Alternately, the reed mechanism can be viewed as having a ``lumped,'' pressure dependent impedance given by:
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 ZR is given by:
The reed interface is then modeled with a nonlinear pressure-dependent reflection coefficient (
) given by Eq. (28) and implemented via a scattering junction as shown in Fig. 13. The pressure entering the downstream instrument air column is determined as (Smith, 1986):
The reed scattering junction.
The reflection coefficient defined by the Bernoulli flow expression of Eq. (22) is shown in Fig. 14.
Reed/air column reflection coefficient values for different values of equilibrium tip opening (H).
For values of
, 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 Zc.