# Examining the Quasi-Steady Airflow Assumption in Irregular Vocal Fold Vibration
## 3 Results
### Errors of Glottal Pressure  
Figure 6 illustrates the root-mean-square error (RMSE) of glottal wall pressure between the quasi-steady/pseudo-static and dynamic cases over one vibration cycle. First, the pressure differences between the quasi-steady/pseudo-static and dynamic cases were calculated for the same element node on the medial surface. Then, the RMSE was obtained by taking the square root of the arithmetic mean of the squares of the pressure differences. In Figure 6, it can be observed that the RMSE of glottal wall pressure is small at 100 Hz, but it increases significantly as the frequency increases to 500 Hz. For example, the cycle-averaged RMSE ranges from 0.056 to 0.077 kPa for all cases at 100 Hz without a vocal tract, while it ranges from 0.238 to 0.261 kPa for all cases at 500 Hz without a vocal tract, representing an almost four-fold increase. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Without Vocal Tract**} & \multicolumn{2}{c}{**With Vocal Tract**} \\ \hline \multirow{2}{*}{**Frequency**} & \multirow{2}{*}{**Parameters**} & **(1,0)–(2,1)** & **(1,1)–(2,0)** & **(1,0)–(2,1)** & **(1,1)–(2,0)** \\  & & **Quasi/Pseudo** & **Quasi/Pseudo** & **Quasi/Pseudo** & **Quasi/Pseudo** \\ \hline \multirow{6}{*}{100 Hz} & Peak flow (\%) & \(-1.18/-2.18\) & \(-1.18/-3.28\) & \(-0.33/2.51\) & \(3.49/5.35\) \\  & Mean flow (\%) & \(-2.24/-3.94\) & \(-3.36/-1.74\) & \(-0.61/-0.39\) & \(-2.14/2.61\) \\  & Phase shift (degree) & \(-11.34/-20.37\) & \(-8.82/-13.93\) & \(-25.48/-27.83\) & \(-24.89/-28.04\) \\  & MFDR (\%) & \(0.49/-16.55\) & \(-9.51/-20.48\) & \(-13.83/-22.94\) & \(-13.19/-15.81\) \\  & H1–H2 (\%) & \(-5.84/-9.23\) & \(-9.53/-0.20\) & \(-0.11/-7.65\) & \(-10.72/-1.31\) \\  & H1–H4 (\%) & \(-7.19/-5.87\) & \(8.54/11.47\) & \(-17.07/-13.15\) & \(8.76/10.09\) \\  & Spectral slope (\%) & \(12.30/14.75\) & \(27.64/44.44\) & \(8.75/13.00\) & \(14.78/6.05\) \\ \hline \multirow{6}{*}{500 Hz} & Peak flow (\%) & \(0.09/1.46\) & \(2.44/-6.79\) & \(28.12/32.95\) & \(37.71/29.57\) \\  & Mean flow (\%) & \(-5.58/-10.26\) & \(-11.01/-8.56\) & \(14.07/10.68\) & \(4.46/10.75\) \\ \cline{1-1}  & Phase shift (degree) & \(-21.27/-44.63\) & \(-27.62/-34.76\) & \(-59.11/-77.93\) & \(-61.89/-66.04\) \\ \cline{1-1}  & MFDR (\%) & \(0.79/-34.05\) & \(-2.18/-25.48\) & \(12.84/-16.15\) & \(17.69/-3.63\) \\ \cline{1-1}  & H1–H2 (\%) & \(-31.45/-43.76\) & \(-42.60/-21.55\) & \(-26.88/-41.37\) & \(-50.66/-32.46\) \\ \cline{1-1}  & H1–H4 (\%) & \(-21.90/-14.58\) & \(-19.46/-23.71\) & \(-2.43/8.23\) & \(-15.88/-19.58\) \\ \cline{1-1}  & Spectral slope (\%) & \(11.75/36.04\) & \(9.09/26.84\) & \(12.03/36.99\) & \(6.18/5.44\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Errors of important aerodynamic and sound spectrum parameters between quasi-steady/pseudo-static and dynamic cases. Phase shift is calculated as the phase difference in the peak flow between the quasi-steady/pseudo-static and dynamic cases. H1–H2 and H1–H4 are the spectral amplitude differences between the first two harmonics and the first and fourth harmonics, respectively. Spectral slope is determined using linear regression in MATLAB to fit the first 20 harmonics of each case. By comparing the cases with and without the vocal tract, a significant increase in the RMSE of glottal wall pressure is also observed. Figure 6 indicates that the RMSEs of glottal wall pressure are almost at the same level for the two different sets, suggesting that glottal shapes do not have a significant effect on the errors of glottal pressure. Although a more significant error is commonly observed during the flow deceleration stage for pseudo-static cases, the difference in the cycle-averaged RMSE of glottal wall pressure between the quasi-steady and pseudo-static cases is small. Overall, the trend of errors of glottal pressure is consistent with the observations made above, that errors are small at low frequencies without a vocal tract.