  Consider the wave propagation in a two-layered medium consisting of a
homogeneous compressible air or fluid on top of a homogeneous isotropic elastic
solid. The interface between the two layers is assumed to be an unbounded rough
surface. This paper concerns the time-domain analysis of such an
acoustic-elastic interaction problem in an unbounded structure in three
dimensions. Using an exact transparent boundary condition and suitable
interface conditions, we study an initial-boundary value problem for the
coupling of the Helmholtz equation and the Navier equation. The well-posedness
and stability are established for the reduced problem. Our proof is based on
the method of energy, the Lax--Milgram lemma, and the inversion theorem of the
Laplace transform. Moreover, a priori estimates with explicit dependence on the
time are achieved for the quantities of acoustic pressure and elastic
displacement by taking special test functions for the time-domain variational
problem.
