For small /3, ReGj- Fi, so that 

 (4>')~((5C/C%){f')q'BRF,; 



km 



20 



(118) 



this is simply the conventional geometrical optics ex- 

 pression for phase fluctuations. Intensity fluctuations, 

 however, depend on the difference between Fj and ReGi, 



Fi-ReGi=X—^ = f rftsec=e«Vi(A)(l-«-(/3,;,)) 

 TT ai,„no " -T) 



(119) 

 and thus vanish in the small /3 limit. Indeed, for small 

 fi, using Eq. (110), we find 



(.2)_<0= 



-((€>^^r* 



xsec^enVi(A) 



(120) 



and this does not coincide with any geometrical optics 

 expression. 



It is also of interest to compute the spectra of phase 

 and intensity fluctuations. For this purpose, we return 

 to Eqs. (66) and (75), and to Eq. (91), but we do not now 

 carry out the integral over dw. For a given value of cti, 

 we integrate over a ray path, keeping in mind that there 

 is a complicated set of forbidden ray sections, depend- 

 ing on the value of oj relative to 0)^, , h ', and fi' (Fig. 5). 

 For very low frequencies the ray is too steep to permit 

 "stationary-phase interaction" with internal waves. 

 For the high frequencies ui may exceed n(z) along some 

 portions of the ray, and internal wave solutions do not 

 exist. The phase and intensity spectra are given by 





H{n-w)H{u 



R(l+Reg-(ft;,))] 

 ^L2{l-Re^(/3,;,))J 



(121) 



For the important range ojj^ «w< >V the entire integra- 

 tion path is permitted and the spectra vary as w"'. 



VIII. COMPARISON WITH NUMERICAL 

 EXPERIMENTS 



As a first application, and test, of the results we have 

 obtained we shall make a comparison with a set of 

 "numerical experiments." ^ These consist of numerical 

 solutions of the parabolic wave equation in the same 

 sound channel we have discussed here, and with a se- 

 quence of internal wave realizations from a two-dimen- 

 sional projection of the spectrum described in Sec. VI. '" 

 The "numerical experiments" use an acoustic fre- 

 quency of 100 Hz, and propagate sound up to ranges of 

 100 km; the remaining parameters are the "standard 

 ones" listed in Sees. IV- VI. In all cases the acoustic 

 transmitter is located on the sound axis, at a depth of 

 1000 m. The receiver consists of a vertical array of 

 hydrophones, 700 m long, centered on the ray in ques- 

 tion, which allows an angular resolution of l|°. 



3 

 4 



FIG. 5. Internal wave contributions toward frequency spectra 

 of acoustic phase and intensity come from "permitted" sec- 

 tions of the ray path (heavy lines). There are no internal 

 wave contributions to frequencies less than the inertial fre- 

 quency (/) and larger than the apex bouyancy frequency (V) be- 

 cause no such internal waves e.xist. The entire ray contributes 

 toward the central band in between (Mi)mi, (typically 10 u,„) 

 and the buoyancy frequency at the lower turning point (for deep 

 rays III does not exist). For lower frequencies, the upper 

 sections of the ray are too steep to permit "stationary -phase 

 interaction" (II), and high frequencies exceed the buoyancy 

 frequency of the deep ray section (IV). 



A. Phase fluctuations 



Solid lines in Fig. 6 show the results of the "numeri- 

 cal experiment" for 128 realizations (to which one may 

 assign a statistical error of perhaps ±20°). The dotted 

 lines are the predictions of the theory outlined in Sec. 

 Vn, and specifically of Eq. (118). Evidently the agree- 

 ment is satisfactory. Overall magnitudes differ be- 

 tween theory and experiment by about (20-30)% (except 

 for the - 1° ray) and the general shapes coincide as well. 

 For the steep rays (±9° and to a lesser extent ±5°) the 



227 



