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

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 $\RR^3 \setminus \overline{Γ_1\cup Γ_2}$ 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 $\RR^3 \setminus \overline{Γ_1\cup Γ_2}$ 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.

18 pages, 2 figures