In this paper, we consider an open tube of diameter , on the side of which a small hole of size 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 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. 389–408DOI 10.4171/JST/17