TM NO. 377 

 Since 2-_Ji,ij-fy 5 condition (A- 59) is equivalent to 



, ^., Pi?/? 6<^ < IT 

 PoiU)^^ . (A.61) 



i 



Therefore^ if the probability density of the phase angle given by equation (A- 59)* 

 depicted by figure ¥-=13 (upper plot)^ is uniform, the ensemble of sine waves is 

 stationary and the probability density of the amplitude is given by equation 

 (A-6o), This density distribution is plotted in figure V-13 (lower plot). 



Relationships for Motions Associated with a Turbtilent Velocity Field 



Reynolds Stresses "■= To examine the concept of the Reynolds stress existing 

 in the surface ocean layer^ one must consider the time variable motions of the 

 water particles at a point immediately beneath wind-driven ocean waves that 

 are being subjected to a surface shear stress by the wind. 



The terms u and w are the time=varying horizontal and vertical^velocity components 

 at a point in the water col"umn of uniform density. The u and w are time mean 

 values given by the following integrals: 



■T/i 



and _ / /'"^''i- 



^-^J ^^"^^ (A-63) 





i/i it ' 



The terms u' and w' are the instantaneoxis deviations from the time mean 

 velocities . Thus ; 



'It' = ^' £0 , 



Consider a small area dA in a horizontal plane between two layers of water 

 which are being subjected to a horizontal shear. The mass of water dm crossing 

 this area vertically during the time dt is given by: 



where ^ is the density - constant, 



A-12 



