## 5 Conclusions  
A number of calculations with lumped element models have used the inertance of the air in the vocal tract [1; 2; 6; 7; 8; 9; 10; 11; 12] to describe the effects of the airflow in the vocal tract (Equation (1)). In most cases, these calculations focused on a number of properties of phonation with little emphasis on the questions of the validity of using Equation (1) to describe the inertance. Thus, it is important to know under what circumstances the inertance of the air within the vocal tract is adequate to describe these effects. This study was undertaken in order to examine this question. The inertance of the vocal tract depends directly on the vocal tract length \(L_{VT}\) and inversely on the vocal tract area \(A_{VT}\). Thus, Equation (1) predicts how the vocal tract area and the vocal tract length should affect phonation properties near the threshold, such as the onset pressure and frequency near onset. In this work, Equation (1) is used in conjunction with the SWM developed by [PERSON] [2], since  
Figure 11: (Color online) (**a**) Measured and calculated threshold pressures for model 113 as a function of vocal tract length. (**b**) Measured and calculated frequencies as a function of vocal tract length. the SWM introduces a smaller number of parameters than the two-mass model [1] or the three-mass model [46]. Qualitative arguments are presented in the Introduction chapter to show that increasing the vocal tract area should raise the onset pressure. These qualitative arguments are examined by the SWM calculations presented in Figures 8a and 10a. Reasonable agreement between the calculations and measurements is presented in Figures 8a and 10a for the four smaller areas, but the discrepancy is considerable for the two largest areas. An examination of the supraglottal flow reveals changes in the locations of the vortical structures downstream of the vocal folds, which may be the reason. However, a better understanding of the fluid dynamics would require a more fundamental calculation based on the Navier-Stokes equation to allow a more thorough examination of the discrepancy and further clarify the role that the inertial properties of the air in the vocal tract play in determining the phonation onset. The agreement between the SWM calculations and the measurements for the frequencies at threshold is much better, as shown in Figures 8b and 10b, provided that the nonlinear stiffness effects are included. The qualitative arguments discussed in the Introduction also predict that increasing the length of the vocal tract should lower the onset pressure. These arguments are supported by the SWM calculations. The results presented in Figure 11a show reasonable agreement between the SWM calculations and the measurements. The frequency calculations presented in Figure 11b show good agreement between the calculations and the measurements for most of the larger lengths, but the wrong trend at the two smallest lengths. Again, the nonlinearity in the stiffness is essential, since otherwise, the variation of the calculated frequencies with length would be much larger than that obtained in the measurements. The experimental models and the SWM reproduce the same behavior: the phonation threshold pressure is reduced by increasing the vocal fold length and decreasing the vocal tract area. Due to their inherent simplicity and the versatility of the models, they should have a potential role in optimization or deep learning algorithms. On the basis of endoscopic high-speed video footage from patients, the model can be applied to estimate the subglottal oscillation threshold pressure similar to the methodology reported by [PERSON] et al. [47; 48]. Assuming a high accuracy of the estimation, the methods make an important parameter available for evaluating the effort that a patient has to make to phonate. Conceptualization, L.F., A.L. and S.K. ; Methodology, L.F., A.L. and S.K. ; software, L.F.; validation, A.L. and S.K. ; formal analysis, L.F., G.K. and S.K. ; investigation, L.F., A.L. and S.K. ; resources, L.F. and S.B.