model, 
Second order effects in the short crested model 
For reasons to be discussed later, it may be necessary to proceed 
to second order in this model and determine the effects of equations (7) on 
the wavy surface especially if breaking waves are to be studied. 
The long crested linear model 
If S(w, 8) becomes concentrated at the angle 9 = 0 and degenerates 
to a function of w only, equations (13) become 
a 2 72 
xi= x(a5.0, t)) = a finer 8.8 sin(~—a - wt + €) N2S(w) dw 
fo) 
co 62 2 
Z =eziq,O,1t)i= sont fe° §/g cos(=—a - wt + €) V2S(w) dw 
fo) 
An alternate notation is given by 
k_6 
x=a-Za e ™ sin(k a-wtte_) 
n n n n 
(21) k 6 
z=6-Za e " cos(k a-wtte ) 
n n n n 
For 6 equal to zero, the inverse of the first equation is 
(22) oh et (oll (54,1) 
This implies a free surface given by 
(23) Zaz (ascetic) az (cent) 
Second order effects in the long crested model 
The long crested model is more amenable to an investigation of 
second order effects because it is simpler. We consider the problem of 
two waves, and then generalize to the randomized process. For two waves 
the linear solution is given by 
