Time-domain synthesis of conical bore instrument sounds

The Non-Linear Reed Function

• Single-reed mechanism modeled as a memory-less nonlinear function (reed mass ignored).

• Reed channel volume flow vs. pressure difference characteristic solved in terms of a traveling-wave reflection function (Smith, 1986):
  
  $$p_{o}^{+} = r(p_{\Delta}) \left[ p_{o}^{-} - \frac{p_{b}}{2}... ...= \frac{Z_{r}(p_{\Delta}) - Z_{o}}{Z_{r}(p_{\Delta}) + Z_{o}},$$
  
  
  $p_{o}$ is the pressure at the air column input, $p_{b}$ is the player's breath pressure, $p_{\Delta} = p_{b} - p_{o}$ is the pressure difference across the reed, $Z_{r}$ is a time-varying reed ``impedance'', and $Z_{o}$ is the real wave impedance at the air column input.

  Figure 2: Bernoulli static flow expression (top); Simplified reed reflection function (bottom).