## A Two-Port Tonehole Model

• 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, Za, 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, Zs, 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
 = (1) = (2)

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
 (3)

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

• The values of Za and Zs vary according to whether the tonehole is open (o) or closed (c) as
 Zs(o) = (4) Zs(c) = (5) Za(o) = -jZc(a/b)2 kta(o), (6) Za(c) = -jZc(a/b)2 kta(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
 (8)

where Zc 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
 (9)

where
 (10) (11)

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.

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