How opening a hole affects the sound of a flute

TL;DR

This paper analyzes how opening a small hole in a flute's tube affects its resonant frequencies, showing that as the hole size diminishes, the spectrum converges to that of a simplified one-dimensional model.

Contribution

The paper introduces a rigorous mathematical analysis of the spectral effects of small holes in flute-like tubes, providing explicit convergence results to a 1D operator.

Findings

Spectrum converges to a 1D operator as hole size approaches zero

First-order approximation describes the note of a flute with an open hole

Mathematical model links physical hole size to acoustic resonance changes

Abstract

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.