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 ($p_{\Delta}$) acting across it, Eq. (21) can be rewritten using Eq. (19) as
  

  $$u = w \left(H - \frac{A_r \cdot p_{\Delta}}{k}\right) \left(\frac{2 p_{\Delta}}{\rho} \right)^{1/2} \mbox{sgn($p_{\Delta}$)}$$

  $$= w H \left(1 - \frac{p_{\Delta}}{p_{c}}\right) \left(\frac{2 p_{\Delta}}{\rho} \right)^{1/2} \mbox{sgn($p_{\Delta}$)},$$ (22)

  where $p_{c} = k H/A_{r} = \mu_r H \omega^{2}_{r}$ 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, $u = u_d$ and we can use Eq. (22) and the relation
    $$u_d(t) = u^{+}_d(t) + u^{-}_d(t) = Z^{-1}_{c} [p^{+}_d(t) - p^{-}_d(t)],$$ (23)
  to find a set of solutions for various values of $p_{\Delta}$.

• The approach of McIntyre et al. (1983) is to assume a constant upstream pressure ($p_u$) and obtain solutions for the outgoing downstream pressure $p_{d}^{+}$ based on incoming pressures $p_{d}^{-}$. This amounts to finding the intersection of the flow curve and a straight line corresponding to Eq. (23), as illustrated in Fig. 11 below.

  Figure 11: Flow and bore characteristic curves.

  

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

• Note that
  

  $$p_{d}^{+} - p_{d}^{-} = p_{d}^{+} - p_{d}^{-} + [p_u - p_u] + [p_{d}^{-} - p_{d}^{-}]$$

  $$= p_u - 2 p_{d}^{-} - p_u + [p_{d}^{+} + p_{d}^{-}]$$

  $$= p_{\Delta}^{-} - p_{\Delta},$$ (24)

  where $p_{\Delta}^{-} = p_u - 2 p_{d}^{-}$ 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 $Z_{c} \, u_{d}(p_{\Delta}) = p_{\Delta}^{-} - p_{\Delta}$.

• For values of $p_{\Delta }^{-}$ greater than $p_{c}$, 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.

  Figure 12: Solutions at reed/air column junction for input $p_{\Delta }^{-}$ ($p_{c} = 2280$ Pascals).