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## THE HALF-HOLE MODEL

Figure 1 shows a cross-sectional view of a woodwind tonehole. In the low-frequency limit, the hole dimensions are usually small in comparison with the acoustic wavelength, thus the acoustic behaviour may be characterised by a lumped acoustic element. For an open hole, the behaviour is approximately that of a pure inertance, while for a closed one it approximately corresponds to a pure compliance [10].
For intermediate tonehole states (partially open holes or half-holes''), the tonehole volume can be divided into an open part'' that behaves as an inertance, and a closed part'' that behaves as a compliance. These volumes operate in parallel, thus the half-hole load impedance is:
 (2)

Figure 2 shows the network equivalent of this model.
The half-hole compliance () and inertance () are given by:
 (3)

where the parameter expresses the tonehole state, defined as the ratio between open and total tonehole volume. The tonehole height is defined such that its product with the tonehole surface equals the geometric volume [3]:
 (4)

The tonehole effective length is similar to , though it includes inner and outer length-correction terms. The value for given in [3] is frequency-dependent, though at low frequencies the following approximation is sufficiently accurate:
 (5)

An additional effect of inserting a hole in a woodwind bore is that the effective acoustic length of the bore is slightly reduced on both sides of the hole [3, 10]. This length-correction depends on the tonehole series equivalent length, for which we found a simplified expression that applies to both open and closed tonehole state:
 (6)

The total main bore negative length correction for a tonehole with series equivalent length is [3]. Thus if the lengths of the main bore sections on each side of the tonehole are and , they should be corrected to and , respectively. Because the length-correction is very small, this formulation differs only slightly from the series impedance formulation in [3].

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