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We study the inverse problem of determining both the source of a wave and itsspeed inside a medium from measurements of the solution of the wave equation onthe boundary. This problem arises in photoacoustic and thermoacoustictomography, and has important applications in medical imaging. We prove that ifthe 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 harmonicfunction $\varphi \in C(\bar{\Omega})$, which holds without any knowledge ofthe source. We also show that if the wave speed $c$ is known and only assumedto be bounded then, under a natural admissibility assumption, the source of thewave can be uniquely determined from boundary measurements.

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