 

 than a single hydrophone to select different rays, the 

 theoretical calculations described in Sec. VII must be 

 somewhat modified. There intensity fluctuations were 

 calculated for a fixed receiver position; here we must 

 calculate fluctuations for a fixed receiver angle but hav- 

 ing a variable vertical position. This amounts to re- 

 placing the quantity A]\ in Eq. (120) by a different geo- 



metrical factor B\\, 

 with respect to z of the optical path length from the 

 transmitter to a receiver located at a fixed range and 

 seeing a fixed vertical angle, rather than one located at 

 a fixed range and height. This quantity has been eval- 

 uated numerically, and then Eq. (120) has been used, 

 in order to obtain the theoretical values shown in the 

 table. 



The quantity £][ can be evaluated analytically for lin- 

 ear and quadratic sound channels (with circular and 

 sinusoidal rays) which approximate the real sound chan- 

 nel for deep and near-axial rays, respectively. We 

 find 



„-! jj-i 1 sinKxcosK(R- x) 

 B,, - X , B„ 



K 



cosKR 



(122) 



for deep and near-axial rays, respectively, where Zir/K 

 is the wavelength of the sinusoidal rays, so that n/K is 

 the range of one loop. We remind the reader that 



1 x{R-x) 



,., 1 sinKxs\nK(R- x) 



A., = - 



K 



sinXR 



are the corresponding quantities for a fixed-point re- 

 ceiver. Thus, for near-axial rays, iJj^ becomes in- 

 finite when the receiver is located at the turning point 

 of the rays. It is here that all rays are parallel; this 

 is the analogue of a caustic for a beam former receiver. 

 As we have already remarked in Sec. HI, when the re- 

 ceiver is placed near a caustic, our approximate expres- 

 sions [Eq. (75)] fail; to correct it would require keep- 

 ing the effect of horizontal spreading in Eq. (74). We 

 are therefore able to compare our calculated values 

 with the "experimental" ones only if we avoid placing 

 the receiver near a (beam former type of) caustic. For 

 near-axis rays, where we can use Eq. (122), these caus- 

 tics occur at ranges fl= (« + 5)jr//f; since iT/if=fl = 20km 

 for near-axis rays, there are caustics of ranges of 10, 

 30, 50, ... km. For off-axis rays, the positions of caus- 

 tics must be determined numerically. The entries in 

 Table ID are made by avoiding these. 



The agreement between theory and "experiment" is 



TABLE III. RMS intensity fluctuations in dB. The upper 

 (lower) number in each entry is the theoretical ("experimental") 

 value. 



228 



