EHD self-modulation of capillary-gravity waves on a liquid layer of uniform depth

The electrohydrodynamic self-modulational of capillary-gravity waves on the surface of a dielectric fluid layer of finite depth subjected to a tangential electric field is investigated by using the method of multiple scales. A nonlinear Schrödinger equation for the complex amplitude of quasi-monochromatic travelling waves is derived. The stability characteristics of a wave train are examined on the basis of the nonlinear Schrödinger equation. It is demonstrated, for the pure hydrodynamical case, that the capillary-gravity waves are modulationally stable for the wavenumbers and the liquid depths belonging to three stable regions. The introduction of the electric field has a stabilizing effect for small values of the wavenumber k in the first region; it does not have a significant effect in the second region; and it has a destabilizing effect in the third region. Higher values of the electric field, generate two new regions of stability and a new unstable region, however related to the previous ones. Further increasing the electric field decreases the first new stable region, while the second new stable region decreases and the new unstable region increases. Therefore, the effect of the electric field is different for the different regions of stability, and this effect is more strong if the dielectric constant of the upper fluid is less than the one of the lower fluid.