An Analytical Prediction of the Bifurcation Scheme of a Clarinet-Like Instrument: Effects of Resonator Losses

Abstract

The understanding of the relationship between excitation parameters and oscillation regimes is a classical topic concerning bowed string instruments. The paper aims to study the case of reed woodwinds and attempts to find consequences on the ease of playing. In the minimum model of clarinet-like instruments, three parameters are considered: i) the mouth pressure, ii) the reed opening at rest, iii) the length of the resonator assumed to be cylindrical. Recently a supplementary parameter was added: the loss parameter of the resonator (using the “Raman model”, that considers resonator losses to be independent of frequency). This allowed explaining the extinction of sound when the mouth pressure becomes very large. The present paper presents an extension of the paper by Dalmont et al. (JASA, 2005), searching for a diagram of oscillation regimes with respect to the reed opening and the loss parameter. An alternative method is used, which allows easier generalization and simplifies the calculation. The emphasis is done on the emergence bifurcation: for very strong losses, it can be inverse, similarly to the extinction one for weak losses. The main part of the calculations are analytical, giving clear dependence of the parameters. An attempt to deduce musical consequences for the player is given.