# Examining the Quasi-Steady Airflow Assumption in Irregular Vocal Fold Vibration
## 3 Results
### Strouhal Number and Velocity Ratio Analysis  
To investigate how the variations of flow rate errors are associated with the quasi-steady assumption, the Strouhal number of each dynamic case was calculated. In the equation \(S_{t}=f_{o}L/v_{g}\), \(f_{o}\) and \(L\) are, respectively, the vibration frequency and thickness (0.8 cm) of the vocal fold. \(v_{g}\) is the cycle-averaged upstream flow velocity (taken at glottis inlet) in each case. The value of \(S_{t}\) for each dynamic case is listed in Table 2. At 100 Hz vibration frequency, \(S_{t}\) is of the order of 0.03 for all cases, while \(S_{t}\) increased nearly fivefold (approximately 0.15) at 500 Hz frequency. Note that the value of \(S_{t}\) at 100 Hz frequency falls well within the range (0.01 to 0.1) accepted for QSFA [12; 14], whereas the value of \(S_{t}\) at 500 Hz frequency is beyond the range. Therefore, the very large errors of flow rate at 500 Hz frequency are due to the breakdown of QSFA. In other words, the unsteady effects of glottal flow play an important role at a high phonation frequency and cannot be neglected.  
The ratio of flow velocity to vocal fold vibration velocity, \(v_{g}/v_{v}\), was also calculated during each of the 16 phases for the dynamic cases. In the calculation \(v_{g}/v_{v}\), \(v_{g}\) is the mean flow velocity at the glottis inlet, and \(v_{v}\) is the averaged vibration velocity of the medial surface on the side of mode with one half wavelength in the longitudinal direction. Figure 5 plots \(v_{g}/v_{v}\) versus t/T during the open phase for each case, and the absolute percent errors of flow rate between the quasi-steady/pseudo-static and dynamic simulations are displayed in the form of error bars at each data point (the length of error bars denote the magnitude of errors). Figure 5 shows that \(v_{g}/v_{v}\) is generally in an inverse relationship with flow rate error. For t/T < 0.2 and t/T > 0.5, \(v_{g}/v_{v}\) is relatively small but accompanied by a large flow rate error. While in the interval 0.2 < t/T < 0.5, \(v_{g}/v_{v}\) has a high value, but the error is significantly reduced. It is observed that, in the cases with the frequency of 500 Hz and the supraglottic tract included, this inverse relationship is not as evident as in other cases, probably because the air inertia effect in the vocal tract dominates the flow rate error. Figure 5 also shows that for the same phase, \(v_{g}/v_{v}\) has a much smaller value at 500 Hz than at 100 Hz, which is consistent with the above inverse relationship since flow rate errors are much larger at 500 Hz. These observations verify that a large velocity ratio between glottal flow and vocal fold vibration is necessary for the legitimacy of the quasi-steady assumption. Lastly, it is also noted that the variations of flow rate errors with \(v_{g}/v_{v}\) show a similar trend for quasi-steady and pseudo-static cases.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & \multicolumn{2}{c}{**100 Hz**} & \multicolumn{2}{c}{**500 Hz**} \\ \cline{2-5}  & **Without Vocal Tract** & **With Vocal Tract** & **Without Vocal Tract** & **With Vocal Tract** \\ \hline (1,0)–(2,1) & 0.032 & 0.031 & 0.154 & 0.165 \\ (1,1)–(2,0) & 0.031 & 0.030 & 0.145 & 0.145 \\ \hline \end{tabular}
\end{table}
Table 2: The value of the Strouhal number for each dynamic case.