How opening a hole affects the sound of a flute

In this paper, we consider an open tube of diameter $\epsilon>0$, on the side of which a small hole of size $\epsilon^2$ 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 the open parts of the tube. We show that this spectrum converges when $\epsilon$ 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.