Equilibrium surface configurations of a conducting liquid in external spatially periodic electric field

The problem of possible equilibrium configurations of the free surface of a conducting liquid in an external spatially periodic electric field is considered. It is shown that, under identical conditions, the problem admits solutions with different wavelengths, including the wavelength corresponding to the field period and the half shorter wavelength corresponding to the period of the electrostatic pressure oscillations. In the former case, an exact partial solution is found that describes a surface profile with allowance for the capillary forces.