## 3 Results and Discussion
### Baseline Case Dynamics
#### 3.1.2. Vocal Fold Dynamics  
Glottal area waveform, which measures the minimum glottal cross-section area versus time during sustained vibrations, typically obtained from videostroboscopy images, is an important clinical measure of vocal fold dynamics. Different from the glottal flow rate waveform, which includes the effect of both vocal folds' kinematics and air inertial effect, glottal area waveform is determined only by the vocal fold kinematics; therefore, it is widely used to quantify the vocal fold opening amplitude, speed, and time. Figure 4 shows the phase-averaged glottal area over one vibration cycle in the baseline case. Due to the artificially enforced minimum glottal gap, the glottal area did not drop to zero during glottic closure. The open quotient (\(\tau_{\text{o}}\)) of the vibration, which is defined as the ratio of the \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Computed Value** & **Typical Range [31]** \\ \hline \(f_{0}\) (Hz) & 201 & 65–260 \\ \(\tau_{\text{s}}\) & 1.45 & 1.1–3.4 \\ \(\tau_{\text{o}}\) & 0.67 & 0.4–0.7 \\ \(q_{peak}\) (mL/s) & 220 & 200–580 \\ \(q_{mean}\) (mL/s) & 113 & 110–220 \\ MFDR (L/s\({}^{2}\)) & 240 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Voice-quality-related quantities computed from the baseline model and their typical physiological ranges; \(f_{0}\) is the fundamental frequency; \(\tau_{\text{s}}\) is the skewness quotient; \(\tau_{\text{o}}\) is the open quotient; \(q_{peak}\) and \(q_{mean}\) are the mean and peak glottal flow rates, respectively; MFDR is the maximum flow declination rate. Figure 3: The phase-averaged flow rate over one vibration cycle in the baseline case. duration of open glottis to the duration of the vibration cycle, was computed and found to be 0.67. Note that the closed glottis was defined when the time rate of change of the glottal area was less than 2% of the maximum time rate of change of the glottal area. Moreover, \(\tau_{o}\) is another laryngeal dynamics quantity of important implication to voice quality. This parameter typically ranges between 0.4 and 0.7 for normal phonation. A value lower than 0.4 indicates a \"pressed\" voice, and a value greater than 0.7 indicates a \"lax\" voice [36]. The value of 0.67 in the baseline case is within the normal range but implies a lightly lax voice. To show the detailed vibration pattern of the vocal folds, Figure 5 plots the mid-coronal profile of the vocal folds at four different time instants during one vibration cycle. At 0.17 T, when the vocal folds were opening, the glottis formed a convergent shape. At 0.36 T, the glottis formed a parallel channel. At 0.55 T, when the vocal folds were closing, the glottis formed a divergent shape. Finally, at 0.74 T, the vocal folds fully closed and remained closed for the rest of the cycle. This alternative convergent-divergent shape change of the glottis clearly indicates the formation and propagation of a mucosal wave on the vocal fold medial surface. This mucosal wave propagation is a hallmark of vocal fold dynamics, found to be essential for the sustained energy transfer from flow to vocal folds [36]. Both the results of the glottal area waveform and vocal fold vibration pattern indicate that our simulation reproduced typical vocal fold vibratory dynamics. Vocal fold vibration is known to be the result of a dynamic entrainment process of a number of vibration modes [37; 38]. To reveal this dynamic process, we applied the proper orthogonal decomposition (POD) method to obtain the primary vibration modes of the vocal fold vibration. The POD method is a common method to use low-dimensional quantities to express complex dynamics. It computes the most energetic vibratory modes  
Figure 4: The phase-averaged glottal area waveform of the baseline case. Figure 5: The vibration pattern of the vocal folds in the baseline case, shown by the mid-coronal profile of the vocal folds at four different time instants over one vibration cycle. and their modal coefficients through the proper orthogonal decomposition of vocal fold kinematics.