  Consider the scattering of an acoustic plane wave by a bounded elastic
obstacle which is immersed in an open space filled with a homogeneous medium.
This paper concerns the mathematical analysis of the coupled two- and
three-dimensional acoustic-elastic wave propagation problem in the time-domain.
A compressed coordinate transformation is proposed to reduce equivalently the
scattering problem into an initial-boundary value problem in a bounded domain
over a finite time interval. The reduced problem is shown to have a unique weak
solution by using the Galerkin method. The stability estimate and an a priori
estimate with explicit time dependence are obtained for the weak solution. The
reduced model problem is suitable for numerical simulations. The proposed
method is applicable to many other time-domain scattering problems imposed in
open domains.
