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:T section transmission-line representation of the tonehole.
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
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 Za and Zs vary according to whether the tonehole is open (o) or closed (c) as
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 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
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).
Digital waveguide tonehole two-port scattering junction.
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.
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.
Digital waveguide two-port tonehole implementation scheme, including delay-line length interpolation filters.