TM No. 377 



variable displacement vector of a particle from a fixed point in the XZ plane, 

 given by: 



IL s // Lx + « L 2 , 



The velocity at an instant of time is given by: 



(V-l) 



4A- = V r // d J± + jfc 4jL? (V-2) 

 dt d-t d-t 



Eckart defines a "random o scil lation" as that motion existing when ty remains 

 approximately constant as \U-|* increases indefinitely. On the other hand, a 

 "random drift" exists when ilLI 7 ' increases indefinitely with time, although the 

 variance of the velocity |Vl l remains constant. When Eckart made this distinction, 

 he may not have been thinking specifically of ocean wave motions; but particles in 

 ocean waves fulfill each criterion, depending upon which coordinate one chooses to 

 observe. For example, along the Z axis the motion is definitely bounded; i.e., 



L% is approximately constant over short periods (for stationary conditions). In 

 other words, the variance of the amplitude of the orbital motions is approximately 

 unchanging during a period ranging from a few minutes to a few hours. Thus, in the 

 vertical, there is a random oscillation condition. Note also that for short 

 periods : 



(i±*y so . (v-3) 



In the X direction, the situation is different. The value of L-x is, in 

 general, always increasing with time. This is due partly to the actual continuous 

 displacement peculiar to surface waves, but mainly to a mean flow of tidal current 

 and perhaps wind drift. Also, over short periods: 



OLL*) ^ CONSTANT. 



(Y-k) 



The reason for considering these definitions is to indicate the degree of 

 anisotropy of the wave motions. It may reasonably be assumed that, over the 

 period of measurement, there will be zero mean vertical motion. However, the mean 

 horizontal motion will usually be non-zero, caused by slowly oscillating or meander- 

 ing currents that contribute to the u (but not to the w) velocity data. The 

 statistics of the random oscillation for vertical motion w(t) should be simpler to 

 interpret than those for horizontal motion u(t), since the latter function can have 

 strong contributions of a random drift nature. Hence, for a given time interval of 



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