equation (3.7). The expression, b(y ),is given in equation (4.2) 
for the particular problem under study. The integral is evaluated 
in Bierens de Haan [1867]. 
It follows that equation (4.3) is just another way to write 
equation (4.1), and if it were integrated (4.1) would be obtained. 
As written, equation (4.3) is more informative than equation (4.1) 
because it is an integral which contains a term which varies sinu- 
soidally as a function of time, and, from Chapter 2, a great deal 
is known about how such waves travel. 
Neither equation (4.1) nor equation (4.3) gives sufficient 
information to determine the solution of the problem completely. 
There are many disturbances of the free surface which could have 
produced the observed variation in time at the point of observa- 
tion. The various spectral components which combine at the point 
x = O and y = O to produce the disturbance might have come from 
many different directions. It will be assumed that most of the 
disturbance came from the negative x direction and is traveling 
in the positive x direction. Thus variation in y does not occur 
and 7 will be a function of x and t alone. This assumption is 
definitely an approximation to what occurs in nature because it 
implies that the crests of the disturbance are infinitely long 
in the y direction. 
The first term in equation (4.3) contributed only a very 
small amount to the total integral because, with, positive, the 
magnitude of the exponential term is small to start with and be- 
comes smaller as php increases. Let these spectral components 
travel in the negative x direction. 
The second term in equation (4.3) contributes a major part 
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