very important. But wave records at present are observed at a 
fixed point as a function of time. If this very simple example 
of a short crested sea surface yields such widely varying re- 
cords and such widely varying values of the average potential 
energy, what assurance is there that actual wave records as a 
function of time represent the sea surface in the neighborhood 
of the point of observation, and that the average of the squared 
wave record is actually related to the potential energy of the 
sea surface? 
For the non-Gaussian case, that is for sea surfaces of the 
form of equation (8.5), the potential energy averaged over time 
varies from place to place. Stated another way, it is a function 
of the point of observation as shown by equation (10.2). 
Gaussian short crested sea surfaces 
For the Gaussian case of a short crested sea surface, it can 
be proved that the potential energy averaged over time at any 
point on the sea surface is given by equation (10.3). This pro- 
perty of the Gaussian case is very important because it shows 
that the current wave records as obtained as a function of time 
do contain important information worthy of more detailed and re- 
fined methods of analysis. 
The proof of the statement made by equation (10.3) is some- 
what lengthy, and some other important results are also obtained. 
Consider first the integral definition of the short crested sea 
surface as given by equation (10.4) in which xy and yz are given 
subscripts to point out that the sea surface is being observed 
as a function of time at a fixed point. In Chapter 9, the inte- 
= 250n= 
