Determining both the source of a wave and its speed in a medium from boundary measurements

We study the inverse problem of determining both the source of a wave and its speed inside a medium from measurements of the solution of the wave equation on the boundary. This problem arises in photoacoustic and thermoacoustic tomography, and has important applications in medical imaging. We prove that if the solutions of the wave equation with the source and sound speed $(f_1,c_1)$ and $(f_2,c_2)$ agree on the boundary of a bounded region $\Omega$, then $$\int_{\Omega}(c_2^{-2}-c_1^{-2})\varphi dy=0,$$ for every harmonic function $\varphi \in C(\bar{\Omega})$, which holds without any knowledge of the source. We also show that if the wave speed $c$ is known and only assumed to be bounded then, under a natural admissibility assumption, the source of the wave can be uniquely determined from boundary measurements.