Single-Reed Theory

Figure 6.3: Schematic diagram of mouth cavity, reed aperture, and bore.

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 (§E.17), [234]). This model has been verified well experimentally until the reed is about to close, at which point viscosity effects begin to appear [94]. It has also been verified that the mass of the reed can be neglected to first order, 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 [407], 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.,

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

where

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

and

$$u_b\isdef u_b^{+}+ u_b^{-}= \frac{p_b^{+}-p_b^{-}}{R_b}$$ (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 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 (6.1), i.e., such that

$$= u_m+u_b= \frac{p_{\Delta}}{R_m(p_{\Delta})} + \frac{p_b^{+}-p_b^{-}}{R_b},$$ 0 (7.4)

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

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 [143,230,289] 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 (6.4) 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 (6.4) can be multiplied through by $R_b$ and written as $0=G(p_{\Delta})+p_{\Delta}-p_{\Delta}^{+}$, or

$$G(p_{\Delta}) = p_{\Delta}^{+}-p_{\Delta},\qquad p_{\Delta}^{+}\isdef p_m- 2p_b^{+}$$ (7.5)

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

$$p_b^{-}= p_m- p_b^{+}- p_{\Delta}(p_{\Delta}^{+})$$ (7.6)

An example of the qualitative appearance of $G(x)$ overlaying $p_{\Delta}^{+}-x$ is shown in Fig. 6.4.

Figure 6.4: Normalized reed impedance $G(p_{\Delta }) = R_bu_m(p_{\Delta })$ overlaid with the ``bore load line'' $p_{\Delta }^{+}-p_{\Delta }= R_bu_b$.