Model Generalizations

The open end of a cylindrical pipe is poorly represented by a load impedance $Z_{L}(\omega) = 0.$
In general, the acoustic properties of the end of the wind-instrument bore are characterized by a non-zero and complex valued load impedance and this impedance can be expressed as a lumped traveling-wave reflectance, as given by the bracketed term of Eq. (36).
Figure 10 represents the digital waveguide implementation of plane-wave pressure propagation in a cylindrical tube terminated by the reflectance $\mathcal{R}_{L}(z).$ Thermal and viscous boundary layer losses are neglected in this model.

Figure 10: Digital waveguide implementation of plane-wave propagation in a cylindrical tube, neglecting viscothermal losses.
$\begin{figure}\begin{center} \begin{picture}(4.75,2) \put(0,0){\epsfig{file = ... ...8){$\mathcal{R}_{L}(z)$} \end{picture} \end{center}\vspace{-0.2in}\end{figure}$
The waveguide structure of Fig. 10 can be further simplified by limiting observation of physical pressure to the input of the tube.

**Figure 10:** Digital waveguide implementation of plane-wave propagation in a cylindrical tube, neglecting viscothermal losses.
$\begin{figure}\begin{center} \begin{picture}(4.75,2) \put(0,0){\epsfig{file = ... ...8){$\mathcal{R}_{L}(z)$} \end{picture} \end{center}\vspace{-0.2in}\end{figure}$

Then, by linearity and time-invariance, the digital filter representing $\mathcal{R}_{L}(z)$ can be “pushed” through the lower delay line to its output and a single delay line used for the simulation, as shown in Fig. 11.

**Figure 11:** Simplified digital waveguide implementation of plane-wave propagation in a cylindrical tube using a single delay line and neglecting viscothermal losses. The pressure observation point is constrained to the entryway of the bore.
$\begin{figure}\begin{center} \begin{picture}(4.9,1.05) \put(0,0){\epsfig{file ... ...8){$\mathcal{R}_{L}(z)$} \end{picture} \end{center}\vspace{-0.1in}\end{figure}$