This and the following sections focus on elaborating an accurate acoustic model of the tuning slot, by which the measured results can be reproduced accurately. To achieve this objective, woodwind tonehole models are examined and a hybrid model of the pipe is set up utilizing one-dimensional transmission line elements and results of simulations carried out by means of the finite element method.
A one-dimensional acoustic model
1D model of a woodwind instrument with a single tonehole is shown in Figure 7.1. The model consists of two acoustic waveguide elements, a two-port tonehole model and two lumped two-pole elements representing the radiation impedances at the mouth and open end of the instrument. The first waveguide represents the instrument from the mouth to the position of the tonehole, while the second waveguide stands for the remaining part of the instrument up to the open end. The configuration of a common cylindrical tonehole is shown in Figure 7.2. Using the general notation of the corresponding literature,adenotes the radius of the main bore,bstands for the (effective) radius of the tonehole andtrepresents the height of the side bore. The ratiob/ais denoted byδ.
This setup has been investigated by a number of researchers. Some of them [24, 45, 53, 82, 83, 91, 92, 108] developed tonehole models which have been applied for the calculation of acoustic properties of woodwind instruments, such as the clarinet, the flute or the recorder. These models are applicable for a relatively large range of parameters δ and t. Tonehole models of several authors are reviewed in the following and their applicability for tuning slot modeling is examined later on.
7.2.1 The T-circuit model
The tonehole can be represented by an equivalent T-circuit in a one-dimensional waveguide model. This model was first introduced by Benade [24] and has been generalized by Keefe [83],
7.2. THE WOODWIND TONEHOLE MODEL 85 b
2a t
Figure 7.2. Configuration of a typical cylindrical tonehole in the main bore of a woodwind instrument. a– main bore radius,b– tonehole radius,t– tonehole chimney height
Za/2 Za/2 Zs
Figure 7.3. Equivalent T-circuit of a tonehole. Zs– shunt impedance,Za
– series impedance
who also published measurement data [82] verifying the theoretical results.
The T-circuit model T-circuit
model is depicted in Figure 7.3. It is composed of a shunt impedanceZsand two
series impedancesZa/2. This general model is the basic structure used by most of the authors, however, each of them gives slightly different formulations for the values of the shunt and series impedances based on theoretical deductions, experimental, or computer simulation results. The same model is also used in wave-digital modeling of woodwind instruments, see e.g. [144].
One of the recent works in the topic is by Lefebvre & Scavone [91, 92]. They have developed a novel formula for the length correction parameters of an open tonehole based on data fit ap-proximations to finite element simulation results. Their formulation provides a better fit than the previous models for lower values ofδ. They report that “The most important discrepancy be-tween current tonehole theories and our simulation results concerns the frequency dependence of the shunt length correction for toneholes of short height . . . ” [91]. This specific issue is addressed later in Section 7.5.
In the following the calculation of the parameters of the T-circuit is discussed. The work by Dalmontet al. [45] is followed and extended regarding the evaluation of tonehole parameters.
The open hole shunt impedanceZsis defined in [45] as
tonehole shunt impedance
Zs=Zi+Zo, (7.1)
withZi andZodenoting the inner and outer correction impedances, respectively. In a low fre-quency approximation, with neglecting the radiation losses for the time being, equation (7.1) can be expanded as
Zs=Zi+Zo= jkZH0(t+ti+tm+tr), (7.2) wherekis the wave number andZH0 =ρ0c/(πb2) is the acoustic plane wave impedance of the side bore, withρ0denoting the average density of air, andcrepresenting the speed of sound. The termsti,tm, andtrdenote the inner, the matching volume, and the radiation length corrections and are discussed later.
7.2.2 Radiation impedance
To take the real part of the radiation impedance into account as well, the outer impedanceZois calculated as a tube of lengtht+tmterminated by the radiation impedanceZr, which gives [45]
outer impedance Zo=ZH0Zr+ jZH0tank(t+tm)
ZH0+ jZrtank(t+tm). (7.3)
Using equation (7.1) with the equivalent length approximation ofZileads to
Zs= jkZH0ti+Zo. (7.4)
2a 2b
t
Matching volume
Internal volume External
volume
a 2b
t
Figure 7.4.The matching volume. Left: Definition using the tonehole model. Right: problem of interpreta-tion in case of small wall thickess
To evaluate the value ofZr radiation
impedance
the geometry of the flange must be known. Two cases can be calcu-lated analytically, (1) when the flange is infinite, or (2) when there is no flange at all, as discussed in Section 3.4. The formulas (3.64) and (3.65) can be utilized to obtain the reflection coefficientRr and the corresponding length correctiontr. When the opening is considered elliptical, the length correction is scaled by a corresponding factorK(ε), as defined in eq. (3.66). Finally, the radiation impedanceZris obtained by using eq. (3.63).
In reality, the geometry of the flange is somewhere in between the infinite flange and un-flanged limits. It can be considered as a rectangular opening with a cylindrical flange, similar—
but not identical—to setup (f) of the paper by Dalmontet al.[46].1
7.2.3 Matching volume correction
The matching volume is depicted in Figure 7.4, where the area corresponding to the matching volume is marked by gray. The equivalent length correction value oftm = Vm/SH, withVm
denoting the matching volume, is approximated by Nederveenet al.[108] as mathcing
volume correction
tm=bδ
8 1 + 0.207δ3
. (7.5)
As it can be seen, the matching volume is an external correction to toneholes. If the wall thickness is very small, the matching volume correction can turn out to be negative, as illustrated in the right hand side of Figure 7.4. This issue is addressed in Section 7.3.
7.2.4 Inner length correction
Different formulas for the inner length correction termti can be found in [53, 83, 92, 108]. Dal-montet al.[45] use the relation obtained by Nederveenet al.[108] for the calculation of various toneholes. The formula reads as
inner length
correction ti= 0.82−1.4δ2+ 0.75δ2.7
b. (7.6)
It is emphasized [83, 91] thattiis difficult to calculate analytically, and for small values oftthe coupling between the inner and outer acoustic fields prevents the separate analysis of outer and inner length corrections.
1Reference [46] is a numerical study that investigates the radiation impedances of tubes with different flanges.