aLaboratoire d'Acoustique de l'Université du Mans - UMR 6613 CNRS
bLaboratoire Manceau de Mathématiques - Le Mans Université
cAix Marseille Univ, CNRS, Centrale Marseille, LMA, UMR 7031, Marseille
Reed musical instruments can be described in terms of conceptually separate linear and nonlinear mechanisms: a localized nonlinear element (the valve effect due to the reed) excites a linear, passive acoustical multimode element (the musical instrument usually represented in the frequency domain by its input impedance). The linear element in turn influences the operation of the nonlinear element. The reed musical instruments are self-sustained oscillators. They generate an oscillating acoustical pressure (the note played) from a static overpressure in the player’s mouth (the blowing pressure). A reed instrument having N acoustical modes can be described as a 2N dimensional autonomous nonlinear dynamical system. A reed-like instrument having two quasi-harmonic resonances, represented by a 4 dimensional dynamical system, is studied using the continuation and bifurcation software AUTO. Bifurcation diagrams are explored with respect to the blowing pressure, with focus on amplitude and frequency evolutions along the different solution branches. Some of the results are interpreted in terms of the ease of playing of the reed instrument. They can be interpreted as a theoretical illustration of the Bouasse-Benade prescription.
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