Skip to main content

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

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.

This is a preview of subscription content, access via your institution.

References

  1. Ahmada B., Challab D.P., Kiranea M., Sini M.: The equivalent refraction index for the acoustic scattering by many small obstacles: With error estimates. J. Math. Anal. Appl. 424(1), 563–583 (2015)

    MathSciNet  Article  Google Scholar 

  2. Ammari H., Kang H.: Boundary layer techniques for solving the helmholtz equation in the presence of small inhomogeneities. J. Math. Anal. Appl. 296(1), 190–208 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  3. Ammari, H., Kang, H.: Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, vol. 1846. Springer, Berlin, 2004

  4. Ammari H., Khelifi A.: Electromagnetic scattering by small dielectric inhomogeneities. Journal de Mathématiques Pures et Appliquées 83(7), 749–842 (2002)

    MathSciNet  Google Scholar 

  5. Bartoli N., Bendali A.: Robust and high-order effective boundary conditions for perfectly conducting scatterers coated by a thin dielectric layer. IMA J. Appl. Math. 67(5), 479–508 (2002)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. Bendali A., Cocquet P.H., Tordeux S.: Scattering of a scalar time-harmonic wave by n small spheres by the method of matched asymptotic expansions. Numer. Anal. Appl. 5, 116–123 (2012)

    Article  Google Scholar 

  7. Bendali A., Fares M., Piot E., Tordeux S.: Mathematical justification of the Rayleigh cavity model with the method of matched asymptotic expansions. SIAM J. Appl. Math. 71(1), 438–459 (2013)

    MathSciNet  Article  Google Scholar 

  8. Bendali A., Lemrabet K.: The effect of a thin coating on the scattering of a time-harmonic wave for the helmholtz equation. SIAM J. Appl. Math. 6(5), 1664–1693 (1996)

    MathSciNet  Article  Google Scholar 

  9. Bendali A., Makhlouf A., Tordeux S.: Field behavior near the edge of a microstrip antenna by the method of matched asymptotic expansions. Q. Appl. Math. 69, 691–721 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  10. Caloz G., Costabel M., Dauge M., Vial G.: Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer. Asymp. Anal. 50(1/2), 121–173 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Cassier M., Hazard C.: Multiple scattering of acoustic wave by small sound-soft obstacles in two-dimensions: mathematical justification of the Foldy–Lax model. Wave Motion 50, 18–28 (2013)

    MathSciNet  Article  Google Scholar 

  12. Challa D.P., Sini M.: Inverse scattering by point-like scatterers in the Foldy regime. Inverse Problems 28(12), 1–39 (2012)

    MathSciNet  Article  Google Scholar 

  13. Challa, D.P., Sini, M.: On the justification of the foldy-lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes. SIAM Multiscale Model. Simul. 12(1), 58–108 (2014)

  14. Cocquet, P.H.: Etude mathématique et numérique de modèles homogénéisés de métamatériaux. Ph.D. thesis, University of Toulouse, 2012

  15. Costabel, M., Dauge, M.: A singularly perturbed mixed boundary value problem. Commun. Partial Differ. Equ. 21(11–12), 1667–1703 (1996)

  16. Cousteix, J., Mauss, J.: Asymptotic Analysis and Boundary Layers. Springer, New-York, 2007

  17. Dyke, M.V.: Perturbation Methods in Fluid Mechanics, annoted edition edn. The Parabolic Press, Stanford, California, 1975

  18. Eckhaus, W.: Matched Asymptotic Expansions and Singular Perturbations, North-Holland Mathematics Studies, vol. 6. North-Holland Publishing Company, Amsterdam, 1973

  19. Foldy L.: The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers. Phys. Rev. 67, 107–119 (1945)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. Gumerov, N.A., Duraiswamy, R.: Fast Multipole Method for the Helmholtz Equation in Three Dimensions. Elsevier, Amsterdam, 2004

  21. Hsiao, G.C., Wendland, W.L.: Boundary Iintegral Equations. Springer, Berlin, 2008

  22. Huang K., Li P.: A two-scale multiple scattering problem. SIAM Multiscale Model. Simul. 8(4), 1511–1534 (2010)

    Article  MATH  Google Scholar 

  23. Jacobsen, F., Juhl, P.M.: Fundamentals of General Linear Acoustics. Wiley, Chichester, 2013

  24. Joly, P., Tordeux, S.: Matching of Asymptotic Expansions for Wave Propagation in Media with Thin Slots i: The Asymptotic Expansion. Multiscale Model. Simul. A SIAM Interdiscipl. J. 5(1), 304–336 (2006)

  25. Laurens, S., Tordeux, S., Bendali, A., Fares, M., Kotiuga, P.R.: Lower and upper bounds for the rayleigh conductivity of a perforated plate. ESAIM: Math. Model. Numer. Anal. 47(4), 675–696 (2013)

  26. Liao, J., Ji, C.: Extended Foldy–Lax approximation on multiple scattering. Math. Model. Anal. 10(1), 85–98 (2014)

  27. Martin, P.A.: Multiple Scattering Intercation of Time-Harmonic Waves with N Obstacles. Cambridge University Press, Cambridge, 2006

  28. Maz’ya V.G., Poborchi S.V.: Extension of functions in sobolev spaces on parameter dependent domains. Mathematische Nachrichten 178(1), 5–41 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  29. Nédélec, J.C.: Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems. Springer, Berlin, 2001

  30. Olver, F.W., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. NIST U.S. Department of Commerce and Cambridge University Press, New-York, 2010

  31. Ramm, A.G.: Wave Scattering by Small Bodies of Arbitrary Shapes. World Scientific Publishing Co. Inc., Singapore, 2005

  32. Ramm, A.G.: Many-body wave scattering by small bodies and applications. J. Math. Phys. 48, 103,511–1–103,511–28 (2007)

  33. Ramm A.G.: Wave scattering by small bodies and creating materials with a desired refraction coefficient. Afr. Mat. 22, 33–55 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  34. Taylor, M.: Partial Differential Equations I, Basic Theory. Springer, New York, 1996

  35. Vogelius M.S., Volkov D.: Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter. Math. Model. Numer. Anal. 34(4), 723–748 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  36. Wang, Z.X., Guo, D.R.: Special Functions. World Scientific Publishing Co. Inc., Teaneck, New-Jersey, 1989

  37. Wilcox, C.H.: Scattering Theory for the d’Alembert Equation in Exterior Domains, vol. 442. Springer, Berlin, 1975

  38. Zhong Y., Chen X.: Music imaging and electromagnetic inverse scattering of multiple-scattering small anisotropic spheres. IEEE Trans. Antennas Propag. 55(12), 3542–3549 (2007)

    ADS  MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Abderrahmane Bendali.

Additional information

Communicated by C. Le Bris

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords

  • Asymptotic Expansion
  • Multiple Scattering
  • Small Body
  • Matching Function
  • Isotropic Scattering