  In this paper, two-dimensional periodic capillary-gravity waves travelling
under the effect of a vertical electric field are considered. The full system
is a nonlinear, two-layered and free boundary problem. The interface dynamics
arises from the coupling between the Euler equations for the lower fluid layer
and an electric contribution from the upper gas layer. To investigate the
electrohydrodynamic wave interactions, we first introduce the naive flattening
technique to transform the free boundary problem into a fixed boundary problem.
Then we prove the existence of the small-amplitude electrohydrodynamic waves
with constant vorticity $\gamma$ by using local bifurcation theory. Moreover,
we prove that these electrohydrodynamic waves are formally stable in linearized
sense. Furthermore, we obtain a secondary bifurcation curve that emerges from
the primary branch at a nonlaminar solution as $E_0$ being close to some
special value. This secondary bifurcation curve consists of ripples solutions
on the interface of a conducting fluid under normal electric fields. As far as
we know, this new phenomenon in electrohydrodynamics (EHD) is first established
mathematically. It is worth noting that the electric field $E_0$ plays a key
role to control the shapes and types of waves on the interface.
