Scientific Article

Minimal blowing pressure allowing periodic oscillations in a model of bass brass instruments

¹ Laboratoire d’Acoustique de l’Université du Mans (LAUM), UMR CNRS 6613, Institut d’Acoustique - Graduate School (IA-GS), CNRS, Le Mans Université, 72085 Le Mans, France
² Aix Marseille Univ, CNRS, Centrale Marseille, LMA, UMR 7031, 13625 Aix-en-Provence Cedex, France
³ Laboratoire Manceau de Mathématiques – Le Mans Université, 72085 Le Mans, France

^* Corresponding author: remi.matteoli@univ-lemans.fr

Received: 6 September 2021
Accepted: 23 November 2021

Abstract

In this study, an acoustic resonator – a bass brass instrument – with multiple resonances coupled to an exciter – the player’s lips – with one resonance is modelled by a multidimensional dynamical system, and studied using a continuation and bifurcation software. Bifurcation diagrams are explored with respect to the blowing pressure, in particular with focus on the minimal blowing pressure allowing stable periodic oscillations and the associated frequency. The behaviour of the instrument is first studied close to a (non oscillating) equilibrium using linear stability analysis. This allows to determine the conditions at which an equilibrium destabilises and as such where oscillating regimes can emerge (corresponding to a sound production). This approach is useful to characterise the ease of playing of a brass instrument, which is assumed here to be related – as a first approximation – to the linear threshold pressure. In particular, the lower the threshold pressure, the lower the physical effort the player has to make to play a note [The Science of Brass Instruments. Springer-Verlag, 2021]. Cases are highlighted where periodic solutions in the bifurcation diagrams are reached for blowing pressures below the value given by the linear stability analysis. Thus, bifurcation diagrams allow a more in-depth analysis. Particular attention is devoted to the first playing regime of bass brass instruments (the pedal note and the ghost note of a tuba in particular), whose behaviour qualitatively differs from a trombone to a euphonium for instance.