ABSTRACT
We consider the inverse problem of reconstructing the wavespeed in a one-dimensional nonhomogeneous medium from appropriate scattering data. The wavespeed is allowed to have jump discontinuities and the medium may be subject to a nonhomogeneous external restoring force. In the frequency-domain this inverse problem leads to a Riemann-Hilbert problem and an associated singular integral equation. Under suitable conditions we prove that the singular integral equation is uniquely solvable and we discuss how its solution leads to the recovery of the wavespeed. We also show that certain characteristic properties of the wavespeed can be reconstructed more quickly, that is, without completely solving the inverse problem first. Some examples illustrating the reconstruction of the wavespeed are presented.
