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.