Quasilinear approximation for the spectrum of wind-generated water waves

According to Miles’ theory of wind-wave generation, water waves grow if the curvature of the wind profile at the critical height is negative. As a result, the wind profile changes in time owing to the transfer of energy to the waves. In the quasilinear approximation (where the interaction of the waves with one another is neglected) equations for the coupled air–water system are obtained by means of a multiple-time-scale analysis. In this way the validity of Miles’ calculations is extended, thereby allowing a study of the large-time behaviour. While the water waves grow owing to the energy transfer from the air flow, the waves in turn modify the flow in such a way that for large times the curvature of the velocity profile vanishes. The amplitude of the waves is then limited because the energy transfer is quenched. In the high-frequency range the asymptotic wave spectrum is given by a ‘–4’ law in the frequency domain rather than the ‘classical’ ‘–5’ law.