How opening a hole affects the sound of a flute

  • Romain Joly

    Université de Grenoble I, Saint-Martin-d'Hères, France


In this paper, we consider an open tube of diameter ε>0\varepsilon>0, on the side of which a small hole of size ε2\varepsilon^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 ε\varepsilon goes to 00 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.

Cite this article

Romain Joly, How opening a hole affects the sound of a flute. J. Spectr. Theory 1 (2011), no. 4, pp. 389–408

DOI 10.4171/JST/17