Stability of capillary–gravity interfacial waves between two bounded fluids

Abstract

Two-dimensional periodic capillary–gravity waves at the interface between two bounded fluids of different densities are considered. Based on a variational formulation, the relation between wave frequency and wave amplitude is obtained through a weakly nonlinear analysis. All classes of space-periodic waves are studied: traveling and standing waves as well as a degenerate class of mixed waves. As opposed to water waves, mixed interfacial waves exist even for pure gravity waves. The stability of traveling and standing waves with respect to three-dimensional modulations is then studied. By using the method of multiple scales, Davey–Stewartson-type equations are obtained. A detailed stability analysis is performed in three cases: pure gravity waves, capillary–gravity waves when one layer is infinitely deep, and capillary–gravity waves when both layers are infinitely deep. The main results for oblique (i.e., combined longitudinal and transverse) modulations reveal a mostly stabilizing effect of the density ratio for traveling waves and a destabilizing effect for standing waves.