## 3 Results and Discussion
Calculations for the frequency of the oscillation at threshold pressure for the six areas of Figure 8a are shown in Figure 8b, where they are compared with the measured frequencies. The fit to the data is very good. This fit requires a nonlinearity parameter (Equation (3)) \(\eta\) = 250 cm\({}^{-2}\), which is also present in the calculation of Figure 8a. Adding a nonlinearity parameter is necessary because of the behavior of the amplitude of oscillation as the areas of Figure 8 increase. For example, at \(A_{VT}\) = 2.70 cm\({}^{2}\), the displacement of the vocal fold from equilibrium varies between \(-\)0.051 cm to 0.043 cm and at \(A_{VT}\) = 11.70 cm\({}^{2}\), the displacement varies from \(-\)0.0143 cm to 0.0135 cm. The amplitude decreases monotonically as the vocal tract area increases, and the effective stiffness of the vocal fold decreases, explaining the frequency decrease of Figure 8b. If the nonlinearity parameter is set to equal to zero, then the trend of the calculations does not match the trend of the data. Given that the spring constant \(k\) and Young's modulus \(E\) both measure the force required to produce a deformation \(\Delta L\) in an object of length \(L\), one would expect there to be a relationship between these two physical quantities. [PERSON] et al. [41] show that this relationship may be expressed:  
\[k=E\ \frac{A}{L}, \tag{12}\]  
where \(A\) is the area of the cross-section of the solid object (perpendicular to \(L\)). The geometry of Model 113 and Model 114 was the same, and thus, the effective stiffness of these models should be proportional to the ratio of their Young's moduli, (\(E_{113}\) = 4.4 kPa and \(E_{114}\) = 2.5 kPa). On the basis of these considerations, one would expect \(k\) = 65,900 g/s\({}^{2}\) for Model 114. Figure 9: Flow structures in the supraglottal channel for vocal tract areas (**a**) 2.7 cm\({}^{2}\), (**b**) 11.7 cm\({}^{2}\), and for (**c**) no vocal tract. Additional information about the flow structures can be found in [40]. In analyzing the damping of an oscillator, a dimensionless parameter \(\zeta\) is sometimes introduced. This parameter measures how close the damping [42; 43] is to critical damping (\(\zeta=1\)). It also connects the damping constant \(B\) to the oscillator stiffness \(k\) and the mass \(m\) by the Equation [1]:  
\[B=2\ \zeta(k\ m)^{1/2}. \tag{13}\]  
Using the parameters listed in Figure 8, the value of \(\zeta\) is 0.541. With the goal of minimizing the number of free parameters, the same value of \(\zeta\) is chosen for Model 114. Lowering the value of the mass by about 10% from that used in Figure 8 gives a damping constant \(B=109\) g/s for Model 114. The change of mass is necessary to fit the measured value of the frequency (110 Hz) for Model 114 at \(A_{VT}=2.70\) cm\({}^{2}\). The results based on Equations (12) and (13) are shown in Figure 10a, where they are compared with measurements. The fit of Figure 10a is of a similar quality to that of Figure 8a; that is, the calculated values for threshold pressure are in reasonable agreement with the measurements for the four smaller areas of Figure 10a. However, the same discrepancy appears at the two larger areas, and the cause of this discrepancy is likely to be the same as that for Figure 8a. Choosing the nonlinear parameter \(\eta=190\) cm\({}^{-2}\) gives a good fit to the frequency data presented in Figure 10b. The mixture of Ecoflex and Silicone Thinner used to make the vocal fold model for the length experiments is the same as that used for the 113 experiments, as described in Section 2.2. Thus, its Young's modulus is expected to be 4.4 kPa, and all of the elastic and damping parameters are expected to be the same as those listed in Figure 8. The mass flow generator, the silencer, and the subglottal channel are the same as those used for the experiments of Figure 8, and the pressure sensor is located 5.0 cm upstream from the vocal tract, as before. The pressures and frequencies are determined in the same way as described in Section 2.3.