Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers

Daniel Bouche; Youngjoon Hong; Chang-Yeol Jung

doi:10.3934/dcds.2017048

Article Contents

2017, Volume 37, Issue 3: 1159-1181. Doi: 10.3934/dcds.2017048

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Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers

1.
CMLA, ENS Cachan, CNRS, Universit, Paris-Saclay, 94235 Cachan, France
2.
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago 851 S Morgan St, Chicago, IL 60607, USA
3.
Department of Mathematical Sciences, School of Natural Science Ulsan National Institute of Science and Technology, UNIST-gil 50, Ulsan 689-798, Republic of Korea

* Corresponding author
* Corresponding author

Received: November 2015

Revised: October 2016

Published: May 2017

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Abstract

We study the asymptotic behavior of the two dimensional Helmholtz scattering problem with high wave numbers in an exterior domain, the exterior of a circle. We impose the Dirichlet boundary condition on the obstacle, which corresponds to an incidental wave. For the outer boundary, we consider the Sommerfeld conditions. Using a polar coordinates expansion, the problem is reduced to a sequence of Bessel equations. Investigating the Bessel equations mode by mode, we find that the solution of the scattering problem converges to its limit solution at a specific rate depending on k.

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Mathematics Subject Classification: 34E20, 35B40, 78A05, 78A45.

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Figure 1. Scattering by a circular object

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Figure 2. Sound-soft scattering from a circular object as in Figure 1 at (Top) $k=50$, (Bottom) $k=500$: (a) $\Re (u_{inc}^{(k)}) = \cos(ikx)$: incident wave traveling to the right, (b) $\Re u_s^{(k)}$: scattered wave, (c) $\Re u^{(k)} = \Re u^{(k)}_{inc} + \Re u^{(k)}_s$: total solution. This simulation is generated using the MPSPACK (see [6])

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