In the theory of waves of finite height, it is not possible 
to take two solutions, add them, and find a solution for the 
combined effect of the two profiles. Thus the known solutions 
for waves of finite height are all purely periodic, and they do 
not apply to the sea surface if the sea is irregular. 
Linearized Equations 
Since the irregularity of the sea surface will be of the 
greatest interest in this paper, the equations must be linearized 
if known mathematical techniques are to be applied to the analysis. 
The assumption can be made that » is so small that the square of 
pm and its partial derivatives can be neglected compared to the 
magnitude of m and its partial derivatives. Under this assumption 
equations (2.1), (2.2), (2.3), (2.4) and (2.5) can be replaced 
by equations (2.6), (2.7), (2.8), (2.9) and (2.10) which are 
also given in Plate I. 
Equations (2.6) through (2.10) are very much simpler than 
the first set of equations. The non-linear terms have been omitted 
from equations (2.2) and (2.4), and to the same degree of validity, 
it is possible to evaluate the free surface boundary conditions 
at z = O instead of at z =) os These equations are linear. If 
@, is a solution of equations (2.6), (2.7), (2.8), (2.9), and 
(2.10)'5 and if > is a solution of the same equations, then 9, + 95 
is also a solution. In addition, P, + P5 and 7),+ 7» are 
defined. Strictly speaking the equations hold exactly only for 
waves of infinitesimal amplitude. In what follows, they will 
be applied to waves of finite height with the reservation that 
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