## 3 Results and Discussion
### Underlying Mechanism--Correlation to the Flow Resistance  
Same as [PERSON] and [PERSON] [18], we hypothesized that the above-observed effects of SGS on voice production are primarily due to its effect on the flow resistance. To examine this hypothesis, we computed the flow resistances across the SGS (trans-stenosis resistance) and glottis (trans-glottis resistance) for all of the SGS severity cases. The flow resistance is defined as \(\frac{|\Delta P|}{Q}\), where \(\Delta P\) is the mean pressure drop across the SGS or glottis, and \(Q\) is the corresponding incompressible glottal flow rate. The total flow resistance is defined as the sum of the two. Figure 13a shows the cycle-averaged flow resistances versus the SGS severity in which the SGS location is 3.13 cm. Figure 12: The comparison of the phase-averaged (**a**) flow rate waveform and (**b**) glottal area waveform in the simulation cases with the varied SGS location and severity. (**c**) The fundamental frequency, sound intensity, and similarity of the corresponding modes in the cases with the varied SGS location and severity. It is noticed that, up to the SGS area reduction of 75%, the trans-stenosis resistance stays nearly zero, and as a result, the total flow resistance is the same as the trans-glottis flow resistance. Moreover, the change of the SGS area reduction up to 75% does not affect the trans-glottis flow resistance. It indicates that, when the SGS area reduction is below 75%, the SGS does not generate additional flow resistance nor affect the trans-glottis flow resistance. When the SGS area reduction is 75%, a very slight increase in the trans-stenosis flow resistance and total flow resistance can be seen. Beyond 75% SGS severity, the trans-stenosis resistance starts to increase rapidly with the increasing SGS severity. In the meanwhile, the trans-glottis resistance slightly drops. Because the increase of the trans-stenosis resistance is much more than the drop of the trans-glottis resistance, the total flow resistance also rapidly increases with the SGS severity. When the SGS area reduction reaches 99%, the trans-stenosis resistance has become comparable to the trans-glottis resistance; as a result, the total flow resistance is about twice its values in the baseline case. These observations indicate that, when the SGS area reduction is beyond 75%, the SGS starts to generate additional flow resistance, which increased rapidly with the SGS area reduction. The effect of the SGS on the flow resistance is found to be consistent with its effect on the flow rate and vocal fold vibration. The SGS is found to start to lower the flow rate at 75% area reduction and lower the vocal fold vibration amplitude at 90% area reduction. These are consistent with its threshold to increase the flow resistance. That the threshold of SGS severity to affect vocal fold vibration appears higher than that to affect the flow rate is likely because the flow rate is more sensitive to the changes in flow resistance. Furthermore, [PERSON] and [PERSON] [28] reported that a decrease in trans-glottis pressure drop could result in a slight decrease in the \(f_{0}\). Our results show that increasing the SGS severity beyond 75% decreases the flow rate and trans-glottis flow resistance, suggesting that the trans-glottis pressure would also be decreased. It thus explains the drop of the \(f_{0}\) with the SGS severity beyond 75%. To investigate why the threshold SGS severity to increase the flow resistance occurs at 75% area reduction, we computed the area ratio between the glottis and SGS and plot its relationship with SGS severity in Figure 13b. The area ratio is defined as the cycle-averaged glottal area to the SGS area. The figure shows a similar relationship between the area ratio and SGS severity to that of the trans-stenosis flow resistance: The area ratio remains nearly zero until the SGS severity reaches 75%; after that, it rapidly increases with the SGS severity.