Oscillation threshold of woodwind instruments

We give a theoretical study of the nature of the bifurcations occurring at the oscillation threshold of woodwind instruments, or of physical systems obeying similar non-linear equations of motion. We start from the simplest description of the acoustical behavior these instruments, a mathematical model containing two equations only, one of which is linear but includes delays, while the other is non-linear but has no delay, and discuss its predictions concerning the characteristics of thc small oscillations. In particular we study the nature of the bifurcation occurring at threshold; if the bifurcation is direct, the amplitude of the oscillations increases progressively when the control parameter exceeds a threshold value; but, if the bifurcation is inverse, very small oscillations are not necessarily stable and the oscillation may jump discontinuously to a finite amplitude. While direct bifurcations correspond better to what naive intuition would expect, the surprising result of our calculations is their occurrence is by no means the general rule. We also discuss the shape (spectral content) of the small oscillations, and show that they do not always become quasisinusoidal in the limit of infinitely small solutions, in contrast with what is often assumed in the literature (Worman rule). Frequency shifts are investigated as well near threshold. More generally, we show how, despite of the simplicity of the equations of motion themselves, the characteristics of the non linearities of the excitator and of those of the resonator combine to produce a variety of possible behaviors which are not necessarily intuitive.