## 2 Models and Methods
### Surface Wave Model and the Vocal Tract
Neglecting \(k_{ex}\) in Equation (6), specializing to the rectangular case where \(\xi_{01}=\xi_{02}=\xi_{0}\), and carrying out an expansion in inverse powers of \(\xi_{0}\) there allows Equation (6) to be simplified to the following form:  
\[P_{g}(t)\ =\ 2\ P_{sub}\ \tau\ \dot{\xi}(t)/(k_{ent}\xi_{0})\, \tag{7}\]  
provided that only the lowest-order terms in \((\xi_{0})^{-1}\) are kept. Thus, the driving force on the right-hand side of Equation (3) simplifies to a term that has the effect of a negative damping term. Equating the coefficient of this term to that of the damping term on the left-hand side of Equation (3) yields:  
\[P_{th}=k_{ent}\ B\ c\ \xi_{0}/L_{g}T^{2}, \tag{8}\]  
which is essentially [PERSON]'s equation for the threshold pressure [2], since the transglottal coefficient \(k_{t}=k_{ent}\), when \(k_{ex}\) is neglected. Equation (8) requires careful interpretation. If the entrance loss coefficient \(k_{ent}\) is assumed to be near one (\(k_{ent}=1.37\) was a typical value for much of the earlier work), then Equation (8) predicts that the threshold pressure becomes very small in the limit of small glottal half-widths. However, none of the data collected in the experiments that [PERSON] et al. did to test the predictions of the SWM showed this behavior [3; 5]. [PERSON] and [PERSON] [6] showed that the key to resolving this discrepancy was a careful analysis of the behavior of the entrance loss coefficient at small glottal widths, since this coefficient could become very large in this limit, where viscous effects become large. In fact, they showed that the behavior of the entrance loss coefficient in the limit of small glottal widths was consistent with an inverse relationship, that is:  
\[k_{ent}\;=\;E/\xi_{0}\;+\;F\;. \tag{9}\]  
In Figure 2, calculations based on the analytic expressions presented in Equations (8) and (9) are compared with measurements done by [PERSON] and [PERSON] [5], when no vocal tract was present in their experiments. As emphasized earlier [6], the properties of Equations (8) and (9) allow one to account for the observed slope of the line expressing the threshold pressure as a function of the glottal half-width as well as its intercept at zero half-width. For this fit to the data, the parameters \(B\) = 112 g/s, \(c\) = 400 cm/s, \(E\) = 0.072 cm, and \(F\) = 0.600 are required. The geometrical parameters, \(L_{g}\) = 2.22 cm and \(T\) = 1.1 cm, are determined from the dimensions of the vocal fold model used in the experiments. When the vocal tract is added, a numerical solution is required, since the time derivative in Equation (2) prevents obtaining an analytic connection between the glottal flow rate and the subglottal pressure. The numerical solution involves the simultaneous solution of Equations (2)-(5). The inertance term of Equation (2) does not introduce additional free parameters, since the vocal tract length and its area are determined from the experiment. For [PERSON] and [PERSON]'s experiment, \(L_{VT}\) = 16.51 cm and \(A_{VT}\) = 2.82 cm\({}^{2}\). The results of the numerical solution are shown in Figure 2, where they are compared with [PERSON] and [PERSON]'s vocal tract measurements. The fits to the data collected with the vocal tract as well as those collected without the vocal tract are very good. The largest differences (about 13%) occur at a glottal width of 0.10 cm, and most of the differences are substantially smaller. The numerical solution of Equations (2), (3), (5), and (6) requires values for the mass and stiffness parameters. Following the lead of [PERSON] and [PERSON] [1; 25; 26; 27], these were chosen to be \(k\) = 88,000 g/s\({}^{2}\) and \(m\) = 0.15 g. In order to make the calculation as simple as possible, the nonlinearity parameter was set equal to zero. It is worth noting that the use of Equations (2) and (4) for the vocal tract calculation eliminates the necessity for the inverse \(\xi_{0}\) expansion, and thus removes the small amplitude requirement.