The nonlinear evolution equations of fourth-order for two surface gravity waves including the effect of air flowing over water

Abstract

Abstract The nonlinear evolution equations of fourth-order have been established for two surface gravity waves in infinite depth of water including the effect of air flowing over water. In the present paper, we have applied a general approach depending upon the integral equation due to Zakharov. Based on these equations, the stability analysis has been studied in the appearance of a uniform gravity wave packet with another wave packet of the identical group velocity. Graphs have been drawn for the instability growth rate of the uniform wave packet with shorter wave number versus the perturbed wave number for several values of the amplitude of the wave packet of larger wave-number and several values of wind velocity. We have observed from the figures that the instability growth rate of the second wave packet enhances with the enhancement of nondimensional wind velocity for the settled value of the amplitude of the first wave packet.