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

4.3. RESULTS AND DISCUSSION 65
10 11 frequency (Hz)
10 10
Z s
(Pa s m 3 )
10 9
b/a = 0.5
10 8
10 7
b/a = 1.0
500 1000 1500 2000 2500 3000 3500
Figure 4.10: The shunt impedance Z s for closed finger holes of length 1.0 mm and b/a = 0.5 to
1.0, a = 7.5 mm.
hole has zero compliance and the shunt impedance is infinite.) Nevertheless a clear linear
trend is apparent, and there is no clear dependence on t.
The data of Figure 4.11 were fitted with the formula
t finger /b =−0.76b/a. (4.11)
This equation is independent of t, although for longer holes the length correction might be
expected to be reduced a little, since a longer hole intersects with a larger diameter cylinder
at the outside of the instrument, resulting in less finger protrusion. The fit formula (4.11) is
sufficient for modelling the classical flute, and the influence of t might only be important for
modelling instruments with holes meeting a nearly flat surface, as is the case for the bassoon.
4.3.2.2 Series impedance and length correction
The series length correction for closed finger holes is shown in Figure 4.12. For large holes
and thin walls, t a
(c) is positive, suggesting that the extent of the finger protruding into the hole
more than compensates for the cavity ordinarily created by a closed tone hole, and the flow
narrows near the closed finger hole. A simple empirical correction to the equation of Dubos
et al. (1999) (2.36) allows a reasonable fit to the experimental data of Figure 4.12. The extent
of finger protrusion into the hole is quantified in relation to the length t 0 . For any given hole
diameter ratio b/a,whent = t 0 the closed finger hole has no net effect on the flow and t (c)
a = 0.