{u^= — B Eniio -IT JT-J TTT 



2 J„ OTT Ci) (o) -Win) 



Tjrfuj Z^ 



(/+/J- 



_2_ 1 



3 "^ 27rVT 



0.76, from a),„ to 2 0.1 In 

 ■ 1 , from Win to « , 



with the j contributions as before. 



*The work on random vector statistics was started during the 

 1974 JASON Summer Study under the auspices of Stanford 

 Research Institute, supported by the Advanced Research 

 Projects Agency, Subsequent analysis has been supported by 

 the Office of Naval Research. 



'For jV/tami—/t/ichigan project, starting with J. C. Steinberg 

 and T. G. Birdsall, "Undei-water Sound Propagation in the 

 Straits of Florida, "J. Acoust. Soc. Am. 39, 301-315 (1966). 

 For a recent paper see J. G. Clark and M. Kronengold, 

 "Long-period fluctuations of CW signals in deep and shallow 

 water, "J. Acoust. Soc. Am. 56, 1071-1083 (1974). 



^J. G. Clark, "Ray Propagation in an Undei-water Acoustic 

 Channel with Time Varying Stratification, " Tech. Rep. IML 

 70107 (University of Miami, Rosenstiel School of Marine and 

 Atmospheric Science, 1970); 1. Dyer, "Statistics of Sound 

 Propagation in the Ocean, "J. Acoust. Soc. Am. 48, 337— 

 345 (1970); J. G. Clark, N. L. Weinberg, and M. J. Jacob- 

 son, "Refracted, Bottom-Reflected Ray Propagation in a 

 Channel with Time-Dependent Linear Stratification, " J. 

 Acoust. Soc. Am. 53, 802-818 (1973). H. A. DeFerrari, 

 "Effects of horizontally varying internal wavefields on multi- 

 path interference for propagation through the deep sound 

 channel," J. Acoust. Soc. Am. 56, 40-46 (1974). H. 

 DeFerrari and R. Leung, "Spectrum of phase fluctuations 

 caused Ijy multipath interference, " J. Acoust. Soc. Am. 58, 

 604-607 (1975). 



^Surface scattered arrivals differ in frequency from cr by 

 roughly ±0.1 Hz (tlie frequency of ocean waves) and are re- 

 moved lay narrow-band filtering. Bottow-scattered arrivals 

 are greatly attenuated. 



*In fact the phases were computed 32 times for each 5 min 

 period and then averaged [T. Birdsall (private communica- 

 tion)]; this may account for the reasonable behavior of phase 

 spectra (as compared to intensity spectra) at the high fre- 

 quency limit. It is the reason for the phase jumps between 

 5 min readings (Sec. I). 



*C. J. R. Garrett and W. H. Muiol-t, "Space-Time Scales of 



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

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

 nal Wave Observations from a Midwater Float: Part II, " 

 J. Geophys. Res. (in press) (1976). 



•"W.H. Munk and F. Zachariasen, "Sound Propagation Through 

 a Fluctuating Stratified Ocean: Theory and Observation, " 

 J. Acoust. Soc. Am. (in press) (1976). 



'Units are a bother, l^ut the usual way out of plotting loga- 

 rithmically and laljeling decibels won't do. F^iu) has the 

 dimension of frequency, and the spectrum is a plot of fre- 

 quency versus frequency, derived as follows: In Eq. (41) 

 and (43), Ixith and u) are in radians per second, and FJ((j) 

 gives the contribution, per unit band (rps), to <^^) in (rps)-; 

 hence Fj(a)) has the dimensions (rps)Vrps = rps. The mea- 

 sured time series is 6$: the phase difference (in cycles) 

 during an interval 6t. The contributions to ((6*)^) are dis- 

 tributed among 64 frequency titands between and (26/)"', 

 each of width (128 6()"' cps = 27r(128 6<)'' rps. The plotted 

 spectrum is then 



F„(w)= A(6()' 27r(128 6()-' Fl(w) 



v6t 

 '256 



Kl.27.^)' 



with 1' in sec"', 6(=300 sec, and wi.Hz) = 

 The plotted intensity spectrum is 



F,(i)=(r^„VT^F,(c.) 



(27r/86 400) cx)(cpd). 



1286t 



'N. L. Weinberg, J. Clark, and R. P. Flanagan, "Internal 

 tidal influence on deep-ocean acoustic-ray propagation, " J. 

 Acoust. Soc. Am. 56, 447-458 (1974). 



"l. Magaard and W. D. McKee, "Semi-diurnal Tidal Currents 

 at 'Site D,'- Deep-Sea Res. 20, 997-1009 (1973). C. 

 Wunsch, "Internal Tides in the Ocean, " Rev. Geophys. 

 Space Phys. 13, 167-182 (1975). C. N. K. Mooers and D. 

 A. Brooks, "Tidal and Longer Period Fluctuations of Inter- 

 nal and External Fields in the Florida Current, Summer 

 1970, " Deep-Sea Res. (in press) (1976). M. G. Briscoe, 

 "Preliminary Results from the Tri-moored Internal Wave 

 Experiment (IWE.X), " J. Geophys. Res. 80, 3872-3884 

 (1975). 



'"For midstation, the total (i^^) greatly exceeds both fort- 

 nightly ($'), see Fig. 4. 



"B. Zetler, W. Munk, H. Mofjeld, W. Brown, and F. 

 Dormer, "MODE tides, " J. Phys. Oceanogr. 5, 430-441 

 (1975). 



' M. Parke and M. Hendershott (personal communication) . 



247 



