TM No. 377 



approximate wave height H. The vertical bars with the letters u and w or T/,Uf 

 indicate the positions of the auto-spectra of the particular \A. } oO pair (given 

 in appendix B) . 



The cross spectra in figures V-HO and V-Ul are representative of measure- 

 ments made either near the surface during low sea state conditions (serial 029, 

 071, 072, and 120), or at depth during higher sea states (serial 106 and 109). 

 Thus, the cross-spectral densities and the accompanying covariances have 

 relatively small values. The data of serial 08l, O83, and 102 (figure V-U2) 

 are derived from more intense wave conditions, and are associated with larger 

 negative spectral densities and covariances. 



Linear cross-spectral scales were used to avoid the difficulty of plotting 

 curves having both positive and negative values on a logarithmic axis. These 

 curves indicate that most of the cross-spectral energy is concentrated below 500 

 mcps (above 2 sec period). 



The low energy observations in figures V-U0. and V-Ul show covariances 

 ranging from +8.9 cm 2 sec -2 to -7.6 cm 2 sec" 2 . These values are equivalent to 

 the area between the curve (solid line) and the zero axis. 



In general, the C uw curves attain a maximum absolute value at a similar 

 frequency position as the auto-spectra peaks. In the case where the covariance 

 is negative, the C U w peak is negative, and similarly for a positive covariance 

 (see serial 029). The Q uw curves in figures V-U0 and V-Ul tend to vary in their 

 relationship with the curves of C uw . Thus, serial 071, 072 and 106 indicate a 

 quasi-mirror image relationship between Qu W and Cuw; whereas serial 109 and 120 

 seem to display the C^ peak shifted from the Qu W peak. The coherence R displays 

 maximum values at or near the indicated cross-spectral peaks. The stronger the 

 cross-spectrum peaks, the greater is the magnitude of R. 



Figure V-U2, which displays the cross-spectra of higher energy wave systems, 

 shows a much more clearly defined spectrum relationship. Both Cuw and Q uw peak 

 at the same frequencies as the auto- spectra. The Q uw spectra show a much larger 

 peak magnitude than the Cuw The coherences corroborate the energy concentration 

 at these wave frequencies, attaining values from 0. 70 to 0.95. In general, the 

 higher energy waves have clearly defined cross-spectra, whereas the low energy 

 records exhibit small cross-spectral peaks and unstable patterns. A slight 

 shift between the peaks of C U w and Q uw is shown by all cross-spectra except 

 serial 029 and 102. 



The co- spectra from the BBELS data are similar to the data from Narragansett 

 Bay shown in figure IV-2. Although not shown, the Qu W for the Narragansett Bay 

 data also peaked strongly at the ambient wave frequency indicated by the auto- 

 spectra. 



The Q uw function registers the degree of out-of-phase spectral energy 

 associated with u and w (see chapter III). If the u and w velocity components 

 were represented by a sin t'-& and a cos <r'-h , respectively, then C uw would be ( 



llj-0 



