A stability analysis for interfacial waves using a Zakharov equation

Publisher Website
Google Scholar
Abstract
An amplitude equation for weak interactions of waves is derived to describe the time evolution of disturbances on an interface between fluids of differing densities, with rigid upper and lower boundaries. This equation is analogous to the Zakharov equation for water waves and is used to investigate the stability of a periodic wave. It is found that, for small wave steepnesses, the instabilities are due to resonant quartets with perturbation wavenumbers of the order of, or less than, that of the main wave. A second instability is found for large perturbation wavenumbers and moderately high wave steepnesses. This is restricted to the case when the Boussinesq parameter is small. It is shown that this is a Kelvin–Helmholtz instability caused by a wave-induced jump in the fluid velocity across the interface.