Nonlinear dynamics of musical reed and brass wind instruments

Abstract

A musical wind instrument transforms a constant pressure input from the player's mouth into a fluctuating pressure output in the form of a radiating sound wave. In reed woodwind and brass instruments, this transformation is achieved through a nonlinear coupling between two vibrating systems: the flow control valve formed by the mechanical reed or the lips of the player, and the air column contained by the pipe. Although the basic physics of reed wind instruments was developed by Helmholtz in the nineteenth century, the application of ideas from the modern theory of nonlinear dynamics has led to recent advances in our understanding of some musically important features of wind instrument behaviour. As a first step, the nonlinear aspects of the musical oscillator can be considered to be concentrated in the flow control valve; the air column can be treated as a linear vibrating system, with a set of natural modes of vibration corresponding to the standing waves in the pipe. Recent models based on these assumptions have had reasonable success in predicting the threshold blowing pressure and sounding frequency of a clarinet, as well as explaining at least qualitatively the way in which the timbre of the sound varies with blowing pressure. The situation is more complicated for brass instruments, in which the player's lips provide the flow valve. Experiments using artificial lips have been important in permitting systematic studies of the coupling between lips and air column; the detailed nature of this coupling is still not fully understood. In addition, the assumption of linearity in the air column vibratory system sometimes breaks down for brass instruments. Nonlinear effects in the propagation of high amplitude sound waves can lead to the development of shock waves in trumpets and trombones, with important musical consequences.