Resonant capillary–gravity interfacial waves

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Abstract
Two-dimensional space-periodic cabillary–gravity waves at the interface between two fluids of different densities are considered when the second harmonic and the fundamental mode are near resonance. A weakly nonlinear analysis provides the equations (normal form), correct to third order, that relate the wave frequency with the amplitudes of the fundamental mode and of the second harmonic for all waves with small energy. A study of the normal form for waves which are also periodic in time reveals three possible types of space- and time-periodic waves: the well-known travelling and standing waves as well as an unusual class of three-mode mixed waves. Mixed waves are found to provide a connection between standing and travelling waves. The branching behaviour of all types of waves is shown to depend strongly on the density ratio. For travelling waves the weakly nonlinear results are confirmed numerically and extended to finite-amplitude waves. When slow modulations in time of the amplitudes are considered, a powerful geometrical method is used to study the resulting normal form. Finally a discussion on modulational stability suggests that increasing the density ratio has a stabilizing effect.