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

CHAPTER 4. FINGER HOLE IMPEDANCE SPECTRA AND LENGTH CORRECTIONS 52
U 1
U 2
Z a
/2 Z a
/2
p 1
p 2
Z s
Figure 4.1: A T-junction representation of a finger hole showing pressure p and volume flow U
at the input (subscript 1) and output sides of the hole (subscript 2).
electro-acoustic analogy the hole may be represented by the T-junction shown in Figure 4.1,
comprised of the two impedances Z a (the ‘series’ impedance) and Z s (the ‘shunt’ impedance).
This lumped-element model will accurately describe the acoustic effect of the hole provided
the wavelength is significantly greater than the hole dimensions, and provided the hole is symmetrical
with respect to the bore axis. Asymmetrical holes may be represented by a T-junction
with three different impedances (Poulton 2005). Although all of the holes measured in this
chapter are symmetrical, the methods may be easily extended to measure the shunt and series
impedances of asymmetrical tone holes.
Coltman (1979) measured the shunt reactance for flute tone holes using a tube closed at
one end by a piston driver and at the other by a rigid microphone. The tone hole and key
mechanism were placed at the centre of the tube. In the two lowest modes of this system (the
so-called ‘slosh’ and ‘butt’ modes) the two quarter-wave ends oscillate in either the same or
opposite directions. In the ‘slosh’ mode the resonance frequency is almost unaffected by the
presence of the hole (since for this mode there is a pressure node at the hole position), whereas
in the ‘butt’ mode some acoustic flow escapes through the hole and the frequency is reduced.
The reactance of the tone hole is derived easily from these two resonance frequencies. With
such a simple system Coltman found the open hole shunt reactance (expressed as a length
correction) for a wide range of hole–key combinations. Coltman also measured the change
in air column effective length caused by closed tone holes. (This is related to the closed hole
shunt and series reactances.) Clearly, any frequency-dependence of the end correction cannot
be measured with this system.
Keefe (1982a) measured the shunt reactance and resistance and the series reactance of
open and closed tone holes. Using a system similar to that of Coltman, Keefe measured both
‘saxophone-’ and ‘clarinet-like’ holes. ‘Saxophone-like’ holes have a diameter much greater
than their height while the height of ‘clarinet-like’ holes is equal to or greater than their diameter.
The input impedance of a closed pipe with the tone hole in the centre was measured. The
tone hole impedance components were derived from the frequency and bandwidth of peaks in
the impedance spectrum. Keefe found that the measured reactances agreed with theory (Keefe
1982b) and with the flute data obtained by Coltman (1979). The shunt resistance of an open
hole was also measured under linear conditions although, as Keefe points out, the acoustic
level is much lower than under playing conditions, when nonlinear losses become important.