A Two-Port Tonehole Model

The fundamental acoustic properties of toneholes have been extensively studied and reported by Dubos et al. (1999); Nederveen et al. (1998); Lefebvre and Scavone (2012); Keefe (1990a); Dalmont et al. (2002); Keefe (1981).

The model described by Keefe (1990a) is an accurate representation for a tonehole unit, assuming adjacent tonehole interactions are negligible.

In this description, acoustic variables at the tonehole junction are related by a transfer matrix of series and shunt impedance parameters.

Keefe's original derivation of the tonehole parameters was based on a symmetric $T$

section, as shown in Fig. 1 (Keefe, 1981).

**Figure 1:** section transmission-line representation of the tonehole.
$\begin{figure}\begin{center} \begin{picture}(3,1.2) \put(0,0){\epsfig{file = f... ...95){$Z_{a}/2$} \put(1.43,0.47){$Z_{s}$} \end{picture} \end{center} \end{figure}$

The series impedance terms, $Z_{a},$ result from an analysis of anti-symmetric pressure distribution, or a pressure node, at the tonehole junction. In this case, volume flow is symmetric and equal across the junction.

The shunt impedance term, $Z_{s},$ results from an analysis of symmetric pressure distribution, or a pressure anti-node, at the tonehole, for which pressure is symmetric and equal across the junction.

The transfer matrix that results under this analysis is given by

$\displaystyle \left[\begin{array}{c} P_{1} \\ U_{1} \end{array}\right]$	$\textstyle =$	$\displaystyle \left[\begin{array}{cc} 1 & Z_{a}/2 \\ 0 & 1 \end{array}\right] ... ... 1 \end{array}\right] \left[\begin{array}{c} P_{2} \\ U_{2} \end{array}\right]$
	$\textstyle =$	$\displaystyle \left[\begin{array}{cc} 1 + \frac{Z_{a}}{2 Z_{s}} & Z_{a} \left(1... ...} \end{array}\right] \left[\begin{array}{c} P_{2} \\ U_{2} \end{array}\right].$	(1)

obtained by cascading the three matrices that correspond to the three impedance terms.

Based on the approximation that $\vert Z_{a}/Z_{s}\vert \ll 1,$ Eq. (1) can be reduced to the form

$\displaystyle \left[\begin{array}{c} P_{1} \\ U_{1} \end{array}\right] = \left[... ...1 \end{array}\right] \left[\begin{array}{c} P_{2} \\ U_{2} \end{array}\right],$ (2)

which is the basic tonehole unit cell given by Keefe for transfer-matrix calculations.

The values of $Z_{a}$ and $Z_{s}$ vary according to whether the tonehole is open (o) or closed (c) as

$\displaystyle Z_{s}^{(o)}$	$\textstyle =$	$\displaystyle Z_{c}(a/b)^2 \,\left(jkt_{e} + \xi_{e}\right),$	(3)
$\displaystyle Z_{s}^{(c)}$	$\textstyle =$	$\displaystyle -jZ_{c}(a/b)^2\cot(kt),$	(4)
$\displaystyle Z_{a}^{(o)}$	$\textstyle =$	$\displaystyle -jZ_{c}(a/b)^2 kt_{a}^{(o)},$	(5)
$\displaystyle Z_{a}^{(c)}$	$\textstyle =$	$\displaystyle -jZ_{c}(a/b)^2 kt_{a}^{(c)}.$	(6)

Definitions and descriptions of the various parameters in Eqs. (3) – (6) can be found in (Keefe, 1990a).

To render these relationships in the digital waveguide domain, it is necessary to transform the plane-wave physical variables of pressure and volume velocity to traveling-wave variables as

$\displaystyle \left[\begin{array}{c} P_{1} \\ U_{1} \end{array}\right] = \left[... ...{1}^{-} \\ Z_{c}^{-1} \left( P_{1}^{+} - P_{1}^{-} \right) \end{array}\right],$ (7)

where $Z_{c}$ is the characteristic impedance of the cylindrical bore, which is equal on both sides of the tonehole.

Waveguide pressure variables on both sides of the tonehole are then related by

$\displaystyle \left[\begin{array}{c} P_{1}^{-} \\ P_{2}^{+} \end{array}\right] ... ...rray}\right] \left[\begin{array}{c} P_{1}^{+} \\ P_{2}^{-} \end{array}\right],$ (8)

where

$\displaystyle \mathcal{R}^{-} = \mathcal{R}^{+}$	$\textstyle \approx$	$\displaystyle \frac{Z_{a} Z_{s} - Z_{c}^{2}}{Z_{a} Z_{s} + 2 Z_{c} Z_{s} + Z_{c}^{2}},$	(9)
$\displaystyle \mathcal{T}^{-} = \mathcal{T}^{+}$	$\textstyle \approx$	$\displaystyle \frac{2 Z_{c} Z_{s}}{Z_{a} Z_{s} + 2 Z_{c} Z_{s} + Z_{c}^{2}},$	(10)

calculated using Eqs. (1) and (7) and then making appropriate simplifications for $\vert Z_{a}/Z_{s}\vert \ll 1.$

Figure 2 depicts the waveguide tonehole two-port scattering junction in terms of these reflectances and transmittances. This structure is analogous to the four-multiply Kelly-Lochbaum scattering junction (Kelly and Lochbaum, 1962).

**Figure 2:** Digital waveguide tonehole two-port scattering junction.
$\begin{figure}\begin{center} \begin{picture}(3,1.64) \put(0.2,0){\epsfig{file ... ...){$\mathcal{T}^{-}(z)$} \end{picture} \end{center} \vspace{-0.15in} \end{figure}$

For the implementation of the reflectances and transmittances given by Eqs. (9) – (10) in the digital waveguide structure of Fig. 2, it is necessary to convert the continuous-time filter responses to appropriate discrete-time representations.

Results of this approach are shown in Figure 3 and are compared with reproduced results using the technique of Keefe (Keefe, 1990a) for a simple flute air column with six toneholes.

**Figure 3:** Calculated reflection functions for a simple flute air column [see (Keefe, 1990a)]. Transmission line model vs. DW two-port model with one hole closed (top), three holes closed (middle), and six holes closed (bottom).
$\begin{figure}\begin{center} \epsfig{file = figures/2pttonehole.eps,width=4in} \end{center} \end{figure}$

The implementation of a sequence of toneholes in this way is diagrammed in Figure 4.

**Figure 4:** Digital waveguide two-port tonehole implementation scheme, including delay-line length interpolation filters.
$\begin{figure}\begin{center} \epsfig{file = figures/discrep.eps,width=3.8in} \end{center} \end{figure}$