Nonlinear EHD stability of the travelling and standing waves of two superposed dielectric bounded fluids in relative motion

The nonlinear modulation of the interfacial waves of two superposed dielectric fluids with uniform depths and rigid horizontal boundaries, under the influence of constant normal electric fields and uniform horizontal velocities, is investigated using the multiple-time scales method. It is found that the quasi-monochromatic travelling waves can be described by a nonlinear Schrödinger equation in a frame of reference moving with the group velocity. The stability of uniform and periodic solution of the nonlinear Schrödinger equation is tested by means of three-mode model, involving the interaction of a finite number of Fourier components. A set of coupled differential equations is obtained, describing the effect of sideband modulations. The Benjamin–Feir instability appears clearly related to the linearized technique, and is the first stage of the nonlinear recurrence phenomenon. On the other hand, the complex amplitude of quasi-monochromatic standing waves near the cut-off wavenumber is governed by a similar type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation makes it possible to estimate the nonlinear effect on the linear cut-off wavenumber.