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IMA Journal of Applied Mathematics Advance Access published online on September 20, 2006
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxl019
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© The Author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Received September 16, 2005
Accepted August 16, 2006

Article

Scattering by a perfect conductor in a waveguide: energy-preserving schemes for integral equations

D. Volkov 1 * and G. A. Kriegsmann 2

1 Mathematical Sciences Department, Worcester Polytechnic Institute,100 Institute Road, Worcester, MA 01609, USA
2 Department of Mathematical Sciences, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA

* To whom correspondence should be addressed.
D. Volkov, E-mail: darkovolkov{at}yahoo.ca; darko@wpi.edu


  Abstract

The scattering matrix for a perfectly conducting electrical cylinder (or a sound hard obstacle) in a waveguide is unitary. This is a well-known result which is a consequence of the conservation of power. When a numerical method is employed to approximate the reflection and transmission coefficients of the cylinder, an approximate scattering matrix can be constructed. An integral equation of the second kind for an unknown density can be solved, and the density can then be used for computing the entries of the approximate scattering matrix. We show that this approximate matrix is unitary for cylinders of symmetric cross-section, regardless of the order of the approximation. In the nonsymmetric case, the approximate scattering matrix still satisfies a conservation of energy condition, albeit in an unfamiliar form. As the order of the approximation is increased, conservation of energy is also satisfied in the more familiar form to machine accuracy.

Keywords: guided waves; scattering matrix; integral equations; continuous and discrete preservation of energy.
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