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Physical Audio Signal Processing
    Woodwinds
       Single-Reed Instruments
          Single-Reed Theory

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Single-Reed Theory

Figure 6.3: 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. 6.3. 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 (§F.5.7), [255]). This model has been verified well experimentally until the reed is about to close, at which point viscosity effects begin to appear [100]. It has also been verified that the mass of the reed can be neglected to first order,7.1 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 [440], 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.,

$\displaystyle u_b+u_m= 0$ (7.1)

where

$\displaystyle u_m(p_{\Delta}) \isdef \frac{p_{\Delta}}{R_m(p_{\Delta})}$ (7.2)

and

$\displaystyle u_b\isdef u_b^{+}+ u_b^{-}= \frac{p_b^{+}-p_b^{-}}{R_b}$ (7.3)

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