Flute acoustics: measurement, modelling and design - School of ...

CHAPTER 5. IMPEDANCE SPECTRA OF THE FLUTE AND CLARINET 74
the player’s embouchure. The embouchure hole is usually slightly elliptical or oval in crosssection
and is tapered so that it is larger in cross-sectional area closer to the bore. The modern
flute measured in this chapter has a headjoint with a bore 168.0 mm long. The tuning slide
section of the headjoint is cylindrical, 37.0 mm long with a bore diameter of 19.0 mm. The bore
tapers smoothly towards the cork, where the diameter is 16.8 mm. The McGee classical flute
headjoint has a cylindrical bore, 158.5 mm long with a bore diameter of 19.0 mm.
Acoustically, the headjoint thus comprises the embouchure hole, an upstream bore section
stopped by the cork and a downstream bore section which attaches to the body of the flute.
Despite such geometrical simplicity, the headjoint is arguably the most complex section of the
flute to model, and may require the use of empirical parameters for a good fit.
The waveguide model was first tested on a simple T-junction, shown schematically in Figure
5.6 (for details of the model implementation, see Chapter 8). The input arm to the junction
was of diameter 7.8 mm, equal to the diameter of the small bore impedance head. The
T-junction was attached to the impedance head and the impedance was measured both with
the output arm open and closed. The microphones are thus positioned sufficiently distant from
the junction to measure only plane waves, and there is no diameter discontinuity at the reference
plane of the impedance head. In a real headjoint there are two diameter discontinuities—
one where the embouchure hole attaches to the impedance head and another at the junction
between the embouchure hole chimney and the bore of the instrument—and these discontinuities
are close enough that the evanescent modes at each can interact. This simple T-junction
should be a little easier to model than a real headjoint. The measured impedance spectra along
with the predictions of the model are shown in Figures 5.7 and 5.8. The prediction of the model
is quite good for this simple T-junction, although the attenuation factor at high frequencies
appears to be a little higher than in the model.
The input impedance spectra of the modern and classical flute headjoints were measured
in order to test and refine the model. The headjoints were measured both open to the air and
closed by a brass stop. The measured spectra for the open headjoints are shown in Figures 5.9
and 5.10, along with the predictions of the model, using published tone hole length corrections
at the junction with the bore ((2.34), (2.35) and (2.39) in Chapter 2). The prediction of the model
is quite good at impedance maxima but not at impedance minima (which occur at a higher
frequency and are lower in impedance than measured). The model deviates similarly from
experiment for closed headjoints (Figures 5.11 and 5.12).
Clearly, the discontinuity created at the input to the instrument by the measuring system
changes the measured input impedance of the headjoint. This could in principle be accounted
for using multi-modal theory, although this approach is complicated by the consideration that
higher modes at the inside and outside of the embouchure hole interact. A simpler approach
was found to be sufficient.
The model deviates from experiment most severely at impedance minima, when flow into
the embouchure hole is greatest. However, the flow does not follow closely the walls of the
embouchure hole because of the input discontinuity. If the embouchure hole is modelled as
a cone with entry diameter equal to the diameter of the impedance head and exit diameter as