Reed Scattering Theory

Alternately, the reed mechanism can be viewed as having a “lumped,” pressure dependent impedance given by:

$\displaystyle Z_{r}(p_{\Delta}) = \frac{p_{\Delta}}{u(p_{\Delta})}.$ (25)

Under the assumption that the reed motion is adequately described as “stiffness dominated” and in-phase with the driving pressure and that the resulting volume flow through the reed slit is also frequency-independent (ignoring the inertance of the air in the reed channel), this characterization can be evaluated in the time domain as a memory-less non-linearity.

Viewing the reed/air column junction using scattering theory as seen from the air column side, pressure-wave reflection from an impedance of $Z_R$

is given by:

$\displaystyle Z_{r}$	$\textstyle =$	$\displaystyle \frac{P^{+} + P^{-}}{(P^{+} - P^{-})/Z_c}$	(26)
$\displaystyle \frac{P^{-}}{P^{+}}$	$\textstyle =$	$\displaystyle \frac{Z_{r} - Z_c}{Z_{r} + Z_c} = \frac{1 - Z_c/Z_r}{1 + Z_c/Z_r}$	(27)

The reed interface is then modeled with a nonlinear pressure-dependent reflection coefficient ( $\rho(p_{\Delta})$ ) given by Eq. (27) and implemented via a scattering junction as shown in Fig. 13. The pressure entering the downstream instrument air column is determined as (Smith, 1986):

$\displaystyle p_{d}^{+}$	$\textstyle =$	$\displaystyle p_{d}^{-} \cdot \rho(p_{\Delta}) + p_{u}^{+} \left[ 1 - \rho(p_{\Delta}) \right]$
	$\textstyle =$	$\displaystyle p_{u}^{+} - \left[ p_{u}^{+} - p_{d}^{-} \right] \rho(p_{\Delta}).$	(28)

**Figure 13:** The reed scattering junction.
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The reflection coefficient defined by the Bernoulli flow expression of Eq. (21) is shown in Fig. 14.

**Figure 14:** Reed/air column reflection coefficient values for different values of equilibrium tip opening ().
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For values of $p_{\Delta} > p_{c}$ , the pressure reflection coefficient has unity gain. This corresponds to reflection from a rigidly stopped end, as mentioned previously.

A coefficient value of zero corresponds to a junction without discontinuity.

A coefficient of -1.0 corresponds to an ideally open-pipe end (a reed mechanism of zero impedance).

For normal reed mechanisms with a narrow-slit geometry, it seems impossible that the reed/bore junction could ever be characterized by a reflection coefficient of zero or lower. It is expected that the impedance of the reed slit, even when unblown, has some minimum value (probably frequency-dependent, but we'll ignore that for now) greater than $Z_{c}$ .