  This paper is concerned with the time-dependent acoustic-elastic interaction
problem associated with a bounded elastic body immersed in a homogeneous air or
fluid above an unbounded rough surface. The well-posedness and stability of the
problem are first established by using the Laplace transform and the energy
method. A perfectly matched layer (PML) is then introduced to truncate the
interaction problem above a finite layer containing the elastic body, leading
to a PML problem in a finite strip domain. We further establish the existence,
uniqueness and stability estimate of solutions to the PML problem. Finally, we
prove the exponential convergence of the PML problem in terms of the thickness
and parameter of the PML layer, based on establishing an error estimate between
the DtN operators of the original problem and the PML problem.
