# Effects of Vertical Glottal Duct Length on Intraglottal Pressures in the Convergent Glottis
## 4 Discussion
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Vertical Glottal Duct** & **Glottal Minimal Diameter** & **Glottal Angle** & **Transglottal Pressure** \\  & **Length** & & & \\ \hline Intraglottal pressure & Large & High & High & High \\ Transglottal pressure & Small & High (Especially for Re \(<\) 500) & Small & Middle \\  & & Middle & High (Especially for Re \(<\) 500) & Small & Small \\  & & Middle & High (Especially for Re \(<\) 500) & Small & Small \\  & & Middle & High (Especially for Re \(<\) 500) & Small & Small \\  & & & Volume flow & None & High & Middle \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The effects of vocal fold physiological parameters on intraglottal aerodynamic parameters for convergent glottal angles. The results of this study appear to provide partial support for the equation for the phonatory threshold pressure, PTP, given by [PERSON] [6],  
\[P_{th}=\frac{2k_{t}Bc\zeta_{01}^{2}}{L_{g}T^{2}(\xi_{01}+\xi_{02})} \tag{5}\]  
where \(k_{t}\) is the transglottal pressure coefficient [37; 38; 39], \(B\) is the damping factor for motion of the vocal fold, \(c\) is the speed of the mucosal wave, \(L_{g}\) is the glottal length (anterior-posterior direction), \(T\) is the glottal thickness (inferior-superior direction, the same as the vertical glottal duct length), \(\xi_{01}\) is the prephonatory inferior glottal half-width, and \(\xi_{02}\) is the prephonatory superior glottal half-width. In the study reported here, it was found that as the length of the vertical glottal duct increased, there was greater pressure applied to longer portions of the medial vocal fold surfaces from the glottal entrance, thus adding pressure to separate the vocal folds during glottal opening. This should be consistent with lowering the PTP value, as indicated by Equation (5), although it is recognized that the greater pressure may act against a greater mass. Increasing the minimal diameter in this study (equivalent to increasing \(\xi_{02}\)) reduced the intraglottal pressures, which should then tend to increase the PTP for phonation, which is contradicted by Equation (5). Equation (5) also predicts that increasing the convergent glottal angle should increase the PTP value. However, in this study, increasing the convergent angle increased the intraglottal pressures, which should tend to reduce the PTP value. Previous studies also found that there are both negative and positive relations between vocal fold thickness and phonation threshold pressure. [PERSON] et al. (1997) [26] found that a higher PTP was always obtained for a thinner vocal fold. They also found that PTP is lowest for the divergent glottis and highest for the convergent one. [PERSON] (2005) [24] also predicted that increases in vocal fold adduction and vocal fold thickness would decrease PTP. However, in a continuum vocal fold model study, [PERSON] (2010) [28] showed that an increase in the medial surface thickness reduced the coupling strength and therefore led to increased phonation threshold pressure, which contradicts the prediction of Equation (5) and experimental observation by [PERSON] et al. (1997) [26]. In their two-layer physical model experimental study, [PERSON] and [PERSON] (2011) [27] further declared that a lower PTP can be achieved by reducing medial surface thickness or increasing the cover-layer depth. Overall, as pointed out in [PERSON] (2010) [28], this seeming contradiction is \"likely due to the multivariable dependence of phonation threshold\". The variety of values of other vocal fold properties (geometry, body-cover stiffness ratio, etc.) may not only vary the vocal fold thickness but also may have different or even opposite influence on the phonation threshold pressure. Further theory and experimental studies are needed to clarify these discrepancies. The numerical model in this study used some simplifying assumptions that can be considered as limitations.