Role of Domtnant Wave tn Spectrum of Wind-Generated Waves 
putng equal to 0,333. m, The water surface maxima are Rayleigh 
distributed, and thus the water surface variance o* is related to the 
average height H 1/3 of the highest one third waves through the re- 
lation (Longuet-Higgins (1953)) : 
MECC De ey 
Oe ae ae 
(4) 
where the wave height is defined in the usual manner as the distance 
between trough and crest of a wave. Equation 4 is remarkable be- 
cause it connects the physical concept of a wave with the mathemati- 
cal description of the surface, and through this with the spectrum. 
Equation 4 is not sufficient to define the magnitude of the 
spectral density at wm. We must find an additional condition which 
puts bounds on the growth of the individual wave. This condition is 
imposed by the breaking of the dominant wave, much as in the con- 
cept on which Phillips model is based. But unlike the breaking of 
all component waves, the similarity spectrum presupposes a limi- 
ting growth pattern imposed on the whole spectrum by the dominant 
wave. An equilibrium spectrum exists also for the similarity spec- 
trum, but in a different sense than that used by Phillips. The physi- 
cally unrealistic condition of breaking is no longer imposed on each 
of the mathematical component waves. But there is no doubt that a 
limiting form of a spectrum exists. It is well known thatat given fetch 
and with constant wind velocity, there is found (at least in the labo- 
ratory), a condition at which the time function representing the wa- 
ter surface elevation becomes truly stationary. Every spectrum 
that is determined from this time function is a sample from the sa- 
me ensemble. There is also sufficient evidence to suggest that even 
in the duration-limited case of water wave generation - that is, ina 
case where a wind of constant u has started to blow over a water 
surface which was initially at rest -, there exist only such waves on 
the water surface which have grown to the maximum possible heights. 
Only in those instances when the wind had calmed after generating 
equilibrium waves, or when wind starts blowing over non-equilibrium 
waves that have entered from another storm area in the form of 
swell, or when wind blows over waves that are left from a previous 
storm, there will be spectra that are not of the equilibrium type. 
These spectra are likely to have a shape that cannot be described 
by the similarity law. We conclude that the similarity shape is al- 
ways valid for the maximum possible waves, and thus the spectrum 
of the water surface at equilibrium contains the maximum spectral 
densities, i.e. the equilibrium spectrum is the envelope to all pos- 
sible wave spectra at any one fetch and wind speed. 
LST? 
