Nonlinear gravity–capillary waves with forcing and dissipation

We present a study of nonlinear gravity–capillary waves with surface forcing and viscous dissipation. Based on a viscous boundary layer approximation near the water surface, the theory permits the efficient calculation of steady gravity–capillary waves with parasitic capillary ripples. To balance the viscous dissipation and thus achieve steady solutions, wind forcing is applied by adding a surface pressure distribution. For a given wavelength the properties of the solutions depend upon two independent parameters: the amplitude of the dominant wave and the amplitude of the pressure forcing. We find two main classes of waves for relatively weak forcing: Class 1 and Class 2. (A third class of solution requires strong forcing and is qualitatively different.) For Class 1 waves the maximum surface pressure occurs near the wave trough, while for Class 2 it is near the crest. The Class 1 waves are associated with Miles' (1957, 1959) mechanism of wind-wave generation, while the Class 2 waves may be related to instabilities of the subsurface shear current. For both classes of waves, steady solutions are possible only for forcing amplitudes greater than a certain threshold. We demonstrate how parasitic capillary ripples affect the dissipative and dispersive properties of the solutions. In particular, there may be a significant deviation from the linear phase speed for gravity–capillary waves. Also, wave damping is strongly enhanced by the parasitic capillaries (by as much as two orders of magnitude when compared to the case with no capillary waves). Preliminary experimental results from a wind-wave channel give good agreement with the theory. We find a sharp cut-off in the wavenumber spectra of the solutions which is similar to that observed in laboratory measurements of short gravity–capillary waves. Finally, for large wave amplitudes we find a sharp corner in the wave profile which may separate an overhanging wave crest from a train of parasitic capillaries.