## The Reed-Spring Solution

• 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
 u = = (23)

where 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)], (24)

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.

• However, it is preferable to maintain the dependence on , which allows us to modify the mouth pressure in a time-domain simulation.

• Note that
 pd+ - pd- = pd+ - pd- + [pu - pu] + [pd- - pd-] = pu - 2 pd- - pu + [pd+ + pd-] = (25)

where 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.