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Approximation by Multipoles of the Multiple Acoustic Scattering by Small Obstacles in Three Dimensions and Application to the Foldy Theory of Isotropic Scattering

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Abstract

The asymptotic analysis carried out in this paper for the problem of a multiple scattering in three dimensions of a time-harmonic wave by obstacles whose size is small as compared with the wavelength establishes that the effect of the small bodies can be approximated at any order of accuracy by the field radiated by point sources. Among other issues, this asymptotic expansion of the wave furnishes a mathematical justification with optimal error estimates of Foldy’s method that consists in approximating each small obstacle by a point isotropic scatterer. Finally, it is shown how this theory can be further improved by adequately locating the center of phase of the point scatterers and the taking into account of self-interactions. In this way, it is established that the usual Foldy model may lead to an approximation whose asymptotic behavior is the same than that obtained when the multiple scattering effects are completely neglected.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, UMR5219, University of Toulouse, CNRS, INSA, 135 avenue de Rangueil, 310770, Toulouse Cedex 01, France

    Abderrahmane Bendali

  2. CERFACS, 42 Avenue Gaspard Coriolis, 31057, Toulouse Cedex 01, France

    Abderrahmane Bendali

  3. PIMENT, Physique et Ingénierie Mathématique pour l’Énergie, l’environnemeNt et le bâtimenT, 2 Rue Joseph Wetzell, Sainte-Clotilde, 97490, La Réunion

    Pierre-Henri Cocquet

  4. Project-Team Magique 3D, INRIA and University of Pau, Avenue de l’Université, BP 1155, 64013, Pau, France

    Sébastien Tordeux

Authors
  1. Abderrahmane Bendali
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  2. Pierre-Henri Cocquet
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  3. Sébastien Tordeux
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Correspondence to Abderrahmane Bendali.

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Communicated by C. Le Bris

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Bendali, A., Cocquet, PH. & Tordeux, S. Approximation by Multipoles of the Multiple Acoustic Scattering by Small Obstacles in Three Dimensions and Application to the Foldy Theory of Isotropic Scattering. Arch Rational Mech Anal 219, 1017–1059 (2016). https://doi.org/10.1007/s00205-015-0915-5

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  • DOI: https://doi.org/10.1007/s00205-015-0915-5

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