  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 bifur-cation 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.
