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

5.3. RESULTS AND DISCUSSION 81
10 8
10 7
|Z| (Pa s m −3 )
10 6
10 5
10 4
model
experiment
0 1 2 3 4
f (kHz)
Figure 5.12: Impedance spectra (experiment and model) for a closed classical headjoint before
adjustment of parameters.
measured, the model is a much closer fit to experiment, for both the modern and classical flute
headjoints.
Several small refinements were made to the model to improve the fit. A small length correction
was applied to the embouchure hole to achieve closer fitting of impedance minima. The
correction
t emb = 0.5γ 2 b, (5.1)
where b is the inside radius of the embouchure hole and γ = b/a for bore radius at the embouchure
hole a, was found to fit both headjoints sufficiently. The necessity of such a length
correction is not unexpected, given that the effective cone angle for the embouchure hole (accentuated
as it is by the small entry diameter) is relatively large. The length corrections given in
Chapter 2 and used in the model were all calculated for cylindrical holes, and cannot be applied
to conical holes without modification.
With the above-mentioned corrections applied, the model overestimates the height and
depth of impedance maxima and minima. For this reason a series resistance and a shunt conductance
were added at the input to the network model. The value of each allowed precise
fitting of the extrema. The values (in SI units)
R emb = (6.9 × 10 −6 Hz −1 )Z 0 f and (5.2)
G emb = (1.3 × 10−4 Hz −1 )f
(5.3)
Z 0
for (respectively) the series resistance and shunt conductance were used. Again, using these