How opening a hole affects the sound of a flute

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Romain Joly

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How opening a hole affects the sound of a flute A one-dimensional mathematical model for a tube with a small hole pierced on its side Romain Joly Institut Fourier UMR 5582 CNRS/Universite de Grenoble 100, rue des maths B.P. 74 38402 Saint-Martin-d'Heres, France May 2011 Abstract In this paper, we consider an open tube of diameter ? > 0, on the side of which a small hole of size ?2 is pierced. The resonances of this tube correspond to the eigenvalues of the Laplacian operator with homogeneous Neumann condition on the inner surface of the tube and Dirichlet one on the open parts of the tube. We show that this spectrum converges when ? goes to 0 to the spectrum of an explicit one- dimensional operator. At a first order of approximation, the limit spectrum describes the note produced by a flute, for which one of its holes is open. Key words: thin domains, convergence of operators, resonance, mathematics for music and acoustic. AMS subject classification: 35P15, 35Q99. 1 Introduction and main result In this paper, we obtain a one-dimensional model for the resonances of a tube with a small hole pierced on its side. Our arguments are based on recent thin domain techniques of 1

approximation

?2? ≤

effective length

main result

dirichlet boundary

has been known

domain problems

dimensional objects

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pefav

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