Uniformly valid expansion for finite amplitude surface waves in electrohydrodynamics in a fluid of infinite depth

A weakly nonlinear theory of capillary gravity waves on the surface of a fluid of infinite depth in the presence of an electric field is investigated. We have used slow scales perpendicular to the free surface in addition to the usual slow distance and time scales to obtain uniformly valid perturbation solutions. The equations governing the evolution of the amplitude are obtained for the progressive as well as standing waves. The stability analysis for progressive waves reveals that the wave train of constant amplitude is unstable against modulations. It is interesting to observe that there exist three nonlinear stable as well as unstable regions. We also obtain the nonlinear cut-off wave number which separates the region bf stability from that of instability. We find that the electric field has a destabilizing influence on the cut-off wave number.