Stability of a Horizontal Fluid Interface in a Periodic Vertical Electric Field

The stability of the interface in the presence of a periodic electric field is considered. It is shown that the stability is governed by a Mathieu equation, that the interface can be unstable even if the electric field is at all times weaker than that needed for instability in the case of a steady field, and that, when instability occurs, the waves may either be synchronous with the electric field, or have twice its frequency.