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May  2009, 3(2): 211-229. doi: 10.3934/ipi.2009.3.211

The inverse acoustic obstacle scattering problem and its interior dual

Brian Sleeman 1,

1. 

School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom

Received  November 2008 Revised  March 2009 Published  May 2009

This paper addresses possible connections between two classical inverse problems arising in wave propagation. The first is the problem of extracting geometrical information about an unknown bounded domain from a knowledge its eigen-frequencies. The chosen method of investigation being the high frequency asymptotics of the associated counting function. The second problem is the inverse obstacle scattering problem. That is the determination of an unknown obstacle from far field data. This problem is investigated through the high frequency asymptotics of the associated scattering phase. It turns out that there is a remarkable similarity between the asymptotic expansions in each of these problems. We discuss a number of ideas and techniques along the way including representations of the scattering matrix and the Kirchoff approximation. We also show how to solve scattering problems for polygonal obstacles. Whether there is a deep physical connection between interior and exterior scattering problems remains a challenging area of research.
Keywords: Eigenvalue problem, acoustic obstacle scattering, isospectrality, scattering matrix, scattering phase..
Mathematics Subject Classification: Primary:35P20, 78A46;Secondary 34E20, 78A45, 81U2.
Citation: Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems & Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211
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