## 2 Models and Methods  
### Surface Wave Model and the Vocal Tract  
Fulcher and Scherer [6] show that the subglottal pressure \(P_{sub}\) is responsible for accelerating the air in the vocal tract as well as moving the air through the glottal duct created by the medial surfaces bounding the glottis, which is shown schematically in Figure 1. Thus, the subglottal pressure is related to the flow rate, its time derivative, and the glottal areas at the entrance \(A_{1}(t)\) and at the exit \(A_{2}(t)\) by the equation:  
\[P_{sub}=\ I_{VT}\frac{dU_{g}}{dt}+\ \frac{\rho U_{g}^{2}}{2}\left[\frac{k_{ ent}-1}{A_{1}^{2}(t)}\ +\ \frac{1-k_{xx}}{A_{2}^{2}(t)}\right], \tag{2}\]  
where \(k_{ent}\) is the entrance loss coefficient and \(k_{ex}\) is the exit coefficient [23; 24]. The dynamics of the vocal fold coordinate \(\xi\) in the SWM is described by an equation of motion [2; 6] whose driving term includes a factor \(P_{g}\), which is the average of the pressure \(P(z)\) over the medial surface of the vocal fold, that is:  
\[m\ddot{\xi}+\ B\dot{\xi}+\ k\Big{(}1\ +\ \eta\xi^{2}\Big{)}\xi=\ L_{g}TP_{g}=\ L_{g}\int_{-T/2}^{T/2}P(z)\ dz. \tag{3}\]  
Figure 1: A schematic diagram showing the surface wave propagating along the medial surface of the vocal tract. The halfwidth at the glottal entrance is \(\xi_{01}\), and that at the glottal exit is \(\xi_{02}\). In Equation (3), \(m\) is the mass of the vocal fold, \(k\) is the spring constant representing its stiffness, and \(\eta\) allows for a possible nonlinearity in the stiffness. The quantity \(L_{g}\) there denotes the glottal length (anterior-posterior direction), and \(T\) denotes the glottal thickness (inferior-superior direction). The dots in Equation (3) denote time derivatives. The pressure in the integrand of Equation (3) involves the area \(A(z,t)\). This may be expressed in terms of the vocal fold surface coordinate \(\xi_{Surf}(z,t)\) by the equation, \(A(z,t)=2\ L_{g}\ [\xi_{0}(z)+\xi_{Surf}(z,t)]\), where \(\xi_{0}(z)\) is the prephonatory shape of the glottal surface. Since the surface coordinate satisfies a wave equation, the combination of variables \(z\) and \(t\) takes a simpler form, that is, \(\xi_{Surf}(z,t)=\xi_{Surf}(t-z/c)\), where \(c\) is the velocity of the surface wave. Fundamental to the SWM is the expansion of \(\xi_{Surf}(z,t)\) as a power series in \(z\). This requires the assumption that the phase difference between the glottal entrance and the glottal exit is not too large. If only the first-order correction is needed, then \(\xi_{Surf}(z,\ t)=\xi(t)-z\frac{\dot{\xi}(t)}{c}+\ldots\), where \(\xi(t)\) is the coordinate of the center of the vocal fold, which is the same as the coordinate of Equation (3). Following [PERSON]'s assumption of a trapezoidal prephonatory shape and that the phase difference of the top and bottom edges of the vocal fold is not too large, the integral in Equation (3) can be done analytically. This yields:  
\[P_{g}(t)=P_{sub}-\left(\rho U_{g}^{2}(t)/2\right)\left[(k_{ent}-1)/A_{1}^{2}(t )\ +\ 1/A_{1}(t)\ A_{2}(t)\right]. \tag{4}\]  
Since \(z=-T/2\) for the glottal entrance and \(z=T/2\) for the glottal exit, the areas in Equation (4) may be written as:  
\[ \begin{array}{c}A_{1}(t)\ =\ 2\ L_{g}\Big{[}\xi_{01}\ +\ \xi(t)\ +\ \tau\ \dot{\xi}(t)\ +\ \ldots\Big{]};\\ A_{2}(t)\ =\ 2\ L_{g}\Big{[}\xi_{02}+\ \xi(t)\ -\ \tau\ \dot{\xi}\ (t)\ +\ \ldots \Big{]},\end{array}  \tag{5}\]  
where \(\tau=T/(2c)\) is the time for the surface wave to propagate from the center of the medial surface to the glottal exit. If no vocal tract is involved, then one can eliminate the factor of \(U_{g}^{2}\) in Equation (4) by using Equation (2). Then Equation (4) takes the form:  
\[P_{g}(t)=P_{sub}\ [1-k_{ex}-A_{2}(t)/\ A_{1}(t)]/\Big{[}1-k_{ex}+(k_{ent}-1)\ A_{2} ^{2}(t)/\ A_{1}^{2}(t)\Big{]}. \tag{6}\]  
Measurements show that for most of the glottal diameters, the entrance coefficients are much larger than the exit coefficients [23; 24].