Abstract:

We study the wave equation on a bounded domain Rm or on a compact Riemannian manifold with boundary. Let us assume that we do not know the coefficients of the wave equation but are only given the hyperbolic Neumann-to-Dirichlet map that corresponds to physical measurements on a part of the boundary. We show that it is possible to construct a sequence of Neumann boundary values so that at a time t0 the corresponding waves converge to zero while the time derivative of the waves converge to a delta distribution. A key feature of this result is that it does not require knowledge of the coefficients in the wave equation, that is, of the material parameters inside the media. However, we assume that the point where the energy of wave focuses is known in travel time coordinates, and satisfies a certain geometrical condition. This is joint work with Anna Kirpichnikova, Matti Lassas and Lauri Oksanen.