The equilibrium range in the spectrum of wind-generated waves

Abstract

Consideration of the structure of wind-generated waves when the duration and fetch of the wind are large suggests that the smaller-scale components of the wave field may be in a condition of statistical equilibrium determined by the requirements for attachment of the crests of the waves. A dimensional analysis, based upon the idea of an equilibrium range in the wave spectrum, shows that for large values of the frequency ω, the spectrum Φ(ω) is of the form $\Phi (\omega) \sim \alpha g^2\omega^{-5}$ where α is an absolute constant. The instantaneous spatial spectrum Ψ (k) is proportional to k⁻⁴ for large wave numbers k, which is consistent with the observed occurrence of sharp crests in a well-developed sea, and the loss of energy from the wave system to turbulence and heat is proportional to $\rho _w u^3_*$, where ρ_w is the water density and u the friction velocity of the wind at the surface. This prediction of the form of Φ(ω) for large ω with α = 7·4×10⁻³, agrees well with measurements made by Burling (1955).