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

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 ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ under the assumption that ${\text{RR}}^3\setminus \overline{{\mathrm{\Gamma }}_1\cup {\mathrm{\Gamma }}_2}$ is connected. If that is not the case we provide counterexamples that demonstrate non-uniqueness, even if ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are smooth and “almost” flat. Finally, we show in the case where ${\text{RR}}^3\setminus \overline{{\mathrm{\Gamma }}_1\cup {\mathrm{\Gamma }}_2}$ is not necessarily connected that after excluding a discrete set of frequencies, ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ can again be differentiated from overdetermined boundary data.