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

64 CHAPTER 4. FINGER HOLE IMPEDANCE SPECTRA AND LENGTH CORRECTIONS
0
0.05
0.1
t = 1 mm
t = 2.5 mm
t = 5.0 mm
theory
0.15
t a
/ a
0.2
0.25
0.3
0.35
0.4
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
b / a
Figure 4.9: The series length correction t a for open holes compared with (2.37).
are coupled.
Figure 4.9 indicates that some modifications to the fit formula (2.37) may be required for
open finger holes but (2.37) is probably sufficient to use in the current work.
4.3.2 Closed Finger holes
4.3.2.1 Shunt impedance and finger length correction
The shunt impedance of a closed finger hole may similarly be considered as the sum of the
inner radiation impedance and the input impedance of the hole tube section terminated by
the load Z finger , the impedance of the player’s finger. The load impedance is derived from the
measurement of Z s as in §4.3.1.1 and the finger length correction is given by
t finger = cot−1 (Z finger )
−ikZ 0
. (4.10)
The correction t finger is expected to be a negative quantity, since the finger protrudes into the
hole, increasing in size with hole diameter.
The shunt impedance Z s for closed finger holes of length 1.0 mm with b/a = 0.5 to 1.0 is
shown in Figure 4.10. Similar spectra were obtained for hole lengths 2.5 and 5.0 mm. From
these spectra, the length correction due to the player’s fingers may be derived. Due to the errors
below 1.5 kHz, frequencies below this limit were not included in the calculation of the length
correction.
The measured finger length correction is shown in Figure 4.11. Some data are missing,
due to the measured impedance being above the dynamic range of the measuring system. (For
example, if the volume occupied by the finger is equal to the volume of the hole, then the closed