x = R*. For downward rays, /Ij^ is reasonably well ap- 

 proximated simply by x{R - x)/R. 



V. FLUCTUATIONS IN SOUND VELOCITY 



In the presence of an internal wave field with vertical 

 displacements f , a particle momentarily at z comes 

 from a rest height z - J. The resulting velocity fluc- 

 tuation at a fixed depth z is 



5C/C = 6C/C = a5T + 05S+y5P , 



where 



5T=i-B,Tf, 5S=J-a,S, 5P = pgi-Ap/p 



are the internal wave-produced fluctuations in T, S, P. 

 The wave-induced pressure fluctuations at a fixed depth 

 are reduced by a factor Ap/p(=:10'') over the pressure 

 fluctuation pgS experienced by a fixed water particle. 

 Henceforth the effect on 6C of the internal wave associ- 

 ated pressure fluctuations will be neglected, and so 



6C=t{8.C-8,C^) = J8.Cp ; 



(87) 



e. g. , the internal waves convect the potential velocity 

 gradient as defined in Eq. (79): 



5C/C = nHz)f^i/g. (88) 



The rms vertical displacement is given by (GM72) 



rms(J} = rms(fo)(«A!o)"''^ . rms(£o) = 7. 3 m , 



relative to its near-surface value, and so increases 

 with depth as n""^; accordingly 



rms(5C/C) = rms(6C/C)o(nA!o)"^ , 



with 



(89) 



rms(6C/C)o = iinlrms(Co)/g , 



decreases exponentially with depth with a scale |B 

 = 0.67 km. For orientation, set s = l and /j=24.5; typi- 

 cal values are given in Table I. In very deep water 

 5C/C is of order 10"^, and accordingly the rms fluctua- 

 tions in sound velocity are a few cm/sec. The rms 

 horizontal velocity components associated with internal 

 waves are (GM72) 



rms(M) = rnis(Mo)(n/wo) . "o = 4. 7 cm/sec , 



leading to the values in the last column. The last two 

 columns give relative perturbations in sound propaga- 

 tion associated with vertical displacement and horizon- 

 tal particle velocity, respectively. The latter effect is 

 much smaller (except in very deep water), and will be 

 ignored subsequently. On the other hand, the u effects 

 dominate at and below inertial frequencies, so that 

 planetary waves with their quasihorizontal particle mo- 

 tions affect sound transmission by Mach refraction. 



VI. INTERNAL WAVE MODEL 



Fluctuations in the vertical structure of temperature 

 and salinity were discovered by Petterson, Helland- 

 Hansen, and Hansen soon after the turn of the century. 

 Since that time there has been a vast literature on the 

 subject (over 500 references were compiled by Roberts') 

 consisting mostly of reports on temperature and current 

 fluctuations at moored instruments, and of a few hori- 

 zontal temperature profiles from tows behind ships. In 

 the past three years, the technology of continuous ver- 

 tical profiling of currents with freely dropped instru- 

 ments has been developed, providing additional infor- 

 mation. A three-dimensional trimooring (IWEX) was 

 installed in 1973 off the American east coast, and we 

 may expect some very useful additional results. 



On the basis of this myriad of observations, Garren 

 and Munk have contrived successive models^ (GM72, 

 GM75) of internal wave spectra. They placed particular 

 emphasis on multiple recordings, separated vertically 

 on the same mooring or horizontally on neighboring 

 moorings, which had shown that fluctuations of frequen- 

 cies as low as 1 cph were uncorrelated for vertical 

 separations exceeding a few hundred meters, and for 

 horizontal separations exceeding a few kilometers. 

 These coherences were interpreted as a measure of 

 reciprocal bandwidth: for separations larger than the 

 reciprocal bandwidth, different wave numbers interfere 

 destructively, and coherence is lost. The following con- 

 clusions were reached: (i) Observations can be recon- 

 ciled with the dispersion law and wave functions of lin- 

 ear internal wave theory, (ii) Towed records are in- 

 sensitive to the ship's course, and moored records are 

 similar for the two velocity components, thus indicating 

 some degree of horizontal isotropy; the evidence is 

 certainly incompatible with internal waves propagating 

 along narrow horizontal beams, (iii) Coherences are 

 incompatible with a model consisting of just the gravest 

 one or two vertical modes (except at tidal frequencies). 

 The GM72 model had equal contributions from modes 1 

 to 20, and none beyond mode 20. But this is too broad; 

 recent measurements by Cairns' are consistent with a 

 mode weighting according to (.f +j\)'^ with;^ = 3. 

 (iv) The myriad of observations, taken over the years 

 at many depths off the American west and east coasts, 

 Hawaii, near Bermuda and Gibraltar, in the Bay of 

 Biscay, and the Mediterranean, agree to within an or- 

 der of magnitude. This suggests some universality in 

 the internal wave spectrum, perhaps due to saturation 

 effects such as those limiting surface waves of high 

 frequency. 



We use the GM75 spectrum, somewhat modified for 

 the Cairns observations: 



F^(w,j;z) = {iHz))G{uj)mj) , 



4 /, f, ,2_ 2 U/2 /-nit) 



223 



