The coupling between ripplons and the motion of ion pools trapped beneath the surface of 4He

It is well known that the motion of positive or negative ions close to the surface of superfluid He-4 can excite ripplons in the helium surface. We consider a simple toy model in which a lattice of point charges couples to the 4He surface via electrostatic forces. We calculate the coupled mode spectrum of this system and present detailed information on the mode splitting which occurs when the ripplons and the normal modes of the ion pool are degenerate. Such energy gaps should, we believe, be experimentally observable.