The dynamics of two-dimensional standing periodic waves at the interface between
two inviscid fluids with different densities, subject to monochromatic oscillations
normal to the unperturbed interface, is studied under normal- and low-gravity conditions.
The motion is simulated over an extended period of time, or up to the point where
the interface intersects itself or the curvature becomes very large, using two numerical
methods: a boundary-integral method that is applicable when the density of one
fluid is negligible compared to that of the other, and a vortex-sheet method that is
applicable to the more general case of arbitrary densities. The numerical procedure
for the boundary-integral formulation uses a global isoparametric parametrization
based on cubic splines, whereas the numerical method for the vortex-sheet formulation
uses a local boundary-element parametrization based on circular arcs. Viscous
dissipation is simulated by means of a phenomenological damping coefficient added
to the Bernoulli equation or to the evolution equation for the strength of the vortex
sheet. A comparative study reveals that the boundary-integral method is generally
more accurate for simulating the motion over an extended period of time, but the
vortex-sheet formulation is significantly more efficient. In the limit of small deformations,
the numerical results are in excellent agreement with those predicted by the
linear model expressed by Mathieu's equation, and are consistent with the predictions
of the Floquet stability analysis. Nonlinear effects for non-infinitesimal amplitudes
are manifested in several ways: deviation from the predictions of Mathieu's equation,
especially at the extremes of the interfacial oscillation; growth of harmonic waves with
wavenumbers in the unstable regimes of the Mathieu stability diagram; formation
of complex interfacial structures including paired travelling waves; entrainment and
mixing by ejection of droplets from one fluid into the other; and the temporal period
tripling observed recently by Jiang et al. (1998). Case studies show that the surface
tension, density ratio, and magnitude of forcing play a significant role in determining
the dynamics of the developing interfacial patterns.