Introduction 
The present state of theoretical knowledge of wind generated 
gravity waves, when studied as a random process in nature, is based ona 
short crested Gaussian sea surface obtained by linearizing the Eulerian 
equations of motion (Pierson [1952], [1955]; Longuet-Higgins [1957]), on 
a second order extension of the Eulerian equations for long crested waves 
in which the underlying linear waves are Gaussian (Tick [1959]), and on 
various results involving wave spectra that do not specify the probability 
structure of the waves, some of which involve nonlinear properties of the 
waves (Phillips [1958], [1961]). 
The short crested Gaussian model has proved fairly useful 
in explaining the angular spreading and dispersion of swell, the marked 
variation in height from wave to wave in a storm sea, wave refraction, 
bottom pressure fluctuations caused by waves passing overhead, and the 
motions of ships in waves (St. Denis and Pierson [1953], Lewis [1955], 
Cartwright and Rydill [1956]). 
These applications are dependent on either the gross features 
of the waves or on natural processes that appear to linearize the system 
even further. For example, the wave profiles are ina sense averaged 
over the length and beam of a vessel at sea, and the nonlinear effects are 
reduced. The higher harmonics ina nonlinear profile and the high fre- 
quency linear components are also attenuated with depth in such a way 
that the application of Gaussian noise theory to the zero crossings of 
bottom pressure fluctuations leads to useful results (Ehrenfeld, Good- 
man, et al [1958]). 
The spectrum of the time process at a fixed point has been 
most frequently studied. Since the actual wave is recorded, more or less, 
the full nonlinear motion is recorded, and the spectrum of this nonlinear 
motion is estimated. The accuracy of computations in a linear theory 
based on spectra estimated from a nonlinear record is open to question. 
In particular, higher moments of the spectra are quite open to question. 
Theoretical developments in the study of the short crested 
Gaussian sea surface require numerous higher moments of the linear part 
