In this paper, we consider the time-domain scattering problem of a two-dimensional overfilled cavity embedded in the infinite ground plane. The problem is first discretized in time by the $\beta,\ \gamma$ Newmark time-marching scheme. At each time step, the variational formulation of the semidiscrete problem is derived via a nonlocal boundary condition to truncate the infinite problem domain. Existence and uniqueness of the variational solutions are established. Error analysis of the fully discrete problem is performed. Stability criteria of the time-stepping scheme are also obtained.