Inverse Problems for Obstacles in a Waveguide

Abstract

In this paper we prove that a particular entry in the scattering matrix, if known for all energies, determines certain rotationally symmetric obstacles in a generalized waveguide. The generalized waveguide X can be of any dimension and we allow either Dirichlet or Neumann boundary conditions on the boundary of the obstacle and on ∂X.In the case of a two-dimensional waveguide, two particular entries of the scattering matrix suffice to determine the obstacle, without the requirement of symmetry.

The semiclassical structure of the scattering matrix for a manifold with infinite cylindrical end

“…First suppose 1 ∈ spec(h 2 ∆ Y ). We use the Poisson operator P as constructed in (13). Let f ∈ H Y , and denote the outgoing data in Pf by f + .…”

“…A relatively short self-contained introduction may also be found in [12,Section 2]. The papers [9,13] use a detailed microlocal analysis of the scattering matrix applied to a specific function in an inverse problem.…”

On the use of Lamb modes in the linear sampling method for elastic waveguides

“…When a finite number of modes are taken into account, the data may be viewed as a scattering matrix. It should also be noted that some recent communications have considered inverse scattering problems in acoustic waveguide from other points of view than LSM, for example [12,15,16]. The main ingredient of the modal formulation in [8] is the expansion of the acoustic field with respect to the transverse modes in the bounded section of the waveguide.…”

“…When a finite number of modes are taken into account, data may be viewed as a scattering matrix. It should also be noted that some recent communications have considered inverse scattering problems in acoustic waveguide from other points of view than the LSM, for example [12,15,16].…”