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On the scattering of two-dimensional elastic point sources and related near-field inverse problems for small discs

Published online by Cambridge University Press:  08 July 2009

C. E. Athanasiadis ,
G. Pelekanos ,
V. Sevroglou  and
I. G. Stratis
C. E. Athanasiadis
Affiliation:
Department of Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis, 15784 Zographou, Greece (cathan@math.uoa.gr)
G. Pelekanos
Affiliation:
Department of Mathematics and Statistics, Southern Illinois University, Edwardsville, IL 62026, USA (gpeleka@siue.edu)
V. Sevroglou
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, 18534 Piraeus, Greece (bsevro@unipi.gr)
I. G. Stratis
Affiliation:
Department of Mathematics, National and Kapodistrian University of Athens, Panepistimiopolis, 15784 Zographou, Greece (istratis@math.uoa.gr)
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Abstract

The problem of scattering of a point-generated elastic dyadic field by a bounded obstacle or a penetrable body in two dimensions is considered. The direct scattering problem for each case is formulated in a dyadic form. For two point sources, dyadic far-field pattern generators are defined and general scattering theorems and mixed scattering relations are presented. The direct scattering problem for a rigid circular disc is considered, and the exact Green function and the elastic far-field patterns of the radiating solution in the form of infinite series are obtained. Under the low-frequency assumption, approximations for the longitudinal and transverse far-field patterns of the scattered field are obtained, in addition to an asymptotic expansion for the corresponding scattering cross-section. A simple inversion scheme that locates the radius and the position of a rigid circular disc, which is based on a closed-form approximation of the scattered field at the location of the incident point source, is proposed.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 139 , Issue 4 , August 2009 , pp. 719 - 741
Copyright
Copyright © Royal Society of Edinburgh 2009

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