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

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  • 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.

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


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  • Figure 1.  Scattering by a circular object

    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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