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

88 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 5.22: Impedance spectra (experiment and model) for the fingering C♯4 on the modern
flute using (2.30) to model the extra impedance caused by the overhanging keypad.
available for radiation and hence lead to a greater than expected length correction. For these
reasons, an empirical correction may be required. Figure 5.22 shows the model fit for C♯4 and
Figure 5.23 shows the model fit for the fingering A4 (diffuse) with measured key dimensions (h,
the height of the disk above the hole perimeter was taken to be that at the hole axis).
5.3.4.3 Two holes open
In the previous section, the model was used to model the impedance of fingerings with one
open hole. The model was then tested on the fingering D4, in which two consecutive holes
are open. Figure 5.24 shows the prediction of the model compared with experiment using the
corrections applied in the previous section. The model overestimates the height and depth of
impedance extrema, particularly in the region 1.5–2.5 kHz. For this reason, the resistance
R open hole = 0.4Z 0 (kb) 2 (5.5)
was added to the radiation impedance of the open hole (k here is the wavenumber and b is
the radius of the hole). Dalmont et al. (2001) do not address the the effect of an overhanging
keypad on the real part of the radiation impedance; the increase seen here (albeit indirectly) is
physically quite plausible.
Inspection of Figure 5.25 reveals that over the frequency range 1.5–2.5 kHz the model locates
the impedance extrema at frequencies slightly higher than measured. In this frequency
range (encompassing the cut off frequency) the radiation impedance of open holes has the