TM No. 377 



for all frequencies, and Qu W would display a maximum peak at the wave frequency 

 0~ f . Thus, the tendency for a strong qua-spectrum is to be expected for ocean 

 waves. The co-spectra C uw should, on the other hand, be suppressed greatly. 

 The large (and in general negative) covariance functions is indicative of a strong 

 correlation between the u and w motions of the frequencies of the ambient wind 

 waves. 



The causes of the covariances and of the strong co-spectra of the u and w 

 wave motions are difficult to assess. The covariances and cross-spectra are 

 extremely sensitive to real (wave-induced) or artificial (instrument-induced) 

 correlations. It is therefore desirable to examine possible sources (natural and 

 artificial) which could contribute to the measured covariances of the wave motions, 



Wave Models and Their Covariance Properties 



A simple mechanism for transfer of wind-imparted horizontal stress or 

 momentum downward through the wave regime was suggested earlier in this chapter. 

 Such mechanisms are likely to be associated with the gross motions of the wind 

 waves (see Shonting, I96U) . 



Results of the u,w observations have been presented in which relatively 

 large (and generally negative) covariances were obtained. It is evident that 

 instrument bias could contribute, at least partially, to the covariances. This 

 biasing phenomena can be examined by constructing some artificial wave models. 



It is instructive to consider some mechanisms associated with wave motions 

 likely to exhibit covariance properties. Examine first the simplest motions of 

 fluid particles moving in circular orbits in a progressive wave moving past a 

 fixed point. Assuming that these waves are deep water waves, then the equations 

 for the horizontal and vertical motions are given by (11-5) and (II-6). These 

 may be rewritten as: 



n! = A% cos a--* > (v-36) 



and 



U) 



'= At s/Ai crt . (v-37) 



Here 



A* = *4<re K * 



in equations (II-5) and (II-6); and 0" = 2TT T~ , where T is the period of 

 oscillation. 



According to equation (III-E^), the covariance of u 1 and w* is: 



lUl 



