Singular perturbation of reduced wave equation and scattering from an embedded obstacle

TL;DR

This paper analyzes how high-density regions in a medium affect wave scattering, showing that as density increases, the wave behaves as if encountering a sound-hard obstacle, with precise estimates and implications for inverse scattering.

Contribution

It provides a rigorous asymptotic analysis of wave fields in high-density regions, demonstrating their convergence to obstacle-like behavior and deriving sharp estimates for inverse scattering applications.

Findings

Wave field inside high-density region decays exponentially.

Outside wave field converges to that of a sound-hard obstacle.

Normal velocity on the obstacle boundary vanishes as density increases.

Abstract

We consider time-harmonic wave scattering from an inhomogeneous isotropic medium supported in a bounded domain $\Omega\subset\mathbb{R}^N$ ($N\geq 2$). {In a subregion $D\Subset\Omega$, the medium is supposed to be lossy and have a large mass density. We study the asymptotic development of the wave field as the mass density $\rho\rightarrow +\infty$} and show that the wave field inside $D$ will decay exponentially while the wave filed outside the medium will converge to the one corresponding to a sound-hard obstacle $D\Subset\Omega$ buried in the medium supported in $\Omega\backslash\bar{D}$. Moreover, the normal velocity of the wave field on $\partial D$ from outside $D$ is shown to be vanishing as $\rho\rightarrow +\infty$. {We derive very accurate estimates for the wave field inside and outside $D$ and on $\partial D$ in terms of $\rho$, and show that the asymptotic estimates are sharp. The implication of the obtained results is given for an inverse scattering problem of reconstructing a complex scatterer.}