  In this paper we propose a minimal model for free reeds taking into account
the significant phenomena. This free reed model may be used to build models of
free reed instruments which permit numerical simulations. Several definitions
for the section by which the airflow passes through the reed are reviewed and a
new one is proposed which takes into account the entire escape area under the
reed and the reed thickness. To derive this section, it is necessary to
distinguish the neutral section (the only section of the reed which always
keeps its length constant while moving) from the upstream or downstream
sections. A minimal configuration is chosen to permit the instabilities of both
(-,+) and (+,-) reeds on the basis of a linear analysis of instabilities
conditions. This configuration is used to illustrate, with temporal
simulations, the minimal model for both kinds of reeds and to discuss the model
assumptions. Some clues are given about the influence, on the playing frequency
and on the dynamic of the sound, of two main parameters of the geometrical
model: the size of the volume and the level of the excitation. It is shown that
the playing frequency of a (+,-) reed can vary in a large range according to
the size of the volume upstream of the reed; that the playing frequency is
nearly independent of the excitation but that the dynamic of the sound
increases with the excitation level. Some clues are also proposed to determine
the nature of the bifurcation for free reeds: it seems that free reeds may
present inverse bifurcations. The influence of the reed thickness is also
studied for configurations where the reed length or the reed width vary to keep
the mass constant. This study shows that the reed thickness can have a great
influence on the sound magnitude, the playing frequency and the magnitude of
the reed displacement which justifies its introduction in the reed model.This
article has been published in Acta Acustica united with Acustica, Vol. 93
(2007), p. 122-144.
