TM No. 377 



fcV = ^^ $l»A<f> 



*V = "Lin* StN A<p . (v-U 3 ) 



This integration is graphically shown in figure V-U3B (middle curve) where 

 the shaded areas above and below the abscissa sum to a non-zero value. For 

 this phase-shifted model, the Reynolds stress term => ^u*w' , from equation (V-U3), 

 is proportional to the products of the amplitudes of the velocity functions and 

 of the sine of the phase angle shift. The sign of the angle 4^) (i.e., the 

 phase lead or lag) determines the sense of the stress. 



It was assumed that Au = Aw. Reynolds stresses were then calculated fnr 

 various amplitudes of the orthogonal velocity functions and various values of 

 &<jp . In figure Y-kk the stress -a u*w' (in dyne cm"" 2 ) is plotted on the 

 ordinate, and the phase angle A<fy on the abscissa. Some suggested wave parameters 

 are given. The values of A from 20 to 60 cm sec~l are representative of the 

 amplitudes experimentally observed for the wide variety of wind and sea conditions. 

 For example, the larger u or w variances (see figures V-26 and V-27) near the 

 surface are of the order of 1200 cm 2 sec~2 o jf the velocity functions were purely 

 sinusoidal, as in equation (V-37) 5 then the variances would be of the form: 



(V-kh) 



or 





For O^ 7 - = 1200 cm sec -2 , A =■ i+9 cm sec -1 . 



It is obvious from this that extremely large stresses can be associated 

 with a relatively small phase shift between orthogonal velocity components. 



The results of some of the LIMDUM I measurements have already been 

 discussed, along with the vertical ,r spatial correlation" of similar velocity 

 components. Covariance functions are also tabulated in appendix B for the 



LIMDUM I data for parallel u or w pairs. The covariances u']_ u*2 or w c i w'g 

 are always very large and positive because the similar wave components at the 

 two depths are almost in phase. The associated correlation coefficients are 

 likewise large and positive; however, they show a reduction in magnitude with 

 depth. 



Since the LIMDUM I system registers the instantaneous velocities of ui and 

 ug or wj_ and W2, it is possible to examine the data pairs for a phase shift 



1^3 



