-28 
that there probably is a residual vorticity field at third order in the La- 
grangian solution that may account for the difference. 
The particle velocity at the crest for the Gerstner wave is 
dx 2 
66 = = 
(66) u(0, 0) Gia 0t=0 aw ta kw, 
and when the phase speed is set equal to the particle velocity, the limiting 
form 
(67) ak = 1 
is still obtained. 
With ak = 1, the second order phase speed and the particle 
velocities at the crest are exactly double that of the linear theory, but it 
is known that the phase speed due to finite height cannot exceed about 1.2 
times the linear phase speed. These results are all indicative of the need 
for different wave models and for measurements of higher order effects 
in waves. 
Construction of z = z(x) for the second order equations 
From recent results of Tick [1961], it may be possible by start - 
ing with equations (28) to construct the vector process given below by an 
appropriate sequence of operations on the linear realizations constructed 
above. 
x(a, 0, 0) 
z(a, 0, 0) 
x(a, T, 0) 
z(a, T, 0) 
x(a, 0, 61) 
z(a, 0, 61) 
(Gh, Sep 61) 
z(a, 7, 5,) 
x(a, 0, 0) 
x,(a, 0,6) ) 
