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

obtained by cascading the three matrices that correspond to the three impedance terms. - Based on the approximation that
Eq. (2) can be reduced to the form

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

*Z*_{s}^{(o)}= (4) *Z*_{s}^{(c)}= (5) *Z*_{a}^{(o)}= - *jZ*_{c}(*a*/b)^{2}*kt*_{a}^{(o)},(6) *Z*_{a}^{(c)}= - *jZ*_{c}(*a*/b)^{2}*kt*_{a}^{(c)}.(7)

- Definitions and descriptions of the various parameters in Eqs. (4) - (7) 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

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

where

calculated using Eqs. (2) and (8) and then making appropriate simplifications for - 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).
- For the implementation of the reflectances and transmittances given by Eqs. (10) - (11) 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:**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). - The implementation of a sequence of toneholes in this way is diagrammed in Figure 4.

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