assumption about the limits of integration, it is not necessary 
to know N x(¥ot) because all spectral components have a direction 
component in the positive x direction. 
If equation (8.14) represents the free surface everywhere, 
then equation (8.15) represents the sea surface at x = O and 
it can be expanded into the form of equation (8.16). Now the 
left hand side of equation (8.16) is a known function, and if 
the values of a(/# ,©) and b(#,0) were known then the step back 
to equation (8.14) would be simple and the problem would be solved. 
Take the known function 7 (y,t), multiply it by 
cos((p *°/g) sin 0*-y)-cos u*t, and integrate it over y and 
t from minus N to plus N and from minus M to plus M. Consider 
the limit as Mand N approach infinity. The first expression 
in equation (8.17) formulates this operation, and in the second 
expression (8.16) has been substituted for 7(0,y,t). The sec- 
ond, third, and fourth term in the bracket from equation (8.16) 
are not needed because the integration is even, and since, for 
example, cos w*t sinwt is odd the integration is zero. The 
third expression in equation (8.17) can be obtained from a trigo- 
nometric identity. 
The integration under analysis is continued in equation 
(8.18). Two transformations of variable are employed in order 
to get from the second expression to the third expression. The 
second expression can be expanded into one integral which involves 
(sin( - p *)M)/( » - » *) and another which involves 
(sin(# + » *)M)/(p+ p *). In the first integral the transfor- 
mation of variable given in equation (8.19) is used and in the 
- 183 - 
