To what extent can a linear analysis predict the behaviour of a flute model ?
Résumé
Although they have been widely studied for years, some aspects of the behaviour of flute-like musical instruments remain poorly understood. The study of a physical model of the instrument has demonstrated its interest in the understanding of various phenomena, such as the hysteresis related to regime changes or the variations of the frequency with the blowing pressure. As it involves both nonlinear and delayed terms, an indepth study of the state of the art flute model requires specific numerical methods, which are often computationally expensive. The simplification of the model through its linearisation around a non-oscillating trivial solution is thus particularly interesting, due to the simplicity of the calculations. The information provided by such an analysis in terms of oscillation frequency or oscillation thresholds of the different periodic solutions has been highlighted in previous work . Surprisingly enough, the present study shows that this simple linear analysis provides information about the stability zones of the different periodic solutions (i.e. the different registers), and allows to predict, in some cases, the register resulting from a transient of the mouth pressure. Such information can be obtained without solving the nonlinear equations and without computing the steady-state oscillations of the model.
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