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

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Chapter IV
Finger hole impedance spectra and length corrections
4.1 INTRODUCTION
Several theoretical and experimental investigations have examined the impedance effects and
equivalent circuits of open and closed tone holes. From these studies length corrections have
been derived that are in most cases sufficient for predicting the effect of a tone hole on the
tuning of a woodwind instrument—formulae for many of these length corrections are given in
Chapter 2. However, most of these studies are limited to simple keyed tone holes, formed with
a chimney that meets the key at a plane. No simple formulae exist for the equivalent circuit
elements to use in the case of finger holes such as are found on classical flutes and recorders.
These finger holes are drilled directly through the wall of the instrument, are often undercut
and are closed with the player’s fingers. In some cases such holes are as large as the tip of a
finger, and when closed the curved pad of the finger protrudes a significant distance into the
bore of the instrument. In this chapter the equivalent circuit for a finger hole is measured over
a wide range of geometrical parameters. The associated length corrections are calculated and
compared with fit-formulae in the literature. An extensive literature search found no length
corrections to use for a closed finger hole—for this case fit-formulae are derived from the measurements
in this chapter.
In the simplest one-dimensional model, an open tone hole on a woodwind instrument effectively
cuts the instrument off at the position of the hole—no acoustic waves penetrate further
along the instrument and in the electro-acoustical model the hole acts as a short-circuit to
ground. In this model closed holes have no effect. However, only for the case of large holes
at low frequencies is this approximation in any way valid. A slightly more complex model
takes into account the non-zero inertance of the air in an open tone hole. This inertance is
greater for longer holes but is still significant for short holes due to the radiation impedance
of a flanged hole. Two acoustic paths are then possible: through the tone hole to the radiation
field (‘ground’) or past the tone hole and into the rest of the flute. Thus the tone hole may be
represented in the transmission line circuit as a shunt impedance to ground. Closed holes are
represented likewise; however in this case the shunt impedance is a compliance, representing
the extra air volume beneath the closed key. Further refinements may be made to the equivalent
circuit of a tone hole. One important refinement is to add a series impedance to the circuit.
This impedance (mostly a negative inertance) represents the effect of flow widening at the tone
hole.
The effect of an open or closed tone hole on the resonant properties of woodwind instruments
may be described according to its effect in two idealised situations: when the hole is at
a pressure antinode and there is maximal acoustic flow to the outside air through the hole; and
when the hole is at a flow antinode and there is no net acoustic flow through the hole. In the