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Single-Reed Theory

Figure 9.40: Schematic diagram of mouth cavity, reed aperture, and bore.

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 $ p_m$ , and the bore pressure $ p_b$ is defined located at the mouthpiece. Any pressure drop $ p_{\Delta}= p_m-p_b$ across the mouthpiece causes a flow $ u_m$ into the mouthpiece through the reed-aperture impedance $ R_m(p_{\Delta})$ which changes as a function of $ p_{\Delta}$ since the reed position is affected by $ p_{\Delta}$ . 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 $ R_m(p_{\Delta})$ is a positive real number for all values of $ p_{\Delta}$ . 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.,

$\displaystyle u_b+u_m= 0$ (10.35)


$\displaystyle u_m(p_{\Delta}) \isdef \frac{p_{\Delta}}{R_m(p_{\Delta})}$ (10.36)


$\displaystyle u_b\isdef u_b^{+}+ u_b^{-}= \frac{p_b^{+}-p_b^{-}}{R_b}$ (10.37)

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

In operation, the mouth pressure $ p_m$ and incoming traveling bore pressure $ p_b^{+}$ are given, and the reed computation must produce an outgoing bore pressure $ p_b^{-}$ which satisfies (9.35), i.e., such that

0 $\displaystyle =$ $\displaystyle u_m+u_b= \frac{p_{\Delta}}{R_m(p_{\Delta})} + \frac{p_b^{+}-p_b^{-}}{R_b},$ (10.38)
$\displaystyle p_{\Delta}$ $\displaystyle \isdef$ $\displaystyle p_m-p_b= p_m- (p_b^{+}+p_b^{-})$  

Solving for $ p_b^{-}$ is not immediate because of the dependence of $ R_m$ on $ p_{\Delta}$ which, in turn, depends on $ p_b^{-}$ . 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 $ p_b^{-}$ . 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 $ G(p_{\Delta}) = R_b
u_m(p_{\Delta}) = R_bp_{\Delta}/R_m(p_{\Delta})$ , and note that $ p_b^{+}-p_b^{-}=2p_b^{+}-p_m-(p_b^{+}+p_b^{-}-p_m)\isdeftext p_{\Delta}-p_{\Delta}^{+}$ , where $ p_{\Delta}^{+}\isdeftext
p_m-2p_b^{+}$ . Then (9.38) can be multiplied through by $ R_b$ and written as $ 0=G(p_{\Delta})+p_{\Delta}-p_{\Delta}^{+}$ , or

$\displaystyle G(p_{\Delta}) = p_{\Delta}^{+}-p_{\Delta},\qquad p_{\Delta}^{+}\isdef p_m- 2p_b^{+}$ (10.39)

The solution is obtained by plotting $ G(x)$ and $ p_{\Delta}^{+}-x$ on the same graph, finding the point of intersection at $ (x,y)$ coordinates $ (p_{\Delta},G(p_{\Delta}))$ , and computing finally the outgoing pressure wave sample as

$\displaystyle p_b^{-}= p_m- p_b^{+}- p_{\Delta}(p_{\Delta}^{+})$ (10.40)

An example of the qualitative appearance of $ G(x)$ overlaying $ p_{\Delta}^{+}-x$ is shown in Fig. 9.41.

Figure 9.41: Normalized reed impedance $ G(p_{\Delta }) = R_bu_m(p_{\Delta })$ overlaid with the ``bore load line'' $ p_{\Delta }^{+}-p_{\Delta }= R_bu_b$ .

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-10-15 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University