Wave propagation in an inhomogeneous acoustic medium may be modeled, for example, by the wave operators $\Box+q(x)$, $\rho(x)\partial_t^2-\Delta$ or $\partial_t^2-\Delta_g$, for a function $q(x)$, a positive function $\rho(x)$ or a Riemannian metric $g(x)$, which are homogeneous outside a ball. The medium is probed by plane waves coming from a finite number (dimension dependent) of different directions, and the resultant time dependent waves are measured on the boundary of the ball. We describe our partial results about the recovery of $q,\rho,g$ from these boundary measurements. These are long standing formally determined open problems. These results were obtained in collaboration with Lauri Oksanen and Mikko Salo.
