We study the effect of a time-periodic, lateral acceleration on the two-dimensional flow of a fluid with a free surface subject to surface tension, confined between two plane, parallel walls under conditions of zero gravity. We assume that the velocity of each contact line is a prescribed, single-valued function of the dynamic contact angle between fluid and solid at the wall. We begin by obtaining analytical solutions for the small-amplitude standing waves that evolve when this function is linear, the fluid is inviscid and the lateral acceleration is sufficiently small. This leads to damping of the motion, unless either the contact angles are fixed or the contact lines are pinned. In these cases, we include the effect on the flow of the wall boundary layers, which are the other major sources of damping. We then consider the weakly nonlinear solution of the inviscid problem when the contact angle is almost constant and the external forcing is close to resonance. This solution indicates the possibility of a hysteretic response to changes in the forcing frequency. Finally, we examine numerical solutions of the fully nonlinear, inviscid problem using a desingularized integral equation technique. We find that periodic solutions, chaotic solutions and solutions where the topology of the fluid changes, either through self-intersection or pinch off, are all possible.