# Examining the Quasi-Steady Airflow Assumption in Irregular Vocal Fold Vibration
## 4 Discussion and Conclusions  
This study aimed to assess the accuracy of the quasi-steady assumption in predicting glottal flow and pressure by analyzing errors in various glottal shapes and simulation setups, with and without the supraglottic tract, at vibration frequencies of 100 and 500 Hz. The findings indicate that the assumption yields minor errors at 100 Hz but substantial  
Figure 8: Comparison of convection, pressure, and shear term between the dynamic and quasi-steady/pseudo-static cases. Figure 7: Variation in convection, pressure, shear, and unsteady term over one vibration cycle in each dynamic case. errors at 500 Hz. The momentum budget analysis suggests that the quasi-steady glottal flow closely resembles an unsteady glottal flow at 100 Hz but diverges significantly at 500 Hz, resulting in notable disparities in the momentum equation terms between quasi-steady/pseudo-static and dynamic simulations. Interestingly, our study reveals that unsteady acceleration is non-negligible even at low vibration frequencies, aligning with experimental results from [PERSON] et al. [14], indicating that glottal flow is inherently unsteady. However, for the frequency range of normal speech phonation (~100-200 Hz), the unsteady acceleration contribution is small, and the quasi-steady approximation provides satisfactory predictions of glottal flow and pressure. Therefore, this assumption remains valid for such frequency ranges. As frequency increases, a point is reached where the predicted flow errors are no longer acceptable, and the quasi-steady assumption is no longer valid. Similar to previous studies [9; 10; 12], our results indicate that errors in glottal flow are nonuniform throughout the vibration cycle. Previous findings have shown that errors are more pronounced during the early opening and late closing phases while remaining minimal during the middle interval of the cycle. This disparity in errors is believed to be related to the significance of glottal wall motion, as indicated by [PERSON] and [PERSON] [13]. During the initial and final stages of flow initiation and shutoff, the velocity of the glottal wall can be comparable to that of glottal flow, making the unsteady effects of wall motion noticeable. Conversely, during the stages around the maximum glottal opening, the wall velocity is much lower than the flow velocity, and the unsteady effects of wall motion are negligible. However, in our current study, we found that the maximum flow rate errors do not occur strictly during the stages of flow initiation and shutoff but rather sometime after the initiation point and before the closure point. This difference can be attributed to the fact that the two sets of glottal shapes never generate a fully closed glottis (as shown in Figure 2), meaning that the flow does not genuinely start from zero velocity and end with a complete stop. Nonetheless, our velocity ratio analysis (\(v_{g}/v_{v}\)) still supports the previous argument. Generally, flow rate errors are more significant during the stages where the relative magnitude of airflow velocity and vocal fold velocity is small, while errors decrease as the relative magnitude increases. This study has also shown that the inclusion of the inertance effect of the air column in the vocal tract can lead to a significant increase in the errors of quasi-steady flow at high frequencies. The momentum budget analysis indicates that including vocal tract inertance does not increase the magnitude of the unsteadiness of glottal flow, which suggests that increased errors are caused by the phase shift of the flow rate due to air inertance. According to lumped-element models of the vocal tract inertance by [PERSON] [3] (Chapter 5), there exist nonlinear interactions between the vocal tract inertance and glottal airflow, skewing the flow rate waveform to the right and reducing the peak flow. These effects are also observed in the results of dynamic simulations in our study.