2 
Ww 
(16) mil 2 /je oe 
g Vim n 
The free surface 
Those particles on the free surface are described by the condition 
6 = 0 and (13) becomes 
co oT 2 
x= x(aspst)imiai— 4} fsin(=-(a cos0+B sin@)-wtt+e)cos@ V2S(w, 0) dO dw 
O -7 
oo 6 2 
(i?) ya= yilas8st)= 16 ' — fh Jsin(—a cos0+ 8 sin@) -wt + €) sind V2S(w, 0) d@ dw 
Oo -T 
co oT 2 
Z= 2 (asBet)— el J cos(— (acos@ + Bsin6) - wt + e) rf 2S(w, 6) dO dw 
OT, 
In principle, and perhaps admitting triple values for the inverses 
over limited regions in the x, y,t space, x = x(a,8,t) and y = y(a, 8, t) 
imply inverses of the form 
a=a(x,y,t) and 
B 
so that the free surface can be found from 
(18) 
B(x, y, t) 
(19) PF AGM (oe S418), [Rey e)) = FACS S18) o 
The surface so defined is certainly not the equivalent of the short 
crested Gaussian sea surface. The short crested Gaussian sea surface 
is equivalent to this representation when the amplitude of the particle 
motions becomes so small that a = x and B = y are satisfactory approxi- 
mations to the inverses given by (18). Otherwise, further study suggests that 
z = 2(x, y,t) has many features of actual waves in that the crests can be 
quite sharply pointed and in that the higher nonlinear harmonics that occur 
in the higher order derivations in the Eulerian system are already present 
in this model. 
It is believed that the problems that arise in the adequate probabi- 
listic description of the surface defined by (19) in terms of (17) and (18) 
will be very difficult to solye and that considerable effort will be required. 
All of the problems that arise in the study of Gaussian noise and in the 
study of the short crested Gaussian sea surface have their analogue in this 
