o 

 < 



FIG. 9. Intensity fluctuations as a function of range for sever- 

 al rays. The solid line indicates the fluctuations from a single 

 hydrophone placed at various ranges along the way. The rapid 

 oscillations in the solid line are due to the rapidly changing 

 multipath environment. It is interesting to note that the rms 

 fluctuation from a large number of paths with random phases 

 is expected to be 5.6 dB. The dashed line indicates the fluc- 

 tuations observed in the ray peak determined from a 700-m 

 vertical array (Gaussian a=180 m). The result of selecting a 

 single path is seen to be a reduction in fluctuations and a 

 smoother dependence of these fluctuations on range. 



lations of internal-wave effects on acoustic transmis- 

 sion.* 



ACKNOWLEDGMENTS 



This work is part of a larger study of the sources of 

 acoustic fluctuations in the ocean begun by Walter Munk, 



whose seminal influence and continual encouragement 

 we gratefully acknowledge. Important conversations 

 were had with Roger Dashen, Kenneth Watson, and 

 Fredrik Zachariasen. 



Our work was largely completed during the 1974 

 JASON Summer Study under the auspices of Stanford 

 Research Institute. The support for our work has come 

 from the Advanced Research Projects Agency, and part 

 from the Office of Naval Research. 



APPENDIX A: VERTICAL BEAMFORMING 



The code we are using propagates sound waves from 

 a point source along a vertical plane in an ocean with 

 internal waves. In order to determine the directional 

 character of the arriving signal at some position down- 

 range from the source, we have formed a vertical ar- 

 ray of receivers and combined the signals with phase 

 delays to amplify the waves coming from particular 

 directions. 



Suppose 'l'(^j) are the wave amplitudes at a set of N 

 points at a particular range spaced equally in depth z. 

 The N points span the ocean depth z^ai. so that z„^ 

 = Nd where d is the spacing of the grid of receivers {d 

 is 15.6 m in our case). 



We define the amplitude arriving from a particular 

 direction 6 at a depth z as 



<p(e) 



= J^'i'i^i) exp- i n^ j exp[- i^z j - z) sine] , 



where a is a measure of the vertical aperture of the 

 Gaussian array and k^ is the acoustic wavenumber. 



We have chosen o- 180 m so that the angular resolu- 

 tion of the array is 0. 5' at 100 Hz and the expected in- 

 crease in intensity for a plane wave arrival, due to the 

 large number of hydrophones being summed, is 14.6 

 dB. Also note that sidelobes of the receiving array are 

 eliminated by the use of Gaussian shading — a practice 

 that is easy to implement in our numerical experiments 

 but inefficient in a field experiment. 



'm. Ewing and J. L. Worzel, "Long-range Sound Transmis- 

 sion, " Geol. Soc. Am. Mem. 27, Part m (1948). 



^See, e. g. , P. R. Tatro ind C. W. Spofford, "Engineering in 

 the Ocean Environment. ' 1973 IEEE Int. Conf. , 206-216; J. 

 Northrop and J. G. Colborn, J. Geophys. Res. 79, 5633- 

 5641 (1974). 



^See, e.g., R. H. Nichols and H. J. Young, J. Acoust. Soc. 

 Am. 43, 716 (1963): B. E. Parkins and G. R. Fox, IEEE 

 Trans. AU-19, 158 11971); J. G. Clark and M. Kronengold, 

 J. Acoust. Soc. Am. 56, 1071-1083 (1974); G. E. aanford, 

 J. Acoust. Soc. Am. 55, 968-977 (1974). 



■•o. S. Lee. J. Acou.<it. Soc. Am. 33, 677 (1961); J. C. Beck- 

 erle, J. L. Wagar, and R. D. Worley, J. Acoust. Soc. Am. 

 44, 295 (1968); J. C. Beckerle, J. Acoust. Soc. Am. 45, 

 1050 (1969); E. J. Katz, J. Acoust. Soc. Am. 42, 83 (1967); 

 V. A. Polyanskaya, Akus. Zh. 20, 95 (1974). 



''C. Garrett and W. H. Munk, Geophys. Fluid Dyn. 2, 225-264 

 (1972); C. Garrett and W. H. Munk, J. Geophys. Res. 80, 

 291 (1974); W. H. Munk, private communication (1974). 



*F. Zachariasen and W.H. Munk, unpublished. 



's. M. Flatte and F. D. Tappert, "A Computer Code to Calcu- 

 late the Effect of Internal Waves on Acoustic Propagation in 

 the Ocean," SRI publ. (in press). (Note that the internal- 



208 



