These solutions (eqs. 8 and 11) are simply disguised versions of 
the Gerstner wave in deep water as described in Lamb. The angle @ is 
the direction toward which the wave is traveling. When the solution is 
periodic, as above, the original nonlinear equations are satisfied exactly, 
but it can be shown that there is some second order vorticity and that 
there is a second order correction to the pressure given by 
2 2 
i ak 2k6aagk 
(12) P=P,-sp5+-5-gpe = =5— ep 
neers : : 2 2k6 
Another possibility also suggests itself. With X> =a wkte cos 6, 
Yo = Pag sin® , and Zo =0, equations (1), (2), and (3) are still 
satisfied to second order but third order complications arise. These two 
terms are the equivalent of the s econd order current found in irrotational 
waves in the Eulerian system of equations as described in Lamb. 
Extension of these results to finite depth would probably follow the 
results of Biesel [1952] who achieved a remarkable representation for 
periodic breaking waves by means of the Lagrangian equations (see also 
Pierson [1955]).* 
A randomized short crested model 
The expressions for X),y),and zy in either (8) or (11) are solutions 
of equations (6). These equations are linear. Therefore, a sum, over dif- 
ferent parameter values, of solutions of the form of XY and zy will also 
be a solution. If it is assumed that each particle has a displacement from 
its rest position, ao: Bo 6 _, that is described by a stationary Gaussian 
fo) 
vector process with specified coherency relationships, a solution to equations 
(1) that is correct to first order is given by equations (13). 
co oT 2 5) 2 
x=a- f fe® ° cos@ sin(=—(a cos 6+ B sind(- at + €(w,0)) J25(w, 6) d6@ dw 
fo) 
= Ti 
eu Oo 6) ee 
(13) y=6- 4 fe 8 sind Sia ae cos0+B sinO) - wt + €(w,6)) V2S(w, 8) dO dw 
re) 
-T 
T 25 / 2 
z=6+ if (ee 8 cos Cs (acos0+fsin@) -wt + e(w,6)) V2S(w, 6) dé dw 
Oo -T 
*The remarks concerning the Gerstner wave are in error as these results 
six years later show. 
