We consider (in two-dimensional Euclidean space) the scattering of a plane, time-harmonic acoustic wave by an inhomogeneous medium (OMEGA) with compact support and a bounded obstacle D lying completely outside of the medium. The velocity potential u of the total field satisfies

{(DELTA)(,2) + k('2)n(x)}u(x) = 0,

in the exterior of D and a Neumann boundary condition on the boundary of D. The Sommerfield radiation condition at infinity is imposed on the velocity potential of the scattered wave.

We first show the existence and uniqueness of solutions to the direct scattering problem, using the methods of potential theory. We then examine a model inverse scattering problem. It is shown that from a knowledge of the low frequency asymptotic behavior of the scattered wave (i.e., the far field), one can determine the shape of the obstacle in the presence of an unknown inhomogeneous medium. This is done using integral equation methods and conformal mapping techniques to arrive at a generalized moment problem. By assuming a priori that the functions which determine the shape of the obstacle and the local speed of sound in the inhomogeneous medium lie in given compact sets, we show that the inverse problem of determining both the shape of D and the local speed of sound in (OMEGA) is stable, in the sense that the solution of the inverse scattering problem depends continuously on the far field data. A constrained optimization problem is formulated for this purpose from which we may construct both the shape of D and the local speed of sound in (OMEGA).

Finally, we examine an inverse scattering problem for the case of an obstacle with an impedance boundary condition imposed on its surface. Here we give a constructive method for determining both the shape of the obstacle and the surface impedance from low frequency measurements of the far field patterns corresponding to three different incident waves. As before, this is done using integral equation methods and conformal mapping techniques to arrive at a generalized moment problem for the determination of the unknown shape. We examine a Fredholm integral equation of the first kind for the determination of the surface impedance. By restricting a priori the impedance to lie in a certain compact set and by considering a related constrained optimization problem, we show that the problem of determining the impedance is well-posed.