Sea surface "glitter" 
So far no claim that equation (11.19) must hold has been put 
forward. The sea surface could be covered by many small facets 
and at many points the curvatures can be sharp. However, intui- 
tively at least,equation (11.18) should have a meaning everywhere 
and this is not the case unless equation (11.19) holds. If equation 
(11.19) holds, then C, must be zero. In addition, in the same 
series discussed above, Coy Ces Cos Coy and Co must all be zero 
and for some # greater than p K the series must be of the form 
» Ca, (#48) ]*(cose)® = £49(0)/ n° plus higher order terms. 
The results show that there is a tendency for the high fre- 
quency components to produce many sharp facets on the sea surface. 
These facets can be observed in fresh waves from a generating area 
and they are particularly noticeable in the photograph which has 
been chosen for the frontispiece. Any light breeze can super- 
impose a high frequency spectrum on a swell and it is believed 
that these considerations account very nicely for the sea surface 
VoJi1t ber. i 
Final form for the power spectrum 
If it is required that all derivatives of the sea surface, 
the velocity field, and the pressure field have a defined power 
spectrum and a defined power integral, then the requirement posed 
by equation (11.27) must be fulfilled for any integer value of M, 
no matter how large. No polynomial in (1/p )" can satisfy this 
requirement. Therefore [a(m ,0)1° cannot be represented by a 
fraction consisting of polynomials in » in the numerator and 
denominator. The power spectrum must therefore be either some 
18 
