Wind instruments as time delay systems. Part I: modeling*

In this paper wind instruments are modeled as time delay systems. In fact, a wind instrument is usually made of a linear acoustic resonator (the pipe) coupled with a nonlinear oscillator (the mouth of the instrument). The resonator can be modeled through hyperbolic wave equations. Two kinds of instruments are considered : the first one is a slide flute, (i.e. a kind of recorder without finger holes but ended by a piston to modify the pipe length), for which a realistic model of the air jet coupled with the pipe is given. The second example will concern the case of a simplified trumpet-like instrument composed of a valve (including the mechanics of the lips, contrary to the case of the flute), an air jet coupled with the valve dynamics and an acoustic pipe excited by the jet and radiating in the air. The overall system can be described by a so-called nonlinear neutral state space system.