Abstract
The question of whether there exists an approximation procedure to compute the resonances of any Helmholtz resonator, regardless of its particular shape, is addressed. A positive answer is given, and it is shown that all that one has to assume is that the resonator chamber is bounded and that its boundary is {{mathcal {C}}}^2. The proof is constructive, providing a universal algorithm which only needs to access the values of the characteristic function of the chamber at any requested point.
Highlights
IntroductionThis paper provides an affirmative answer to the following question: Communicated by Arieh Iserles
The study of cavity resonances is much older than the study of quantum mechanical resonances
The foundational work is generally ascribed to Helmholtz [23], who in the 1850s had constructed devices which were designed to identify special frequencies from within a sound wave
Summary
This paper provides an affirmative answer to the following question: Communicated by Arieh Iserles. The authors thank the anonymous referees for their insightful comments which helped improve the presentation of this paper. The most famous example is the “sound of the sea” in a seashell: when we hold a seashell against our ear we hear frequencies that are nearly the eigenvalues of the Laplacian in the closed cavity with Neumann boundary conditions. This is known as a Helmholtz resonator [23].
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