This paper is concerned with the mathematical analysis of a time-dependent fluid-solid interaction problem associated with a bounded elastic body immersed in a homogeneous air or fluid above a local rough surface. We reformulate the unbounded scattering problem into an equivalent initial-boundary value problem defined in a bounded domain by proposing a transparent boundary condition(TBC) on a hemisphere. Analyzing the reduced problem with the Lax-Milgram lemma and the abstract inversion theorem of the Laplace transform,we prove the well-posedness and stability for the reduced problem. Moreover, an a priori estimate is established directly in the time domain for the acoustic wave and elastic displacement by using the energy method.