Under the previous assumption that the reed can be adequately modeled as a mechanical spring, the motion of which is in phase with the pressure () acting across it, Eq. (22) can be rewritten using Eq. (20) as
is the pressure necessary to push the reed against the mouthpiece facing and completely close the reed channel.
The flow through the reed orifice is assumed to separate and form a free jet at the air column entrance. In the ensuing region of turbulence, the pressure is assumed equivalent to the reed channel pressure and conservation of volume flow to hold.
In this case, u = ud and we can use Eq. (23) and the relation
ud(t) = u+d(t) + u-d(t) = Z-1c [p+d(t) - p-d(t)],
to find a set of solutions for various values of .
The approach of McIntyre et al. (1983) is to assume a constant upstream pressure (pu) and obtain solutions for the outgoing downstream pressure pd+ based on incoming pressures pd-. This amounts to finding the intersection of the flow curve and a straight line corresponding to Eq. (24), as illustrated in Fig. 11 below.
Flow and bore characteristic curves.
However, it is preferable to maintain the dependence on , which allows us to modify the mouth pressure in a time-domain simulation.
pd+ - pd-
pd+ - pd- + [pu - pu] + [pd- - pd-]
pu - 2 pd- - pu + [pd+ + pd-]
consists of pressure components known or computable from previous values.
We can then find a table of solutions, such as shown in Fig. 12, for the equation
For values of
greater than pc, the reed is closed and the solutions are given by a straight line of slope one, which corresponds to zero flow and pressure reflection without inversion at the junction.
Thus, when the reed is forced against the mouthpiece facing, the air column appears as a rigidly terminated or stopped end.
Solutions at reed/air column junction for input
(pc = 2280 Pascals).