A fourth-order evolution equation for deep water surface gravity waves in the presence of wind blowing over water

Abstract

The stability of a train of nonlinear surface gravity waves in deep water in the presence of wind blowing over water is considered. An evolution equation is derived for the wave envelope that is correct to fourth order in the wave steepness. The importance of the fourth-order term in the evolution equation was pointed out by Dysthe [Proc. R. Soc. London Ser. A 369, 105 (1979)]. From this evolution equation the expressions for the maximum growth rate of the instability and the frequency at marginal stability are derived, and graphs are plotted for those two expressions against the wave steepness.