the higher the ratio of the wave amplitude to the wave length 
the more inaccurate the results. This linearized theory is 
used in nearly all practical wave studies and especially in 
the theory of wave refraction and diffraction. 
General solutions 
The above set of equations is, nevertheless, still compli- 
cated, and the manifold of possible solutions is extremely large. 
The number of known solutions is quite small. 
There are two general types of solutions to those linear- 
ized equations. One is the periodic solution in time, and the 
other is the non-periodic solution. A periodic solution in 
time is a solution such that at any point in the fluid or at 
the free surface, the same conditions are found one period later 
aS were found at the time of the initial observation. The 
conditions must be the same for all time. Thus, the conditions 
for a periodic solution can be stated as in equation (2.11) in 
Plate II. From equation (2.11), it follows that the free sur- 
face is also periodic. 
The concept of periodicity will be investigated in detail 
in the next chapter. It should be noted at this point, that 
the sum of two periodic solutions need not be periodic unless 
some additional conditions are satisfied. 
In addition, a whole class of non=-periodic solutions can 
be obtained from integration by Fourier's Integral Theorem over 
a continuous spectrum of periods. The quotation at the Start 
of this paper emphasizes the fact that the way to obtain non- 
periodic solutions is to build them up mathematically from 
= Ok i= 
