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Transmission Conditions for the Helmholtz-Equation in Perforated Domains


We study the Helmholtz equation in a perforated domain Ω ε . The domain Ω ε is obtained from an open set \(\phantom {\dot {i}\!}{\Omega }\subset \mathbb {R}^{3}\) by removing small obstacles of typical size ε>0, the obstacles are located along a 2-dimensional manifold Γ0⊂Ω. We derive effective transmission conditions across Γ0 that characterize solutions in the limit ε→0. We obtain that, to leading order O(ε 0), the perforation is invisible. On the other hand, at order O(ε 1), inhomogeneous jump conditions for the pressure and the flux appear. The form of the jump conditions is derived.

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  1. Bonnet-Ben Dhia, A. S., Drissi, D., Gmati, N.: Mathematical analysis of the acoustic diffraction by a muffler containing perforated ducts. Math. Models Methods Appl. Sci. 15, 1059–1090 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  2. Bouchitté, G., Schweizer, B.: Homogenization of Maxwell’s equations in a split ring geometry. Multiscale Model Simul. 8, 717–750 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  3. Cioranescu, D., Damlamian, A., Griso, G., Onofrei, D.: The periodic unfolding method for perforated domains and Neumann sieve models. J. Math. Pures Appl. 89, 248–277 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  4. Cioranescu, D., Saint Jean Paulin, J.: Homogenization of Reticulated Structures Applied Mathematical Sciences, vol. 136. Springer-Verlag, New York (1999)

  5. Claeys, X., Delourme, B.: High order asymptotics for wave propagation across thin periodic interfaces. Asymptot. Anal. 83, 35–82 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Conca, C.: ÉTude d’un fluide traversant une paroi perforée I. Comportement limite prè,s de la paroi. J. Math. Pures Appl. 66, 1–43 (1987)

    MATH  Google Scholar 

  7. Kirby, R., Cummings, A.: The impedance of perforated plates subjected to grazing gas flow and backed by porous media. J. Sound Vib. 217, 619–636 (1998)

    Article  Google Scholar 

  8. Lamacz, A., Schweizer, B.: Effective Maxwell equations in a geometry with flat rings of arbitrary shape. SIAM J. Math. Anal. 45, 1460–1494 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  9. Lobo, M., Oleinik, O. A., Perez, M. E., Shaposhnikova, T. A.: On homogenization of solutions of boundary value problems in domains, perforated along manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. Ser. 4(25), 611–629 (1997)

    MathSciNet  MATH  Google Scholar 

  10. Neuss, N., Neuss-Radu, M., Mikelić, A.: Effective laws for the Poisson equation on domains with curved oscillating boundaries. Appl. Anal. 85, 479–502 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  11. Neuss-Radu, M., Jäger, W.: Effective transmission conditions for reaction-diffusion processes in domains separated by an interface. SIAM J. Math. Anal. 39, 687–720 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  12. Neuss-Radu, M., Ludwig, S., Jäger, W.: Multiscale analysis and simulation of a reaction-diffusion problem with transmission conditions. Nonlinear Anal. Real World Appl. 11, 4572–4585 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  13. Rohan, E., Lukeš, V.: Homogenization of the acoustic transmission through a perforated layer. J. Comput. Appl. Math. 234, 1876–1885 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  14. Sanchez-Hubert, J., Sánchez-Palencia, E.: Acoustic fluid flow through holes and permeability of perforated walls. J. Math. Anal. Appl. 87, 427–453 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  15. Schweizer, B.: The low-frequency spectrum of small Helmholtz resonators. Proc. R. Soc. A 471, 20140339 (2015). 18 pages

    MathSciNet  Article  Google Scholar 

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Support by the German Science foundation under grants DFG SCHW 639/5-1 and SCHW 639/6-1 are gratefully acknowledged.

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Correspondence to Ben Schweizer.

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This article is dedicated to Professor Willi Jäger on his 75th birthday.

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Dörlemann, C., Heida, M. & Schweizer, B. Transmission Conditions for the Helmholtz-Equation in Perforated Domains. Vietnam J. Math. 45, 241–253 (2017).

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  • Helmholtz equation
  • Perforated domain
  • Transmission conditions
  • Neumann sieve
  • Acoustics

Mathematics Subject Classification (2010)

  • 35B27
  • 74Q05