  We consider two formally determined inverse problems for the wave equation in
more than one space dimension. Motivated by the fixed angle inverse scattering
problem, we show that a compactly supported potential is uniquely determined by
the far field pattern generated by plane waves coming from exactly two opposite
directions. This implies that a reflection symmetric potential is uniquely
determined by its fixed angle scattering data. We also prove a Lipschitz
stability estimate for an associated problem. Motivated by the point source
inverse problem in geophysics, we show that a compactly supported potential is
uniquely determined from boundary measurements of the waves generated by
exactly two sources - a point source and an incoming spherical wave. These
results are proved by using Carleman estimates and adapting the ideas
introduced by Bukhgeim and Klibanov on the use of Carleman estimates for
inverse problems.
