# American Institute of Mathematical Sciences

March  2017, 37(3): 1159-1181. doi: 10.3934/dcds.2017048

## 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

Received  November 2015 Revised  October 2016 Published  December 2016

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.

Citation: Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048
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Scattering by a circular object
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])">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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