thing to have in lieu of actual knowledge. It is known from non- 
linear wave theory (Lamb [1932], sec. 250, see footnote on the 
work of Michell, and also Davies [1951]) that, for a purely periodic 
wave of finite height, the ratio of the wave height to the wave 
length cannot exceed one seventh. Michell obtained a value of 1/7.05, 
and Davies' [1951] most recent results are given by a value of 
1/6.914. Equation (11.22)uses the value 1/7 because the result 
is to be only an approximation. Suppose that all of the power in 
the power spectrum for w greater than p» K were concentrated in 
one purely periodic wave with a wave length determined by pw K° 
Then certainly the integral given on the left in equation (11.23) 
is less than the integral given in the middle and there is reason 
to believe that they both must be less than the value, 1/49. The 
longest possible wave length has been taken on the left in this 
equation, and the waves would certainly be very steep if [An(e co yds 
had major contributions for values of # very much larger than p K° 
Of course nothing can be said about how these short waves combine 
with the lonzer waves for # less than # K in the non-linear case, 
but if [A,( ey Ir were identically zero for # less than} x, 
equation (11.23) would still have to hold. It would seem that any 
added disturbance for # less than (ye would only serve to increase 
the instability of the waves for # greater than Hye 
If these arguments are valid, then equation (11.24) follows 
from equation (11.23). It states that the power in [A(py TE from 
wy to infinity must be less than some constant times p rae In 
terms of T,, this power must be less than a200 
For small values of pw a this result gives a larger possible 
13 
