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Oscillation threshold of woodwind instruments

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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.

Document Type: Research Article

Publication date: 1997-01-01

More about this publication?
  • Acta Acustica united with Acustica, published together with the European Acoustics Association (EAA), is an international, peer-reviewed journal on acoustics. It publishes original articles on all subjects in the field of acoustics, such as general linear acoustics, nonlinear acoustics, macrosonics, flow acoustics, atmospheric sound, underwater sound, ultrasonics, physical acoustics, structural acoustics, noise control, active control, environmental noise, building acoustics, room acoustics, acoustic materials, acoustic signal processing, computational and numerical acoustics, hearing, audiology and psychoacoustics, speech, musical acoustics, electroacoustics, auditory quality of systems. It reports on original scientific research in acoustics and on engineering applications. The journal considers scientific papers, technical and applied papers, book reviews, short communications, doctoral thesis abstracts, etc. In irregular intervals also special issues and review articles are published.
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