On the crack inverse problem for pressure waves in half-space

TL;DR

This paper investigates the inverse problem of identifying cracks in a half-space from pressure wave data, establishing conditions for uniqueness and demonstrating non-uniqueness in certain cases.

Contribution

It introduces a method to differentiate between two cracks based on boundary data, and analyzes conditions affecting the uniqueness of the inverse problem.

Findings

Uniqueness in crack differentiation when the complement is connected.

Counterexamples showing non-uniqueness with smooth, flat cracks.

Frequency exclusion can restore uniqueness in certain cases.

Abstract

After formulating the pressure wave equation in half-space minus a crack with a zero Neumann condition on the top plane, we introduce a related inverse problem. That inverse problem consists of identifying the crack and the unknown forcing term on that crack from overdetermined boundary data on a relatively open set of the top plane. This inverse problem is not uniquely solvable unless some additional assumption is made. However, we show that we can differentiate two cracks $\Gamma_1$ and $\Gamma_2$ under the assumption that $\RR^3 \setminus \overline{\Gamma_1\cup \Gamma_2}$ is connected. If that is not the case we provide counterexamples that demonstrate non-uniqueness, even if $\Gamma_1$ and $\Gamma_2$ are smooth and "almost" flat. Finally, we show in the case where $\RR^3 \setminus \overline{\Gamma_1\cup \Gamma_2}$ is not necessarily connected that after excluding a discrete set of frequencies, $\Gamma_1$ and $\Gamma_2$ can again be differentiated from overdetermined boundary data.