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In order to represent a halfhole model in a wave digital modelling context, a decomposition of the instantaneous variables ( and ) into wave variables is required. Taking a threeport modelling approach (as described in [5, 7, 9]), and applying eqs. (1) to the network in Fig. 2, the modelling structure depicted in Fig. 3 results. Because the main bore is modelled as a digital waveguide, both and must equal the main bore characteristic impedance . The scattering equations of the threeport junction that models the wave interaction at the intersection between the main bore and the tonehole are:
with

(8) 
where the lumped element portresistance has to be chosen such that the structure is computable.
Figure 3:
Structure for discretetime modelling of the halfhole model. The delaylines model wave propagation in the main bore.

The continuoustime tonehole ``reflectance'' is:

(9) 
Note that does not correspond to the actual physical tonehole reflectance as seen from the main bore. Substitution of eq. (2) and applying the bilinear transform gives the wave digital reflectance, which has the form of an allpass filter:

(10) 
with
where
is the bilinear operator. In order to avoid a delayfree loop, must be chosen such that the wave entering is not immediately reflected back towards the threeport scattering junction via . This requires setting the filter coefficient , which means that we must choose
. The resulting digital reflectance is:

(12) 
with

(13) 
Both and are computed using the term , so that we can let
(which corresponds to fully closing the tonehole). In order to investigate the effect of the discretisation, the halfhole twoport reflectance (i.e., the reflectance
) of the halfhole was computed for a range of tonehole states. Figure 4 compares the continuoustime halfhole model with its digital version, the ``wave digital tonehole model'', in terms of magnitude response.
Figure 4:
Twoport reflectance of the continuoustime (top) and the discretetime (bottom) halfhole model, for a range of tonehole states (
).

As can be expected, the discretetime model closely approximates the continuoustime model at the lower frequencies. However, the deviation is rather large at the higher frequencies. Fortunately, this discrepancy is relatively insignificant in a full instrument implementation, because the air column reflection function is strongly lowpass due to viscothermal and radiation losses.
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