  This paper concerns the dynamics of a layer of incompressible viscous fluid
lying above a vertically oscillating rigid plane and with an upper boundary
given by a free surface. We consider the problem with gravity and surface
tension for horizontally periodic flows. This problem gives rise to flat but
vertically oscillating equilibrium solutions, and the main thrust of this paper
is to study the asymptotic stability of these equilibria in certain parameter
regimes. We prove that both with and without surface tension there exists a
parameter regime in which sufficiently small perturbations of the equilibrium
at time $t = 0$ give rise to global-in-time solutions that decay to equilibrium
at an identified quantitative rate.
