# Examining the Quasi-Steady Airflow Assumption in Irregular Vocal Fold Vibration
## 3 Results
### Errors of Important Aerodynamics and Sound Spectrum Parameters  
To gain insight into the influence of QSFA on voice outcomes, errors of several important aerodynamic and sound spectrum parameters were computed and compared between quasi-steady/pseudo-static and dynamic cases. The resulting errors are tabulated in Table 3. Peak flow errors were small (\(\leq\)5.35%) for all cases at 100 Hz and 500 Hz without the vocal tract. However, significant errors were observed for the 500 Hz case with the vocal tract, where the dynamic simulation showed a marked decrease in peak flow not seen in the quasi-steady/pseudo-static simulations. This decrease in peak flow at higher frequencies is consistent with previous experimental and numerical findings by [PERSON] and [PERSON] [18], who attributed it to the inertial effects of the air in the trachea and vocal tract that resist changes in flow rate and grow with frequency. Mean flow errors were small (\(\leq\)3.94%) at 100 Hz but more significant at 500 Hz, with a maximum error of 14.07%.  
Phase shift errors, calculated as the phase difference in peak flow between the quasi-steady/pseudo-static and dynamic cases, were substantial for all cases, with negative values indicating delayed peak flow in the dynamic case. The phase shift values significantly increased when a vocal tract was considered due to air inertia in the tract that skews the flow rate waveform to the right. Additionally, as frequency increased from 100 Hz to 500 Hz, a significant increase in phase shift was observed for all cases. The pseudo-static assumption generated larger phase shift errors than the quasi-steady assumption, indicating that glottal wall motion brings the peak flow to an earlier time.  
MFDR errors were relatively small for the quasi-steady case at both 100 Hz and 500 Hz without a vocal tract, while large errors were observed for the pseudo-static case, which could be attributed to larger phase shift errors. MFDR errors increased only for the quasi-steady case when a vocal tract was considered, and no strong correlation between errors and frequency was observed in Table 3.  
Spectral analysis was conducted by measuring the spectral amplitude differences between the first two harmonics (H1-H2) and the first and fourth harmonics (H1-H4), respectively. The spectral slope was calculated by fitting the first 20 harmonics of each case  
Figure 5: \(v_{g}/v_{v}\) versus t/T during the open phase, with the flow rate errors being presented in the form of error bars. Maximum values of \(v_{g}/v_{v}\) for the (1,0)–(2,1) set at 100 Hz are much larger than 500. They are limited to 500 for a better view of other \(v_{g}/v_{v}\) values.  
using linear regression in MATLAB. The results showed that errors of H1-H2 and H1-H4 significantly increased with frequency in simulations without a vocal tract, while errors of spectral slope did not appear to be related to frequency. In simulations with a vocal tract, errors of H1-H2 exhibited the same frequency-dependent characteristic, while errors of H1-H4 did not. These observations suggest that although flow unsteadiness does not have a noticeable impact on sound intensity (MFDR), it still strongly influences sound quality and is, therefore, perceptually significant.  
Overall, applying QSFA leads to small errors in flow rate prediction but large errors in phase shift prediction, with substantial increases in errors when a vocal tract is included or at high vibration frequencies. The pseudo-static assumption does not improve flow rate waveform prediction but actually generates larger errors in phase shift and MFDR predictions. QSFA also influences sound spectrum amplitude differences, e.g., H1-H2 and H1-H4, which can influence sound quality.