  We consider a layer of an inviscid fluid with free surface which is subject
to vertical high-frequency vibrations. We derive three asymptotic systems of
equations that describe slowly evolving (in comparison with the vibration
frequency) free-surface waves. The first set of equations is obtained without
assuming that the waves are long. These equations are as difficult to solve as
the exact equations for irrotational water waves in a non-vibrating fluid. The
other two models describe long waves. These models are obtained under two
different assumptions about the amplitude of the vibration. Surprisingly, the
governing equations have exactly the same form in both cases (up to
interpretation of some constants). These equations reduce to the standard
dispersionless shallow-water equations if the vibration is absent, and the
vibration manifests itself via an additional term which makes the equations
dispersive and, for small-amplitude waves, is similar to the term that would
appear if surface tension were taken into account. We show that our dispersive
shallow water equations have both solitary and periodic travelling waves
solutions and discuss an analogy between these solutions and travelling
capillary-gravity waves in a non-vibrating fluid.
