Cylindrical Sections: Time-Domain Approach

**Figure 14:** Junction of two cylindrical pipe sections.
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In general, abrupt diameter discontinuities rarely occur in vocal tract or wind-instrument air column profiles.

However, these discontinuities will be encountered in the approximation of complex air column shapes using cylindrical pipe sections.

At the boundary of two discontinuous and lossless cylindrical sections, Fig. 14, there will be a change of characteristic impedance which results in partial reflection and transfer of traveling-wave components.

Assuming continuity of pressure and conservation of volume flow at the boundary,

$\displaystyle p_{1}^{+} + p_{1}^{-} = p_{2}^{+} + p_{2}^{-}$ (46)

and

$\displaystyle \frac{1}{Z_{c1}}\left[p_{1}^{+} - p_{1}^{-}\right] = \frac{1}{Z_{c2}}\left[p_{2}^{+} - p_{2}^{-}\right],$ (47)

where $Z_{c1}$ is the characteristic impedance of cylindrical section $1$

Because the characteristic wave impedance of a cylindrical pipe is real, these expressions apply to both time- and frequency-domain wave variables.

Solving for $p_{1}^{-}$ and $p_{2}^{+}$ at the junction,

$\displaystyle p_{1}^{-}$	$\textstyle =$	$\displaystyle \left(\frac{Z_{c2} - Z_{c1}}{Z_{c2} + Z_{c1}}\right) p_{1}^{+} + \left(\frac{2Z_{c1}}{Z_{c2} + Z_{c1}}\right) p_{2}^{-}$
	$\textstyle =$	$\displaystyle \mathcal{R}_{12}\,p_{1}^{+} + \left(1-\mathcal{R}_{12}\right)p_{2}^{-}$	(48)

$\displaystyle p_{2}^{+}$	$\textstyle =$	$\displaystyle \left(\frac{2Z_{c2}}{Z_{c2} + Z_{c1}}\right) p_{1}^{+} - \left(\frac{Z_{c2} - Z_{c1}}{Z_{c2} + Z_{c1}}\right) p_{2}^{-}$
	$\textstyle =$	$\displaystyle \left( 1+\mathcal{R}_{12} \right) p_{1}^{+} - \mathcal{R}_{12} p_{2}^{-}$	(49)

where $\mathcal{R}_{12}$ is the reflectance for the junction of cylinders $1$

and

$\mathcal{R}_{12}$ is given by

$\begin{eqnarray*} \mathcal{R}_{12} &=& \frac{Z_{c2} - Z_{c1}}{Z_{c2} + Z_{c1}} \\ &=& \frac{A_{1} - A_{2}}{A_{1} + A_{2}}, \end{eqnarray*}$

where $A_{1}$ is the cross-sectional area of section $1.$

The relationships of Eqs. (48) and (49) are referred to as scattering equations.

The scattering equations are implemented by the structure shown in Fig. 15a, which was first derived for an acoustic tube model used in speech synthesis (Kelly and Lochbaum, 1962).

**Figure 15:** (a) The Kelly-Lochbaum scattering junction for diameter discontinuities in cylindrical bores; (b) The one-multiply scattering junction [after (Markel and Gray, 1976)].
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Equations (48) and (49) can also be written in the form

$\begin{eqnarray*} p_{1}^{-} &=& p_{2}^{-} + p_{\Delta} \\ p_{2}^{+} &=& p_{1}^{+} + p_{\Delta}, \end{eqnarray*}$

where

$\displaystyle p_{\Delta} = \mathcal{R}_{12} \left[ p_{1}^{+} - p_{2}^{-} \right].$

In this way, the Kelly-Lochbaum scattering junction can be implemented with a single multiply, as shown in Fig. 15b (Markel and Gray, 1976).

Smith (1987) points out that junction passivity is guaranteed for $-1 \leq r_{12}(t) = \mathcal{R}_{12}(\omega) \leq 1.$

The digital waveguide implementation of lossless wave propagation in two discontinuous cylindrical sections is shown in Fig. 16.

**Figure 16:** The digital waveguide model of two discontinuous cylindrical sections.
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In this way, any combination of co-axial cylindrical sections can be modeled using only digital delay lines and one-multiply scattering junctions.

Thermoviscous losses can be implemented with digital filters designed in accordance with previous theoretical analyses.

Because these models are linear and time-invariant, these loss characteristics can be commuted with an open-end filter to maximize efficiency.