= ie 
Also there is a wind drift current in a local sea that is nearly in 
the same direction as the traveling waves. This wind drift current pro- 
duces a field of vorticity in the same sense as the direction of propagation 
of the waves and may well cause breaking to occur for considerably lower 
wave heights than even irrotational theory would predict. 
The work of Longuet-Higgins [1953], [1960] has shaken the con- 
cepts of irrotational motion for gravity waves to their very foundation . 
His results predict that, even in the absence of wind drift currents and 
turbulent effects due to breakers, the mass transport velocity in deep 
water may be considerably stronger than irrotational theory predicts if 
the waves have been running long enough. 
Moreover, as Longuet-Higgins [1953] has pointed out, Duvriel- 
Jacotin has shown that any mass transport velocity can be used as the 
starting point and a wave motion can be superimposed thereon. 
It is also known from classical periodic wave theory that a sharp 
angle at the crest of 120° is another way to define the limiting height of 
a periodic wave of the form z = z(x). It may be possible if the other con- 
cepts discussed above fail, due to the fact that the results are only to 
second order, to study constructions of z = z(x) as a random process to 
see how many waves approach this limiting form. 
With all of these preliminary remarks it would seem that 
equation (29), with perhaps still an arbitrary second order vorticity field 
added, can provide some insight on the growth of a wave spectrum. Con- 
sider two orbital motion spectra with the same variance, one with con- 
tributions with relatively higher frequencies than the other. The mass 
transport at the surface will be stronger for the spectrum with the richer 
high frequency content but at the same time the speeds of the crests will 
be less. The interaction of these effects could conceivably cause the 
higher frequency waves in the spectrum as they are overtaken by and 
ride up on the crests of the longer lower frequency waves (Longuet -Higgins 
and Stewart [1960]) to break and dissipate by turbulent action. At the 
same time the low frequency components can continue to grow and even 
be caused to grow by the breaking of the shorter waves. Considerations 
such as this may explain various wave spectra where forms like Kyoue 
Sow ifasore and Kye have been proposed and where the exponent 
