Single-Reed Theory

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,^{10.19} 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.*,

where

and

is the volume velocity corresponding to the incoming pressure wave and outgoing pressure wave . (The physical pressure in the bore at the mouthpiece is of course .) The wave impedance of the bore air-column is denoted (computable as the air density times sound speed divided by cross-sectional area).

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

Solving for is not immediate because of the dependence of on which, in turn, depends on . A graphical solution technique was proposed [152,246,311] which, in effect, consists of finding the intersection of the two terms of the equation as they are plotted individually on the same graph, varying . This is analogous to finding the operating point of a transistor by intersecting its operating curve with the ``load line'' determined by the load resistance.

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

The solution is obtained by plotting and on the same graph, finding the point of intersection at coordinates , and computing finally the outgoing pressure wave sample as

(10.40) |

An example of the qualitative appearance of overlaying is shown in Fig. 9.41.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University