### ASA 127th Meeting M.I.T. 1994 June 6-10

## 2pMU6. Differential formulation: A powerful tool for woodwind instrument
numerical simulations.

**Noel Gr
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*ESPCI-LOA-10 Rue Vauquelin-75005 Paris, France
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**Ana Barjau
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*Polytech. Univ. of Catalunya, Barcelona, Spain
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**Vincent Gibiat
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*ESPCI, Paris, France
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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.