Spatial bifurcations of interfacial waves when the phase and group velocities are nearly equal

Abstract

Steady waves at the interface between two immiscible and inviscid fluids of differing density are studied. The governing equations are reformulated as a spatial Hamiltonian system leading to new variational principles for uniform states and travelling waves. Analytical methods based on the properties of the Hamiltonian structure and numerical methods are used to find new branches of steady nonlinear interfacial waves in the eighbourhood of the singularity c = cg. While the water-wave problem (upper fluid density negligible) near this singularity has received considerable attention the results for interfacial waves present some new features. The branches of travelling waves when plotted in ($\tilde{F}, \bar{S}$)-space, where $\tilde{F}$ and $\bar{S}$ are related to the energy flux and flow force respectively, show new bifurcations in the context of hydrodynamic waves even at very low amplitudes. The secondary bifurcations are explained by a spatial analogue of the superharmonic instability. An interesting analogy is also found between the spatial bifurcations of travelling waves and the Kelvin–Helmholtz instability. The new branches of waves occur at physically realizable values of the parameters and therefore could have implications for interfacial waves in applications.