Time-domain synthesis of conical bore instrument sounds

Model Approaches: The ``Cyclone''

• A cylindrical-conical segment model:

  Figure 9: ``Cyclone'' physical structure (top) and digital waveguide block diagram (bottom).

  

• The junction filter is derived assuming continuity of pressure and conservation of volume velocity:
  
  $$\mathcal{R}^{-}(z) = \mathcal{R}^{+}(z) = - \frac{\gamma}{\alpha + \gamma} \left(\frac{1 + z^{-1}}{1 - a_{1}z^{-1}} \right),$$
  
  
  where $\alpha$ is the bilinear transform constant and
  
  $$\gamma = \frac{c}{2 x_{o}}, \hspace{0.3in} a_{1} = \frac{\alpha - \gamma}{\alpha + \gamma}.$$