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

66 CHAPTER 4. FINGER HOLE IMPEDANCE SPECTRA AND LENGTH CORRECTIONS
0.1
0.2
t = 1.0 mm
t = 2.5 mm
t = 5.0 mm
fit formula
0.3
t finger
/ b
0.4
0.5
0.6
0.7
0.8
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
b / a
Figure 4.11: The measured length correction t finger and fit formula (4.11).
For closed finger holes the following modified version of (2.36) is used:
t a (c) =
⎧
⎨
−bγ 2
1.78coth(1.84(t−t 0 )/b)+0.940+0.540γ+0.285γ 2 t ≥ t 0
⎩
−bγ 2
1.78/(1.84(t−t 0 )/b)+0.940+0.540γ+0.285γ 2 t < t 0 ,
(4.12)
where for the case t < t 0 the coth(x) function is replaced by its limit lim x→0 coth(x) = 1 x .
The finger protrusion length t 0 was fitted empirically to the experimental data by the equation
t 0 = b[0.55 − 0.15sech(9t/a) + 0.4sech(6.5t/a)(γ − 1)]. (4.13)
Equation (4.12) is plotted along with the experimental data in Figure 4.12.
4.4 CONCLUSIONS AND FURTHER DIRECTIONS
In this chapter the reactive impedance components in the equivalent T-circuit for open and
closed finger holes were measured and compared with fit-formulae in the literature. For open
finger holes existing fit-formulae were found to be sufficient, but for closed finger holes new
fit-formulae were calculated in order to match better the experimental results.
Several refinements to the technique used in this chapter are expected to yield more accurate
results. The dynamic range of the system is in part limited by geometrical differences
among the PVC pipes used. For example, small changes in pipe diameter may swamp the flowwidening
effect of a small finger hole. In further work the test pipes should be more accurately
machined. Alternatively, a finger hole may be formed in a PVC test pipe and subsequently filled