periodic solutions by the use of Fourier's Integral Theorem, 
One purpose of this paper is to show what information is needed 
to carry out this process for the sea surface. It will be 
found that classical wave theory is not quite general enough 
to represent the sea surface. The more recent extensions of 
Fourier Theory to stationary time series have to be employed 
in order to represent waves from a storm at sea adequately. 
Periodic solutions 
To return to the linear equations, then, it becomes necessary 
to study, the nature of purely periodic solutions. In order to 
do this, it is possible to split off the periodicity in time by 
use of the equations (2.12) and (2.13) in Plate II. In equations 
(2.12) and (2.13), Re is read "the real part of." % and 9 are 
complex quantities, and some examples will be given later. 
If equations (2.12) and (2.13) are substituted into the 
linearized equations, a set of reduced equations is obtained 
in which 9 and 7)' are not functions of time. However, (2.12) 
and (2.13) yield the progressive wave solutions. Equation (2.7) 
will not be used for a while and it will not be given in modi- 
fied form. 
The reduced linear equations given in Plate II have been 
solved exactly for a constant depth and for a linearly sloping 
beach. They have not been solved for z = - h(x,y) where h(x,y) 
is an arbitrary function of x and y. 
There are two solutions for constant depth. One solution 
yields an infinite train of traveling infinitely long straight 
parallel wave crests with a free surface which varies sinusoidally 
ay ayes 
