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

86 CHAPTER 5. IMPEDANCE SPECTRA OF THE FLUTE AND CLARINET
10 8
10 7
m
s
(Pa
10 6
|Z|
− 3 )
10 5
model
experiment
10 4
0 1 2 3 4
f (kHz)
Figure 5.19: Impedance spectra (experiment and model) for the modern flute with all holes
closed and with the keypad correction given in (5.4).
have one open hole and the remainder closed, and thus afford a simple test of the capability of
the waveguide model to account for open holes.
Figure 5.20 compares experiment and model for the fingering C♯4 on the modern flute.
This fingering is identical to C4 except that the last hole is open. In Figure 5.20 the open hole
has been modelled using the corrections in §2.2.10 and using a radiative load for a pipe with
circular flange of width 1.0 mm (2.26). Thus any ‘shading’ of the hole by the keypad (and consequent
flattening of the played note) is not yet modelled. As expected, the impedance extrema
predicted by the model are sharper than measured.
The fingering A4 (diffuse) on a modern flute again has only one open hole, but in contrast
to C♯4 the open hole is followed by an array of closed tone holes (being roughly halfway along
the flute) and the keypad over the hole is perforated (the key in question is one of the six keys
normally covered by the player’s fingers). Figure 5.21 shows the prediction of the model without
any shading of the hole by the raised keypad. As in Figure 5.20, the predicted impedance
extrema are sharper than those measured.
Empirical fits by Dalmont et al. (2001) (equations (2.30) and (2.32)) give the extra length
correction (low frequency limit) associated with a disk (which may be perforated) hanging over
a hole. This equation is expected to be a reasonable approximation to the effect of a key on a
flute, although in contrast to the system considered by Dalmont et al. open keys on a flute are
never perpendicular to the hole axis, the surface of the key itself is compliant, and the components
of the flute surrounding the hole and key may contribute to a reduction in the solid angle