TM NOo 3U2 



covariance function for the two biased models^ BR and BSH^ is -3.98 and 

 -2.57 cm^ sec"2 respectively » The covariance for the ideal sinusoidal 

 function is^ of course^, much smaller^ i.e., -0.0^5 em^ sec~2. 



The correlation coefficient, H^jy, for the BR model is -0,21 and for the 

 BSH model is -O.05.. For the U3H model the correlation coefficient is even 

 smaller at -9 x 10"^ . 



The auto-coT'ariance and covariance spectra (which are the Foiorier trans- 

 forms of the respective functions) of the three sets of data are plotted in 

 figures 3^ k^ and 5, The "in phase" or real part of the covariance spectra 

 (tenned co-spectrum) is plotted below the power spectra pairs in these flguxes. 



The power spectra of each velocity component^ i.e., J u and | ■^, are 

 plotted as cm^ sec-2 per cycle per second versus frequency (and period) on 

 the abscissa. The values of | u and | -^ for BR are similar^ with each 

 showing a dominant low frequency peak between 0.2 and 0.3 cps. This peak is 

 well defined and drops sharply at about O.U cps. From 0.4 through to the 

 limiting frequency of about I.7 cps the J -^ fluctuates in a similar manner 

 as ^ u except that the latter displays two peaks, one at 0.8 cps and one at 

 about 1.1 cps. These oscillations of the power spectrum functions are mostly 

 due to chance since the data were generated in a quasi-random fashion. 



The cross spectrum, shows repeated fluctuations ;, displaying negative peaks 

 at 0.3^ 0.8 and 1.25 cps. There is_, however, no extreme peaking indicative 

 of predominant coupling at any one frequency. The co-spectral function is, 

 in general, negative throughout the frequency range. 



The power spectra of the unbiased sinusoidal waves, USH, figure k, show 

 the expected sharply defined peaks at O.p cps, which is the chosen frequency 

 of the sinusoidal velocity components. 



The co-spectrum function displays only a very slight negative pertiorbation, 

 at about 0.5 cps, which is less than 10^ of the magnitude of the cross spectra 

 of the BR. 



Turning to the power spectra of the biased simple harm-onic waves (BSH), 

 figure 5^ again the pronounced peaks of the velocity functions occur at a 

 band between 0.3 to 0.6 cps. The base of the peak for the BSH m.odel is 

 broader than for the USH. 



The co-spectr-jm of the BSH model, figure 5 (bottom), is of special 

 interest since it disjjlays a strong negative peak centered at 0.5 cps. It 

 appears as s^lmost a mirror image of the power spectrums above. 



Thus, by altering the magnitude of the amplitude of the U(t) component 

 by about ^fo in a cyclic fashion, co-variance function is increased by almost 

 two orders of magnitude, and the co-spectra are completely modified to show 



