ESPCI-LOA-10 Rue Vauquelin-75005 Paris, France
Polytech. Univ. of Catalunya, Barcelona, Spain
ESPCI, Paris, France
A woodwind instrument is well described by a linear resonator introducing a delay and a nonlinear excitator without delay. Thus one can model the woodwind instrument with two simple equations (one in the Fourier domain, the other one in the time domain) calling out the acoustical pressure, flow, and admittance of the resonator. Under some conditions, these two equations can be rewritten as a single nonlinear differential equation with delay, the degree of which depends on the resonator's geometry. As an example, a cone can be described by a second-order differential equation which transforms into a first-order one for a cylinder. This differential formulation has some advantages: The minimal phase space dimension transitions to chaos are clearly displayed; moreover, resulting numerical algorithms are much quicker and more stable than those obtained by integral or Fourier transform methods; the control parameter is directly related to the resonator losses (which is easy to control). And last, it's a powerful tool to observe the characteristics of the small oscillations (by increasing a control parameter), which is essential to provide support for the theoretical model.