Time-Domain Analysis of an Acoustic-Elastic Interaction Problem

Consider the scattering of a time-domain acoustic plane wave by a bounded elastic obstacle which is immersed in a homogeneous air or fluid. This paper concerns the mathematical analysis of such a time-domain acoustic-elastic interaction problem. An exact transparent boundary condition (TBC) is developed to reduce the scattering problem from an open domain into an initial-boundary value problem in a bounded domain. The well-posedness and stability are established for the reduced problem. A priori estimates with explicit time dependence are achieved for the pressure of the acoustic wave field and the displacement of the elastic wave field. Our proof is based on the method of energy, the Lax-Milgram lemma, and the inversion theorem of the Laplace transform. In addition, a time-domain absorbing perfectly matched layer (PML) method is introduced to replace the nonlocal TBC by a Dirichlet boundary condition. A first order symmetric hyperbolic system is derived for the truncated PML problem. The well-posedness and stability are proved. The time-domain PML results are expected to be useful in the computational air/fluid-solid interaction problems.