A simplified diagram of the clarinet mouthpiece is shown in
Fig. 9.40. The pressure in the mouth is assumed
to be a constant value
, and the bore pressure
is defined
located at the mouthpiece. Any pressure drop
across
the mouthpiece causes a flow
into the mouthpiece through the
reed-aperture impedance
which changes as a function of
since the reed position is affected by
. To a first
approximation, the clarinet reed can be regarded as a spring flap
regulated Bernoulli flow (§B.7.5), [251]).
This model has been verified well experimentally until the reed is
about to close, at which point viscosity effects begin to appear
[102]. It has also been verified that the mass
of the reed can be neglected to first order,10.18 so that
is a
positive real number for all values of
. Possibly the most
important neglected phenomenon in this model is sound generation due
to turbulence of the flow, especially near reed closure. Practical
synthesis models have always included a noise component of some sort
which is modulated by the reed [435], despite a lack of firm
basis in acoustic measurements to date.
The fundamental equation governing the action of the reed is continuity of volume velocity, i.e.,
In operation, the mouth pressure
and incoming traveling bore pressure
are given, and the reed computation must produce an outgoing bore
pressure
which satisfies (9.35), i.e., such that
It is helpful to normalize (9.38) as follows: Define
, and note that
, where
. Then (9.38) can be multiplied through by
and
written as
, or
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(10.40) |
An example of the qualitative appearance of
overlaying
is shown in Fig. 9.41.