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

2.2. ACOUSTICS OF WOODWIND INSTRUMENTS 25
Figure 2.7: A cylindrical side hole, showing dimensions and the ‘matching volume’ (Dalmont
et al. 2002).
Figure 2.8: The equivalent circuit for a cylindrical side hole (Nederveen et al. 1998).
Z m must be added to the impedance of the hole since the hole does not meet the main bore of
the instrument at a plane. The ‘matching volume’ illustrated in Figure 2.7 may be accounted
for by a length correction t m added to the hole segment. A fitting formula for t m was calculated
by Keefe (1990) and modified by Nederveen et al. (1998) to make it accurate to within ±0.001:
t m = bγ
8 (1 + 0.207γ3 ). (2.34)
If the tone hole is slightly undercut, the correction t m may be modified so that the equivalent
volume includes the volume of the removed wall material.
The impedance corrections Z a and Z i shown in Figure 2.8 arise from higher mode effects
in the vicinity of the hole. The series impedance Z a is a negative inertance and arises from
the flow entering the side hole to an extent (Figure 2.9a). The acoustic mass is thus reduced in
the vicinity of the hole, and the effect is similar to a reduction in the main bore length, scaled
by cos 2 (kL R ), where L R is the distance to the nearest flow antinode. Figure 2.9b shows the
flow and potential lines for an open tone hole with all of the flow shunted by the hole. The
shunt impedance is the impedance of the hole section (including the radiation impedance and