Toward the systematic investigation of periodic solutions in single reed woodwind instruments
Résumé
Single reed woodwind instruments rely on the basic principle of a linear acoustic resonator - the air column inside the cylindrical or conical bore, coupled with a nonlinear exciter - namely the reed and the air jet entering the mouthpiece. The first one is described by its input impedance, which binds the acoustical pressure and flow at the entry of the bore through a linear relation, whereas the second one has a non-smooth, nonlinear characteristic which combines the pressure on both sides of the reed channel, the jet flow, and the reed motion. To find possible playing frequencies, one often analyses the input impedance spectrum in terms of central frequency, height and width of peaks - a method used in various recent publications on bore geometry optimisation. The exciter influence has rarely been taken into account, and in a few restrictive cases only : for precise, fixed value of control parameters ; through time domain simulations, which cannot give all information on the dynamics ; through simplifications of the equations, allowing analytical calculations of some parts of the bifurcation diagram. A more systematic investigation of a given instrument behaviors depending on control parameters requires the framework of dynamical systems and bifurcation theory, as well as specific numerical tools. In the present work, two continuation methods were used to obtain the bifurcation diagram of a clarinet, as comprehensive as possible. Stable and unstable, periodic and static solution branches are shown, revealing instrument characteristics such as oscillation, saturation, and extinction thresholds, as well as dynamic range.
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