TM No. 377 



Covariances and Cross Spectra of Wave Motions 



As discussed in chapter III, estimates were made of the covariances and 

 cross spectra of the u and w components of speed measured with the OMDUM II am 

 OMDUM III wave meter systems. All statistics are listed in table IV-3. 



The u and w data obtained with the OMDUM III system are more reliable 

 because the system was subjected to a relatively thorough calibration and was 

 used with the most stable suspension available (see table V-l). 



Table V-3 summarizes the u, w BBELS observations obtained with the OMDUM III 

 system. In these measurements, the u and w vector components lie in the vertical 

 plane normal to the crests of the wind waves (as depicted in figure 11-10 ) . 

 Listed in this table are: depth D, mean horizontal velocity u, covariance (u r w* ), 

 linear correlation coefficient r, wind speed V, and estimated maximum wave height 

 H. 



In table V-3 there is a relatively wide range of values of u'w* , from +27.5 

 to -l62 cm 2 sec -2 , the algebraic mean being -25.3 cm 2 sec -2 . Of the 28 records, 

 19 covariances were negative. 



The magnitudes of the covariances seem to vary directly as the wind speed 

 and wave height. The light winds and small waves occurring during BBELS-10 and 

 BBELS-13 (071 and 072) observations are associated with relatively small co- 

 variances, whereas high winds and large waves occurring later during BBELS-13 

 (077 through 086) give generally large covariances. 



There is also an indication that the covariance for given sea conditions 

 decreases with depth. Attenuation of u'w* with depth is indicated in serial 102 

 through 110, but this relation is rather tenuous with regard to the other series. 



The mean values u range from +37 to -26 cm sec -1 . The covariances do not 

 appear related to either the magnitude or the sign of u. 



R 



Statistics on the cross spectra of the u and w observations include both 

 covariance spectra (co- spectra, abbreviated C U w) and quadrature spectra (qua- 

 spectra, abbreviated Qa W ) • Listings of these data, together with the coherence 

 and phase angle PHI, are given in appendix B. 



Representative co-spectra and qua-spectra from the observations listed in 

 table V-3 are shown in figures V-UO, V-Ul and V-U2. The co- spectrum axis is 

 located on the left and the qua-spectrum axis on the right of the figures. The 

 solid and broken lines represent C uw and Qu W , respectively. For convenience, 

 the Q uw scale is twice the magnitude of the C^w scale. 



"'Beneath the cross spectra is a plot of the coherence R. This quantity is 

 the square root of the function defined by equation (111-29). Above each plot 

 is the depth of the OMDUM III, the signed value of the covariance (u'w* ), and the 



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