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

CHAPTER 2. THEORY AND LITERATURE REVIEW 26
2.2.10.1 Formulae for the series impedance
Dubos et al. (1999) give the following formulae for the series length correction for an open
and a closed tone hole:
t (o)
a =
t (c)
a =
−bγ 2
1.78tanh(1.84t/b) + 0.940 + 0.540γ + 0.285γ 2 (2.35)
−bγ 2
1.78coth(1.84t/b) + 0.940 + 0.540γ + 0.285γ 2 . (2.36)
For long side holes (t > b) the series length corrections for open and closed side holes are approximately
equal. The length correction is then (Dubos et al. 1999)
t a = −(0.37 − 0.087γ)bγ 2 . (2.37)
Nederveen et al. (1998) give a slightly different formula for the series length correction:
t a = −0.28bγ 2 . (2.38)
Equations (2.37) and (2.38) agree for γ = 1. The difference between the formulae is not particularly
significant since the length correction becomes negligible for small holes.
2.2.10.2 Formulae for the inner impedance
The length correction t i for the inner transitional impedance Z i is given by Nederveen et al.
as
t i = (0.82 − 1.4γ 2 + 0.75γ 2.7 )b (2.39)
and by Dubos et al. (see Dalmont et al. 2002) as
t i = t s − t a γ 2 /4, (2.40)
where t s = (0.82 − 0.193γ − 1.09γ 2 + 1.27γ 3 − 0.71γ 4 )b, and the dependence on t a is due to a
different definition of the impedance components in the T-circuit.
Benade & Murday (1967) measured the inner length correction for an open hole of radius b
drilled through the wall of a pipe with inside radius a. The inner correction E i (= (t m + t i )/b in
the current nomenclature) is given by E i = [1.3 − 0.9(b/a)] for (b/a) ≤ 0.72.
2.2.11 The open hole cutoff frequency
In a simple application of transmission line theory to woodwind musical instruments, it is very
useful to consider an array of open tone holes as somewhat like a high-pass filter. This is because
at sufficiently high frequencies the mass of air in an open hole is too great to accelerate
efficiently, and to an acoustic wave the open hole presents an impedance much greater than
that of the downstream section of the instrument. Benade (1976) derives a theoretical expression
for the cutoff frequency of a continuous waveguide approximating this situation. For an
array of open tone holes spaced uniformly at a distance 2s from each other, where the holes all
have radius b and the bore has radius a, the cutoff frequency is given by
f cutoff ≈ 0.11 b a
c
te s , (2.41)