Assume that the wave is frozen in time over the surface (the x,y 

 spacial plane) . The coordinates (u,v) are a degree rotation of the 

 (x,y) coordinates. The positive u axis lies along the direction from 

 which the wave is traveling. The wave surface n(u,v), shown frozen in 

 time in Figure 1 can be described mathematically by 



n(u,v) = cos(2TrKu + 2tt(^) (2.1) 



where K = 1/A is the wave number of spacial frequency in cycles per 

 unit length along the u axis, and 2tt(J) is a spacial phase shift. 



To make the wave move in time across the spacial plane with a 

 time frequency f = 1/p, where p is the wave period, it is necessary to 

 add a time part to the argument of the cosine function in the model 

 above. The time part is a phase shift dependent only upon time. As 

 time passes, the time part changes causing the cosine wave to move 

 across the (u,v) plane, in this case the ocean surface. Adding the time 

 part we get (where 2v^ is a fixed time phase shift) 



n(u,v,t) = cos(2TrKu + 2Tr(j) + 2?: ft + 27njj) . 



If we combine the effect of the (j) and it phase shifts as a = (f) + i|;, 

 we get 



ri(u,v,t) = cos(2Tr(Ku + ft + a)) (2.2) 



as a simple model of a sinusoidal wave moving in time over the ocean 

 surface . 



Since the coordinates (u,v) are a rotation of the coordinates (x,y) 

 through an angle of 6 degrees , we know 



u = x cos + y sin 9 

 V = -X sin e + y cos 9. 



Using the above relations, and letting H = k cos 9 and m = K sin 9 

 be the spacial frequencies along the x and y axes, respectively, we 

 have 



ri(x,y,t) = A cos(2tt(JIx + my + ft + a)) (2.3) 



as a model for a wave of height 2A moving from a direction 



9 = arctan (m/l) 



with a phase shift of li^a. A wave crest of such a wave system is infi- 

 nite in length. A crest occurs at a set of points (x,y,t) which satisfy 

 the relation 



^x + my + ft = a constant = (n - a) 



