and T s is the period over which the time average is made. 

 Integrating equation (V-13) gives: 



TM No. 377 



m; 



l =-7+£k *»4*Ck-$) 



T ^ 



- r >A 



(V-15) 



Since T_ ^ T, the second term of the expression may be neglected. Thus: 



<^=^[i-iafe^ 



(V-16) 



The evaluation of equation (V-l6) is somewhat arbitrary, since precise 

 information is lacking of the parameters T, L, and A. In order to compare the 

 depth attenuation of the theoretical variance of the trochoidal wave with the 

 observed variance as a function of depth, one must evaluate the coefficient of the 

 exponential term in equation (V-l6). The somewhat arbitrary values of the wave 

 parameters were based on visual observations made at the time of measurements and 

 on the sea state chart (table II-2) derived from statistical observations by Marks 

 (196U). The variance calculated from equation (V-l6) at zero depth is about 

 1100 cm2 sec" 2, using the following values: wave length L = 18.0 m^ period 

 T = U.O sec; and amplitude A = 30 cm. 



This curve is plotted as a solid line in figure V-29, along with the variance- 

 depth distribution from BBELS-5 and 7. The surface value of variance is somewhat 

 arbitrarily chosen, since the values of the parameters can only be estimated. 

 However, the curve clearly displays rapid depth attenuation. Below 1.5-2 meters, 

 the theoretical values of variance fall off much faster than do the measured values, 

 This seems plausible, since the sinusoidal oscillations of the classical waves 

 transfer no momentum downward due to the absence of any Reynolds stresses (i.e., 

 turbulent diffusion). On the other hand, ocean waves cannot be completely 

 irrotational; and, by their very nature, they serve to transfer turbulent energy 

 statistically downward. This is evidenced by the residue of variance existing 

 at depths well below k-6 meters (see figures V-27 and V-28). 



Essentially, the observed variance function characterizes only the gross 

 motional fluctuations in the waves. The lower limit of the size of the eddies 



120 



