Analytical Determination of the Attack Transient in a Clarinet With Time-Varying Blowing Pressure

Abstract

This article uses a basic model of a reed instrument, known as the lossless Raman model, to determine analytically the envelope of the sound produced by the clarinet when the mouth pressure is increased gradually to start a note from silence. Using results from dynamic bifurcation theory, a prediction of the amplitude of the sound as a function of time is given based on a few parameters quantifying the time evolution of mouth pressure. As in previous uses of this model, the predictions are expected to be qualitatively consistent with simulations using the Raman model, and observations of real instruments. Model simulations for slowly variable parameters require very high precisions of computation. Similarly, any real system, even if close to the model would be affected by noise. In order to describe the influence of noise, a modified model is developed that includes a stochastic variation of the parameters. Both ideal and stochastic models are shown to attain a minimal amplitude at the static oscillation threshold. Beyond this point, the amplitude of the oscillations increases exponentially, although some time is required before the oscillations can be observed at the "dynamic oscillation threshold". The effect of a sudden interruption of the growth of the mouth pressure is also studied, showing that it usually triggers a faster growth of the oscillations.