Generation of Wave Groups

The well-known linear stability theory of wind-wave generation is revisited with a focus on the generation of wave groups. As well as recovering the usual temporal instability, the analysis has the outcome that the wave group must move with a real-valued group velocity. This has the consequence that both the wave frequency and the wavenumber should be complex-valued. In the frame of reference moving with the group velocity, the growth rate is enhanced above that for just a temporally growing monochromatic sinusoidal wave. The analysis is extended to the weakly nonlinear regime where a nonlinear SchrÖdinger equation with a linear growth term is discussed.