SECT. 5] 



INTERNAL WAVES 



755 



frequency. Aside from these peaks and a very broad maximum of the 500 m 

 spectra centered at 0.5 c/h, the spectra decrease monotonically, with increasing 

 frequency. There are also unavoidable wiggles due to statistical variations. The 

 decrease above 0.75 c/h is sharper than/"^. The authors point out that this is 

 consistent with the behavior to be expected of internal waves. There should be 

 no waves with frequencies above the maximum of Vaisala's frequency (about 

 5.6 c/h) and few waves with frequencies above the local Vaisala frequency 

 (Fig. 24). Since the local Vaisala frequency is lower at 50 than at 500 m, one 

 expects the 50 -m spectrum to cut off more rapidly than the 500 -m spectrum, as 

 is, in fact, observed. 



35 36 37 



SALINITY {7oo) 



Fig. 24. The variation of temperature, salinity, density and Vaisala's frequency with 

 depth off Castle Harbor. The upper maximum of Vaisala's frequency is associated 

 with the seasonal thermocline, the deeper but weaker maximum with the permanent 

 thermocline. Ng and N^ indicate local values of Vaisala's frequency at the shallow 

 and deep recorders respectively. 



In addition to spectra, the coherence and phase shift between oscillations at 

 the two depths were estimated. The meaning of these terms may be outlined as 

 follows. 



Let y\{t) and y2{t) be the two temperature fluctuations, assumed statistically 

 stationary and oscillating about zero as mean value. The cross- and auto- 

 correlations are 



^12(t) = yi{t)yz{t-r), ru{r) = yi{t) yi{t-r), r22{r) = y2{t) y^it-r). 

 The coherence R and phase shift d are then found from 



(6) 



R{f) exp[i^(/)] = 2(5i>S2)-'/^ ^ ri2(T) ex^{2rrifT) dr, 

 where Si and S2 are the energy spectra of yi and 2/2 respectively : 



/'OO 



>S'i(/) = 4 rii(T) cos (27t/t) dr, 

 jo 



and a similar expression holds for >S'2. With these definitions, 6 is the phase lead 



