a constant value everywhere and thus degenerates into a white 
noise spectrum. Then for x not equal to zero, the free surface 
assumes the form of equation (4.35). 
In this modified form of the Cauchy-Poisson wave problem, 
the amplitude of the waves at a fixed point, x does not increase 
with time and is finite at all x not equal to zero. The disturb- 
ance, as in one of the previous cases, is caused by an infinitely 
high, infinitely long, infinitesimally wide column of water at 
the origin, but in this case it lasts only for an instant of time. 
Thus there is not enough energy to produce an infinitely high 
disturbance at points other than x = 0. 
Physical reality of problem 
The physical reality of the whole problem discussed in this 
section should be considered. If such a wave group were gene- 
rated on the surface of the ocean, would it travel as predicted? 
It might not because no ocean is infinitely deep, because the 
low periods associated with high values of » are really capil- 
lary waves, and because such effects as internal viscosity, and 
the friction of the atmosphere against the moving waves have been 
neglected. 
The wave length of a sinusoidal wave in water of finite 
depth is less than the values employed here. Since the spectrum 
of the waves is defined near p = 0 where the period of the waves 
is infinite, the group will not travel exactly as predicted in 
water of finite depth. 
In figure 1, for the values of the parameters employed, it 
can be seen that that portion of the spectrum which is affected 
* In the y direction. 
*tor the same period. 
=- 75 - 
