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Green's Function for the Damped Wave Equation on a Finite Interval Subject to Two Robin Boundary Conditions

Published online by Cambridge University Press:  05 May 2011

David H. Y. Yen
David H. Y. Yen*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, U.S.A.
*
*Professor Emeritus
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Abstract

The Green's function for the damped wave equation on a finite interval subject to two Robin boundary conditions is studied. The problem for a semi-infinite interval with one Robin boundary is considered first by the Laplace transform method to establish the reflection principle at a Robin boundary. It is seen that the reflected wave generated by an exiting wave at a Robin boundary is a convolution involving the exiting wave and some kernel function. This generalizes the well known classical results for reflections at Dirichlet and Neumann boundaries. The reflection principle at a Robin boundary is then used for the finite interval case by considering multiple reflections and this leads to an infinite sequence of waves. In order to justify the results obtained here we also study the finite interval case by the Laplace transform method. The Laplace transform of the Green's function is expanded, for large values of the transform parameter, into an infinite series of negative exponentials and then inverted term by term. Agreement is reached between these two approaches.

Keywords

Damped wave equationGreen's functionRobin boundary conditions.
Type
Articles
Information
Journal of Mechanics , Volume 19 , Issue 1 , March 2003 , pp. 257 - 262
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2003

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References

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