Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions

Andrea Moiola, Euan Spence

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We consider the Helmholtz transmission problem with one penetrable star-shaped Lipschitz obstacle. Under a natural assumption about the ratio of the wavenumbers, we prove bounds on the solution in terms of the data, with these bounds explicit in all parameters. In particular, the (weighted) H1 norm of the solution is bounded by the L2 norm of the source term, independently of the wavenumber. These bounds then imply the existence of a resonance-free strip beneath the real axis. The main novelty is that the only comparable results currently in the literature are for smooth, convex obstacles with strictly positive curvature, while here we assume only Lipschitz regularity and star-shapedness with respect to a point. Furthermore, our bounds are obtained using identities first introduced by Morawetz (essentially integration by parts), whereas the existing bounds use the much-more sophisticated technology of microlocal analysis and propagation of singularities. We also adapt existing results to show that if the assumption on the wavenumbers is lifted, then no bound with polynomial dependence on the wavenumber is possible.

LanguageEnglish
Pages317-354
Number of pages38
JournalMathematical Models & Methods in Applied Sciences
Volume29
Issue number2
Early online date18 Jan 2019
DOIs
StatusPublished - 28 Feb 2019

Fingerprint

Transmission Problem
Explicit Bounds
Stars
Acoustics
Positive Curvature
Hermann Von Helmholtz
Strictly positive
Strip
Lipschitz
Star
Imply

Keywords

  • Helmholtz equation
  • Lipschitz domain
  • Morawetz identity
  • Transmission problem
  • acoustic
  • frequency explicit
  • resonance
  • semiclassical
  • wavenumber explicit

ASJC Scopus subject areas

  • Modelling and Simulation
  • Applied Mathematics

Cite this

Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions. / Moiola, Andrea; Spence, Euan.

In: Mathematical Models & Methods in Applied Sciences, Vol. 29, No. 2, 28.02.2019, p. 317-354.

Research output: Contribution to journalArticle

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