TABLE Vn. Calculations of (i)]} tor Bermuda. 5, are the 

 inclinations at the axial source of all possible rays to an axial 

 receiver at 1250-km range, consisting of /f upper loops of 

 range R* and R' lower loops of range R' (see Fig. 1)._ F^W) 

 are the dlmensionless contributions per ray loop to (<b\) as 

 read from Fig. 4, leading to the dlmensionless weighted sum 



(Ae/EAe)CfrF;. k-f^). 



XI. CONCLUDING REMARKS 



We end up, after lengthy derivations, with quite sim- 

 ple and transparent formulae for the acoustical fluctua- 

 tions. The formulae make explicit the dependence of the 

 various oceanographic and acoustic parameters. The 

 need is to apply these results to a variety of experimen- 

 tal situations. 



For Project MIMI the measured acoustical fluctua- 

 tions are dominated by the statistics of multipath inter- 

 ference. The observations yield but one parameter 

 which is sensitive to the ocean model: (0f ). Values at 

 midstation and Bermuda are close to those computed for 

 an internal wave model based entirely on oceanographi- 

 cal observations. (There are no free factors in this 

 comparison. ) The agreement could be made even closer 

 by a reasonable adjustment of internal wave param- 

 eters. We conclude that internal waves play an impor- 

 tant and probably dominant role in producing the acous- 

 tic fluctuations. 



The MIMI transmissions are characterized by many 

 deterministic multipaths as determined by the gross 

 profile C(z); the statistical results are not affected by 

 the additional sporadic multipaths resulting from a fine- 

 structure 6C. In contrast, the Cobb Seamount experi- 

 ment has a single deterministic path, but because of the 



high acoustic frequency, sporadic multipaths play an 

 important role in producing high-frequency fluctuations 

 of intensity and phase. Dashen (private communication) 

 has shown that an extension of the present analysis, 

 based on the same ocean model, can account quantita- 

 tively for the high frequencies in terms of sporadic 

 multipathing, but this goes beyond the scope of our pa- 

 per. The mean-square quantities, in contrast, are 

 dominated by low frequencies and can be estimated 

 from singlepath theory. We find measured and computed 

 rms fluctuations to be within a factor of 2. 



ACKNOWLEDGMENTS 



This work has been strongly dependent on the closely 

 related efforts of R. Dashen and S. Flatte. The theo- 

 retical framework was set in an earlier report written 

 by C. Callan and F. Zachaxiasen. We wish to express 

 our gratitude to Callan, Dashen, and Flatte. 



This work was performed during the 1974 and 1975 JASON 

 Sumn.er Studies under the auspices of Stanford Research In- 

 stitute, supported by the Advanced Research Projects Agency. 



'Using e = 10'* cm' sec"' for the dissipation per unit mass, and 

 n = 10"' sec"' for the buoyancy frequency. 



'C. J. R. Garrett and W. H. Munk, "Space-Time Scales of 

 Internal Waves," Geophys. Fl, Dynam. 2, 225-264 (1972); 

 C. J. R. Garrett and W. H. Munk, "Space-Time Scales of 

 Internal Waves: A Progress Report," J. Geophys. Res. 80, 

 291-297 (1975); J. L. Cairns and G. O. Williams, "Internal 

 Wave Measurements from a Midwater Float II, " J. Geophys. 

 Res. (1976)(in press). The model initials suggest some 

 planned obsolescence and have allowed the authors to bring 

 out new models from time to time. 



'C. G. Callan and F. Zachariasen, Stanford Research Institute 

 Technical Report No. JSR-73-10, April 1974 (unpublished). 



*See L. Chernov, Wave Propagation in a Random Medium 

 (McGraw-Hill, New York, 1960); and V. I. Tatarski, The 

 Effects of the Turbulent Atmosphere on Wave Propagation 

 (unpublished). 



^We do not have to be very precise about how we define this 

 angle. 



'W. Munk, "Sound Channel in an Exponentially Stratified Ocean, 

 With application to SOFAR," J. Acoust. .Soc. Am. 55, 220- 

 226 (1974), Fig. 1 (based on Pingree and Morrison). (In that 

 paper, z is positive downward.) See also Fig. 1 inGM72, 

 Ref. 2. 



'j. Roberts, University of Alaska IMS Report No. R73-4 (1973) 

 (unpublished). 



'j. Cairns and G. Williams, "Internal Wave Measurements 

 from a Midwater Float II," J. Geophys. Res. (in press) 

 (1976). 



'S. M. Flatte and F. D. Tappert, "Calculation of the Effect 



. of Internal Waves on Oceanic Sound Transmission, "J. Acoust. 

 Soc. Am. 58, 1151-1159 (1975). 



'^The numerical experiment uses the exact wave functions for 

 the exponential model ocean, weighted according to Eq. (90), 

 whereas our analytical model is based on the corresponding 

 WKB approximation. 



"r. E. Ewart (unpublished). 



'^The computed phase spectrum |Eq. (119)1 is in units of radV 

 rad/s); multiply by (27r)-V(3600/27r) to get cyclesVcph. For 

 the intensity spectrum lEq. (120)1 multiply by (10/logl0)V 

 (3600/27r) to get dsVcph. 



"Roger Dashen has pointed out that the shallow sound axis 

 (2 = — 0, 4 km as compared to the canonical ?= — 1 km) can be 

 expected to produce smaller 6C/C fluctuations, given a 

 canonical internal wave field. The reduction is proportional 



231 



