## Abstract

It is shown that a simple relation, *E*^{*} = 5.1 × 10^{−2} σ_{m}^{*−3}*E*^{*} = *g*^{2}*E*/*u*_{*}^{4}_{m}^{*} = *u*_{*}σ_{m}/*g* the dimensionless angular frequency at the maximum of the energy spectrum, *g* the acceleration of gravity and *u*_{*} the friction velocity of the air. This expression is an alternative form of the relation between dimensionless wave height and period, *H*^{*} ∝ *T*^{*3/2}, which was previously proposed by the author (Toba, 1972) for energy-containing waves, and is extended to individual waves in the wind-wave field in a statistical sense. It is also shown, supported by various data, that the essential part of the one-dimensional energy spectra of growing wind waves should have the form *g*_{*}*u**σ^{−4} for the high-frequency tail of the frequency spectrum, where *g*_{*} is *g* expanded to include the surface tension. This is the form previously proposed by the author (Toba, 1973b) as the one-dimensional spectral form consistent with the above power law relationship, instead of the *g*^{2}σ^{−5} form proposed by Phillips (1958). By use of the power-law relationship for *E*^{*}, it is shown that the proportion of that part of momentum which is retained as wave momentum to the total momentum transferred from the wind to the sea can be expressed by a function of _{m}^{*}*C*/*U*, the ratio between the phase velocity of the energy containing wave and the wind speed. The value of the proportion decreases from about 6% in the form of an error function of *C*/*U*. A prediction equation for the growth of wind waves by a single-parameter representation is proposed, in which the rate of change of *E*^{*} is expressed by a formulation including the error function or by a simple stochastic form. The integration of the equation for the case of fetch-limited conditions is in excellent agreement with data compiled by Hasselmann *et al*. (1973). Reviewing results of recent wind-wave tunnel experiments, emphasis is given on the fact that wind waves are strongly nonlinear phenomena, especially for *C*/*U* ≪ 1. A discussion is presented from this standpoint as to the physical basis for the existence of the simple power law relationship. the spectral form of *g*_{*}*u**σ^{−4} and the stochastic form of the growth equation, and a systematic derivation of these relationships and equations is attempted.