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We consider boundary measurements for the wave equation on a bounded domain$M \subset \R^2$ or on a compact Riemannian surface, and introduce a method tolocate a discontinuity in the wave speed. Assuming that the wave speed consistof an inclusion in a known smooth background, the method can determine thedistance from any boundary point to the inclusion. In the case of a knownconstant background wave speed, the method reconstructs a set contained in theconvex hull of the inclusion and containing the inclusion. Even if thebackground wave speed is unknown, the method can reconstruct the distance fromeach boundary point to the inclusion assuming that the Riemannian metric tensordetermined by the wave speed gives simple geometry in $M$. The method is basedon reconstruction of volumes of domains of influence by solving a sequence oflinear equations. For $\tau \in C(\p M)$ the domain of influence $M(\tau)$ isthe set of those points on the manifold from which the distance to someboundary point $x$ is less than $\tau(x)$.

Inverse obstacle problem for the non-stationary wave equation with an unknown background