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

CHAPTER 4. FINGER HOLE IMPEDANCE SPECTRA AND LENGTH CORRECTIONS 60
0.9
this work
Dalmont et al. 2001
0.8
0.7
δ cyl
/ a
0.6
0.5
0.4
0.3
0 0.2 0.4 0.6 0.8 1
flange ratio
Figure 4.6: The measured length correction δ cyl for radiation from open finger holes of length
2.5 mm compared to the fit-formula of Dalmont et al. (2001). The ‘flange ratio’ is the ratio of
the radius of the finger hole to the radius of the cylindrical flange.
flanged pipe as the flange ratio approaches zero. Therefore, the fit formula will be used henceforth.
4.3.1.2 Series impedance and length correction
The measured series impedance for the 2.5 mm holes is shown in Figure 4.8. Only holes
with b/a > 0.5 are shown, since the series impedance for holes of smaller diameter is below the
dynamic range of the measurement system. In the subsequent analysis frequency components
below 1.8 kHz were ignored, due to the influence of systematic errors. Likewise, the singularity
at approx. 2.1 kHz was removed before analysis.
The series length correction t a was calculated from the series impedance Z a using the equation
t a =
Z a
ik Z 0
. (4.9)
The imaginary part of t a was discarded and the results were averaged over frequency. The measured
length corrections for all sets of finger holes are shown in Figure 4.9. Shown for comparison
is the equation of Dubos et al. (1999) for long finger holes (2.37). This equation overestimates
the size of the correction for small holes and underestimates it for large holes—however
the investigated holes do not satisfy the condition t > b so the equation is unlikely to fit exactly.
Dubos et al. (1999) also gives an equation which scales with t for open tone holes (2.35), but this
equation was found to overestimate the size of t a , particularly at small t. Dalmont et al. (2002)
question the use of this formula, since for short holes the internal and external discontinuities