Eigenmodes of the coupled two - dimensional Wigner - crystal — liquid - surface system and instability of a charged liquid surface

The coupled phonon —surface-wave modes of a two-dimensional Wigner crystal resting on the surface of a liquid are studied. It is shown that, in experimentally accessible situations, the threshold for the formation of the macroscopic "dimple crystal" predicted by Gor'kov and Chernikova is negligibly influenced by the presence of the Wigner crystal, and that the coupled phonon —short-wavelength-surface waves studied recently by Fisher, Halperin, and Platzman, are negligibly influenced by the soft mode which leads to the "dimple crystal".