SECT. 5] INTERNAL WAVES 759 



velocity becomes almost independent of frequency (for frequencies well above 

 the inertial period) and one expects the phase shift between stations to be linear 

 with frequency. The observations show just this relation. From the observed 

 phase shifts, the mean direction of approach (ai2= — 1.5°) is easily found. 



The coherence was generally high at low frequencies and decreased at 

 higher frequencies. The cause of this behavior is threefold: (1) Internal waves 

 do not always come from the same direction ; when the observing stations are 

 many wavelengths apart (i.e. at high frequencies), interference between waves 

 with different directions of approach will vary the phase difference between 

 stations. This reduces coherence. (2) The phase velocity of internal waves of a 

 single frequency varies from time to time because of changes of mode, changes 

 of density structure in the water, changeable tidal currents and effects of finite 

 amplitude of waves. Again, the main effect is to make the phase shift at high 

 frequencies variable and the coherence is reduced. (3) Irregular fluctuations due 

 to turbulence. We shall estimate each of these effects separately, 



B. Relation of Coherence to Beam Width 



Suppose vertical displacements ^i{t) and ^2(0 due to internal waves of a 

 single mode are observed by two observers situated at (0, 0) and {x, y). The 

 relations between the two auto-correlations, the cross-correlation and the 

 directional energy spectrum may be calculated by methods indicated in Chap. 

 15, page 578 and equation (6), provided the waves move without refraction, 

 generation or decay. 



Let E{f, 99) represent the directional energy spectrum (see Chap. 15, page 

 571) which gives the relative energy of waves as a function of frequency and 

 the direction of propagation, 99, measured from the x axis. Then one finds that 

 the coherence R and phase lead of ^2 over ^1 are given by 



R exp(i^) — S~'^ E{f, cp) exp[ — i(</(A; cos (p + yk sin 9?)] dcp, 



where S = E{f, cp) dcp is the energy spectrum regardless of direction, and k 



is the wave number, assumed to be a single valued function of/. Allowance for 

 more than one mode can be made by replacing the integrands by sums over 

 modes. In the present case we do not need to consider this additional complica- 

 tion because inspection of the depth variations of isotherms shows that first 

 mode internal waves are clearly dominant. 



To illustrate the reduction in coherence with angular beam spread, we 

 examine the effect of a fan-shaped beam (Fig. 26) : 



Eif, cp) = {2Acp)-^S{f), 



if I99I < Acp, and zero otherwise. It can be verified that *S'(/) is the spectrum 

 regardless of direction defined above, and that 



R exp(i^) = (Acp)-'^ [cos {yk sin 99) ex-p{ — ixk cos 99)] dcp. 

 Jo 



