SECT. 5] INTERNAL WAVES 761 



beam width of the waves. In the computations it has been assumed that the 

 phase velocity is 22 cm sec~i, independent of frequency, and that the center of 

 the beam is directed at angle ai2= —1.5°, in accordance with the observed 

 phase shifts shown in Fig. 25. The beam width Acp is assumed independent of 

 frequency. 



The observed coherences are all smaller at frequencies between and 0.05 

 c/min than is consistent with any value of beam width. 



By comparison of observed coherence between points 2 and 3 for frequencies 

 between 0.05 and 0.2 c/min, an upper limit to the beam width of Acp = 0.4 radians 

 is indicated. Using this value for Acp the computed coherence between points 1 

 and 2 scarcely decreases with frequency, while the observations show a decrease 

 to "noise level" at /= 0.17 c/min. Therefore processes other than the finite 

 beam width are required to reduce E12. 



C. Reduction of Coherence by Variable Phase Velocity 



The phase velocity of internal waves will be variable if the density gradient 

 changes, if the waves are carried on variable currents, or if the wave steepness 

 is large and variable. 



Let c be the mean phase velocity and Sc its variation from the mean. Then 

 the change of phase difference 86 between points separated by L, due to changes 

 of phase velocity, is less than or equal to 



277/Z|c-i-(c + Sc)-i|. 



The observed phase velocity at low frequencies was 22 cm sec~i. It may be 

 estimated that |Sc|<10 cm sec~i from all causes. Therefore 8^^180° for 

 fL ^0.25 cm sec~i. The coherence between observing points 2 and 3 separated 

 by 153 m would, therefore, not be much affected for frequencies well below 

 0.1 c/min. 



D. Reduction of Coherence by Turbulence 



If turbulent motions are superimposed on the internal waves they can 

 reduce coherence at any frequency. If the spectrum due to turbulence and 

 waves are respectively St and Sw, then it can readily be shown that coherence 

 is reduced in the ratio 



Sw{St+Sw)~^ 



if, as seems reasonable, the turbulence at the two points of observation is not 

 correlated. Since there is no frequency limitation in this relation, the reduced 

 coherence at low frequencies can be due to this cause. The observations give 

 i? = 0.75 for/< 0.05 c/min. From this value we estimate StlStv=0.3. 



E. Summary 



The high frequency reduction of coherence of isotherm depths observed at 

 stations 2 and 3 could be partly due to the arrival of internal waves from a 



