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Physical Audio Signal Processing
    Virtual Musical Instruments
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          Single-Reed Theory

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Single-Reed Theory

Figure 9.40: Schematic diagram of mouth cavity, reed aperture, and bore.
\includegraphics[scale=0.9]{eps/fReedSchematic}

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), [249]). 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 $ 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 [431], 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)

where

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

and

$\displaystyle u_b\isdef u_b^{+}+ u_b^{-}= \frac{p_b^{+}-p_b^{-}}{R_b}$ (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 [151,244,308] 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$.
\includegraphics[width=\twidth]{eps/fReedRelations}


Subsections
Previous: A View of Single-Reed Oscillation
Next: Scattering-Theoretic Formulation

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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