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Transmission Conditions for the Helmholtz-Equation in Perforated Domains

Abstract

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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Acknowledgments

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). https://doi.org/10.1007/s10013-016-0222-y

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Keywords

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

Mathematics Subject Classification (2010)

  • 35B27
  • 74Q05