## 3 Results and Discussion
### Sgs Cases
#### 3.2.4. SGS Cases: Underlying Mechanism  
We examine the correlation between flow resistance, area ratio, and SGS severity effects on voice production. We define flow resistance as \(\frac{|\Delta p_{j}|}{Q}\), where \(\Delta p_{j}\) is the pressure difference across the SGS, glottis, or their summation (represented by \(i\)), and \(Q\) is the corresponding flow rate. Figure 11a plots the cycle-averaged flow resistance across the SGS (\(FR_{SGS}\)), across the glottis (\(FR_{VF}\)) and their sum (\(FR_{Total}\)) for different SGS severity cases. Figure 11b shows the area ratio, defined as the ratio of the glottal area to the SGS area, plotted against SGS severity. For SGS severities below 75%, \(FR_{SGS}\) is negligible, making \(FR_{Total}\) essentially equal to \(FR_{VF}\). However, for higher SGS severities, \(FR_{SGS}\) increases significantly, causing \(FR_{Total}\) to exceed \(FR_{VF}\). Notably, the drop in \(FR_{VF}\) for severe cases remains small. The area ratio results show values below 0.12 for SGS severities of 75% and lower, corresponding to the unchanged flow rate and vocal fold vibration. At 90% SGS severity, the area ratio rises to 0.29, indicating the changed flow rate but minimal changes in vocal  
Figure 11: (**a**) The cycle-averaged flow resistance across the SGS, vocal folds and the summation of these two versus SGS severity in the parametric cases; (**b**) the ratio of the cycle-averaged glottal area to the SGS area versus SGS severity in the parametric cases. Figure 10: (**a**) The Fourier transform of the acoustic pressure at the probe 5 cm outside of the mouth in the parametric cases; (**b**) the sound pressure level and vocal efficiency for different SGS severities. fold vibration compared to the baseline. For higher SGS severities and consequently higher area ratios, both the flow rate and vocal fold vibration change noticeably. These observations suggest that area ratios of 0.12 and lower do not significantly affect \(FR_{Total}\), resulting in the unchanged flow rate and acoustic results. However, for larger area ratios, the change in \(FR_{Total}\) becomes substantial. This implies that in severe SGS cases, both the SGS and vocal fold resistances contribute significantly to altering the flow rate and, consequently, the acoustics. Conceptually, we can consider the airway as a duct with two flow resistances: SGS and vocal fold. Both become significant enough to change the flow rate and acoustics in severe cases. However, the underlying mechanism of the vocal fold vibration changes, which only occurred at the 96% SGS severity case, requires more investigation. By examining Figure 11a more closely, we observe that the change in \(FR_{VF}\) for the 96% case is small. Comparing to our previous study, while the drop in \(FR_{VF}\) for the 96% SGS severity case was slightly larger, the \(FR_{VF}\) did not change significantly for 96% and 99% SGS severity cases. However, there was a significant drop in the vibration of the 99% case compared to the 96% case. This suggests that \(FR_{VF}\) alone is not the main parameter affecting vocal fold vibration. To better understand this mechanism, we focus on the pressure drop across the vocal folds (\(\Delta p_{VF}\)) rather than flow resistance, as the pressure of the flow in the glottis is the primary driver of vocal fold vibration. Figure 12 shows the percentage change in pressure drop across the vocal folds for each SGS severity case compared to the baseline, calculated as \(\Delta p_{VF}=\frac{\Delta p_{VF_{Baseline}}-\Delta p_{VF_{SGS}}}{\Delta p_{ VF_{Baseline}}}\times 100\). The pressure drop remains nearly unchanged for cases up to 75% SGS severity. For the 90% SGS severity case, \(\Delta p_{VF}\) drops by 3%, which, while not causing significant changes in overall vocal fold vibration, does lead to slight changes in the energies of the POD modes as observed earlier. However, for the 96% SGS severity case, the drop in \(\Delta p_{VF}\) is much more substantial at 20%. This analysis helps explain our earlier observations.