Analysis of Transient Acoustic-Elastic Interaction in an Unbounded Structure

TL;DR

This paper analyzes the time-domain acoustic-elastic interaction in an unbounded three-dimensional structure with a rough interface, establishing well-posedness, stability, and explicit a priori estimates for pressure and displacement.

Contribution

It introduces an exact transparent boundary condition and provides a rigorous mathematical analysis of the coupled wave equations in an unbounded domain.

Findings

Proved well-posedness and stability of the coupled problem.

Derived explicit a priori estimates with time dependence.

Established the mathematical foundation for acoustic-elastic interaction analysis.

Abstract

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.