No abstract

This paper is concerned with the scattering problem of time-harmonic acoustic plane waves by an impenetrable obstacle buried in a piecewise homogeneous medium. The so-called generalized impedance boundary condition is imposed on the boundary of the obstacle. Firstly, the well posedness of the solution to the direct scattering problem is established by using the boundary integral method. Then a uniqueness result for the inverse scattering problem is proved; that is, both of the obstacle’s shape and the impedances (μ,λ) can be uniquely determined from far field measurements. Furthermore, a mathematical basis is given to reconstruct the shape of the obstacle by using a modified linear sampling method.

The inverse scattering problem of determining a crack outside a known penetrable cavity from the measurements and point sources placed in a closed curve Λ inside the cavity is considered. The factorization method is used to reconstruct the crack on which the mixed-type boundary conditions are posed. To this end, we need to overcome two key difficulties. One of them is to find the relationship between the two data-to-data operators, which map the boundary data on the crack to the values of scattered fields on the closed curve Λ with outgoing radiation condition and incoming radiation condition, respectively. In addition, it is hard to decompose the near field operator in the usual way owing to the mixed-type boundary conditions. By the aid of an auxiliary operator, we can obtain the desired decomposition form in accordance with the theoretical framework of the factorization method.

No abstract

This paper is concerned with the scattering problem of time-harmonic acoustic plane waves by a union of a crack and a penetrable inhomogeneous medium with compact support. The well-posedness of the direct problem is established by the variational method. An uniqueness result for the inverse problem is proved, that is, both the crack and the inhomogeneous medium can be uniquely determined by a knowledge of the far-field pattern for incident plane waves. The linear sampling method is employed to recover the location and shape of the combined scatterer. It is worth noting that we make the first step on reconstructing a mixed-type scatterer of a crack and an inhomogeneous medium by the linear sampling method.