## 3 Results and Discussion  
Measurements of the threshold pressure for six vocal tract areas ranging from 2.70 cm\({}^{2}\) to 11.70 cm\({}^{2}\) are shown in Figure 8a. The value 1.0 is chosen for the entrance loss coefficient \(k_{ent}\), since the measurements of \(k_{ent}\) for larger pressures [23] tend to approach 1.0. The geometrical parameters are determined from the dimensions of the experiments; that is, \(L_{VT}\) = 19.0 cm, \(L_{g}\) = 1.5 cm, and \(T\) = 0.35 cm. The parameters \(B\) and \(c\) are chosen so that the calculated result is the same at the measured pressure (3520 Pa) at \(A_{VT}\) = 2.70 cm\({}^{2}\), and the parameters \(k\) and \(m\) are chosen so that the calculated frequency fits the measured frequency (139 Hz) at \(A_{VT}\) = 2.70 cm\({}^{2}\). The prephonatory half-width is \(\xi_{0}\) = 0.20 cm for each of the calculations shown in Figure 8. This value was not measured during the experiments, but it was considered to be reasonable from observations of the separation of the vocal folds at the pressure when sustained oscillations become possible [37]. The fit to the first four pressure measurements is very reasonable, but the measured pressures are substantially smaller than the calculations for the two larger areas. One explanation is to add a small exit coefficient for the two larger areas. We have found that adding an exit coefficient of 0.05 at \(A_{VT}\) = 8.7 cm\({}^{2}\) is adequate to reduce the calculated pressure from 4460 Pa to 4040 Pa, which is enough to remove the discrepancy in Figure 8a. This value for the exit coefficient is similar to that calculated from the measurements at larger glottal widths in [23] (Figure 3). A similar reduction occurs at \(A_{VT}\) = 11.7 cm\({}^{2}\). Another possible source of the discrepancy is the nature  
Figure 8: (Color online) (**a**) Measured and calculated threshold pressures for vocal tracts with different areas for vocal fold model 113. (**b**) Measured and calculated frequencies for vocal tracts with different areas for vocal fold model 113.  
of the supraglottal flow pattern, as shown in Figure 9. It was observed to change its character when the cross-section of the vocal tract was large in comparison with the smaller cross-sections [40]. The jet was found to be deflected to either lateral side for \(A_{VT}\) = 2.7 cm\({}^{2}\), due to the appearance of a large vortex that inhibited the jet from proceeding straight ahead. This large vortex increases the flow resistance in the vocal tract channel. As a consequence, the pressure increases in the supraglottal region immediately downstream of the vocal folds. Upon enlarging the vocal tract, the volume of the large vortex and thus its inertia rise, resulting in a further increase of the supraglottal pressure downstream to the vocal folds, and with it an increase of the subglottal threshold pressure. For \(A_{VT}\geq\) 8.7 cm\({}^{2}\), this large vortex disappears, and two vortices located symmetrically on each side of the flow jet allowed the jet to proceed without deflection. As the flow resistance decreases owing to the vanishing of the inhibiting large vortex, the supraglottal pressure and the concomitant threshold pressure decrease for the larger areas in Figure 8a. The calculated threshold pressure values in Figure 8a seem to be leveling off as the area of the vocal tract increases. One would expect this, since for larger areas, the vocal tract inertance becomes small (discussion following Equation (1)) and exerts less influence on the pressure required to produce sustained oscillations. Thus, it should be possible to make contact between the large area limit of Figure 8a and the result for no vocal tract. Further calculations confirm this. For example, at \(A_{VT}\) = 60.0 cm\({}^{2}\), \(P_{th}\) = 4960 Pa, and at \(A_{VT}\) = 80.0 cm\({}^{2}\), \(P_{th}\) = 4980 Pa. The formula listed in Equation (8) yields \(P_{th}\) = 4963 Pa for this limit, and thus, the calculations for these two larger areas are within 0.4% of the expected value.