Minimum models for self-sustained oscillations of conical reed instruments

Abstract

It is now well known that a minimum model for self-sustained oscillations of clarinet-like instruments is the iterated map model, leading to square signals. The reed is assumed to be without dynamics, while losses are ignored (or assumed to be independent of frequency). The generalization to conical instruments is not straightforward. For the present work, the minimum model is used for a truncated cone instrument, but the missing part of the cone is not assumed to be small compared to the wavelength. Thus the result should be a signal without sharp corners. However, without any kind of mouthpiece, no periodic sound can be obtained in a steady-state regime. It will be explained that the choice of a model for the mouthpiece can be done without adding any supplementary parameter (therefore a conical resonator has one parameter only more than a cylindrical one). It is shown that several choices are possible, allowing either the use of: the same inverse nonlinear characteristic than for clarinet-like instruments, or any direct nonlinear characteristic, leading to a great simplicity. Advantages and drawbacks of several solutions are discussed.