Oscillation threshold of a clarinet model: a numerical continuation approach

TL;DR

This study employs a numerical continuation method to analyze the oscillation threshold of a detailed clarinet model, revealing how multiple parameters influence playing regimes and frequency, thus providing deeper insights into reed instrument dynamics.

Contribution

It introduces a numerical continuation approach to study the oscillation thresholds of a clarinet model, extending previous analytical and simulation methods with more comprehensive parameter analysis.

Findings

Reconfirmed the influence of reed resonance frequency on oscillation threshold.

Discovered the combined effect of two parameters on regime selection and frequency.

Demonstrated the usefulness of numerical continuation in musical instrument analysis.

Abstract

This paper focuses on the oscillation threshold of single reed instruments. Several characteristics such as blowing pressure at threshold, regime selection, and playing frequency are known to change radically when taking into account the reed dynamics and the flow induced by the reed motion. Previous works have shown interesting tendencies, using analytical expressions with simplified models. In the present study, a more elaborated physical model is considered. The influence of several parameters, depending on the reed properties, the design of the instrument or the control operated by the player, are studied. Previous results on the influence of the reed resonance frequency are confirmed. New results concerning the simultaneous influence of two model parameters on oscillation threshold, regime selection and playing frequency are presented and discussed. The authors use a numerical continuation approach. Numerical continuation consists in following a given solution of a set of equations when a parameter varies. Considering the instrument as a dynamical system, the oscillation threshold problem is formulated as a path following of Hopf bifurcations, generalizing the usual approach of the characteristic equation, as used in previous works. The proposed numerical approach proves to be useful for the study of musical instruments. It is complementary to analytical analysis and direct time-domain or frequency-domain simulations since it allows to derive information that is hardly reachable through simulation, without the approximations needed for analytical approach.