and sin[( uy *)*/g) sine ]eos p *t, make it possible to find b(#*,e*). 
The spectrum of the disturbance given by a(mu *,0*) and 
b(y *,6*) has now been found from the function 7 (0,y,t) which 
was known. These known values can now be substituted into the 
original formula given by equation (8.28) which is known once 
N(o,y,t) is given. The integration in the square brackets must 
be carried out before the integration over # * and 6*, and in 
order to emphasize this, the y and t which disappear due to the 
process of integration are not starred, and the ones which will 
remain in the final solution are starred. 
a i i ee ee es ee 
ee a ee SS 
Given (x,y) at t equals zero, it is possible to find 
7 (x*,y*,t*) by the methods used above. Formulated in terms of 
Y, where Y is the spectral wave number, equation (8.29) pre- 
scribes a motion such that each elemental wave in the motion has 
a component of travel in the positive x direction. A derivation 
which follows the procedures used above very closely then yields 
the final result as given by equation (8.30). The values of 
a(vy *,0*) and b( VY *,e*) are given in the brackets and can be found 
given 7(x,y). 
The use of vy instead of # is more convenient in the deri- 
vation but not essential. The variable could just as easily have 
been #. The transformation of variables given by equations (8.31) 
and (8.32) then yields equation (8.33). The wave system conse- 
quently has the same form as the previous system. 
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