## 3 Results and Discussion
### Effects of SGS Severity on Glottal Flow Dynamics, Vocal Fold Vibration, and Acoustics
#### 3.2.2. Vocal Fold Dynamics  
Figure 9a plots the phase-averaged glottal area waveform for different SGS severities. Similar to the glottal flow rate, the glottal area is reduced beyond a threshold SGS severity. However, different from the glottal flow rate, the threshold value is at the SGS area reduction of 90%. Specifically, with the area reduction of 90%, 96%, and 99%, the maximum glottal area, respectively, drops 1.2%, 7%, and 30%; the average speed of glottic opening, respectively, drops 1.7%, 15%, and 49%; and the average speed of glottic closing, respectively, drops 1.5%, 10%, and 57% relative to the baseline case. Figure 9b plots the open quotient versus the SGS severity for different cases. The variation of the open quotient is negligible up to 90% severity and increases rapidly beyond that. Our results suggest that, while SGS tends to reduce vocal fold vibration, the effect is noticeable only when the area reduction is beyond 90% (Grades III and IV SGS). Figure 8: (**a**) The phase-averaged flow rate waveform in each case with the varied SGS severity. (**b**) The skewness quotient versus SGS severity. (**c**) The maximum flow deceleration rate (MFDR) versus SGS severity. Figure 9: (**a**) The phase-averaged glottal area waveform for different SGS severities. (**b**) The open quotient versus SGS severity. The POD analysis was performed to investigate the effect of SGS severity on vocal fold vibration modes and their dynamic entrainment. Figure 10a shows the energy percentage of the two most energetic vibration modes in each case. The total energy of the two modes is also plotted. For all of the cases, the two modes captured more than 90% of the total vibration energy. It can be observed that there are no noticeable changes in the modal energy percentage up to 90% of area reduction. Beyond that, at a 96% and 99% area reduction, the energy of the first mode respectively increases by 8.5% and 7.6%; the energy of the second mode respectively reduces by 7% and 1.6%; and the total energy respectively increases by 1.5% and 6%. To evaluate if any changes occurred in the modal shape, the similarity of the shape of the corresponding modes in the different cases is quantified by computing the dot product of the modal shape vectors. The result is plotted in Figure 10b. A value of 1 of the dot product represents two identical modes, and a value of 0 represents two completely orthogonal modes. It is observed that the similarity of the corresponding modes between different SGS severity cases and baseline case remains as high as 98% up to 96% area reduction. A significant drop in the modal similarity on the second mode is observed when the area reduction is 99%. To show the details, Figure 10c plots the extreme states of the first and second modes of the baseline case and those of the case with 99% area reduction. As expected, the shape of the first mode remains highly similar between the two cases. However, in the second mode, a large difference is observed on the superior surface of the vocal fold that the waves on the superior surface appear to have opposite phases between the two cases. It indicates that increasing the SGS area reduction to 99% results in different dynamic entrainment between the waves on the medial surface and superior surface. Figure 10d shows the relationship between vibration frequency and SGS severity. Consistent with vocal fold dynamics, \(f_{0}\) is unaffected up to a 90% area reduction. At 90%, the severity \(f_{0}\) was slightly reduced to 199 Hz, and then it was reduced further to 195 and 183 Hz at 96% and 99% severities, respectively. In sum, our results show that the SGS starts to show noticeable effects on vocal fold vibrations when the area reduction reaches 90%, including reducing the vibration amplitude, increasing the glottal open quotient, and reducing the vibration frequency. The results also suggest that the changes in the vocal fold dynamics are associated with the variation in the energy percentage of the vibration modes or the generation of different vibration modes.