# Examining the Quasi-Steady Airflow Assumption in Irregular Vocal Fold Vibration
## 2 Methods  
### Glottal Shapes and Movement  
Two sets of time-dependent glottal shapes were generated using the surface-wave approach proposed by [PERSON] and [PERSON] [24]. This approach utilizes a combination of (m,n) modes to describe the wave-like motion of the vocal fold surface, where m and n represent the number of half wavelengths along the anterior-posterior and inferior-superior directions, respectively. In normal phonation, the dominant modes are the (1,0) and (1,1) modes, representing the medial-lateral and convergent-divergent motions, respectively. Higher mode numbers correspond to more complex wave motions, which can occur in voice disorders. For instance, a (2,1) mode corresponds to two half-wavelengths in the anterior-posterior direction and one half-wavelength in the inferior-superior direction. In this study, we generated two sets of asymmetric glottal wall dynamics by combining the (1,0) mode (left) and (2,1) mode (right) in the first set and the (1,1) mode (left) and (2,0) mode (right) in the second set. The equations for generating the vocal fold dynamics are briefly introduced below. Taking the right vocal fold as an example, its pre-phonatory medial surface (Figure 1) was defined by a glottal half-width \(\xi_{0R}\) as a function of anterior-posterior (\(y\)) and inferior-superior (\(z\)) directions:  
\[\xi_{0R}(y,z)=(1-y/L)\;[\xi_{0R2}+(\xi_{0R1}-\xi_{0R2}-4\xi_{RR}\;z/T)\;(1-z/T )], \tag{1}\]  
where \(L\) and \(T\) are, respectively, the length and thickness of the vocal fold, \(\xi_{0R1}\) and \(\xi_{0R2}\) are, respectively, the inferior and superior glottal half-widths at the vocal process, and \(\xi_{\mathrm{BR}}\) is a surface bulging parameter, which controls the vertical curvature of the medial surface. The subscript R denotes the right medial surface. The same equation with a subscript \(L\) exists for the pre-phonatory glottal width of the left medial surface (\(\xi_{0L}\)). For the right medial surface, the modal displacement \(\xi_{R}\) at any instant of time (t) is defined as follows:  
\[\xi_{R}(y,z,t)=\xi_{\mathrm{mR}}\;\sin(m\pi y/L)\;[\sin\omega t-n(\omega/c)(z- z_{\mathrm{mR}})\;\cos\omega t], \tag{2}\]  
Figure 1: Diagram of the pre-phonatory configuration of the right medial surface. where \(\xi_{mR}\) is the modal displacement amplitude, \(\omega\) is the angular frequency, \(c\) is the mucosal wave speed, and \(z_{mR}\) is the inflection point for the vertical half wavelength. An equivalent equation exists for the modal displacement of the left medial surface (\(\xi_{L}\)). The overall three-dimensional glottal shape at any moment of time was obtained by superimposing the modal displacements on the pre-phonatory shape of the medial surfaces:  
\[\text{g(y,z,t)}=\xi_{\text{0R}}\ \text{(y,z)}+\xi_{\text{0L}}\ \text{(y,z)}+\xi_{ \text{R}}\text{(y,z,t)}+\xi_{\text{L}}\text{(y,z,t)}. \tag{3}\]  
In this study, the vocal fold thickness, \(T\), and length, \(L\), were adopted from [PERSON] and [PERSON] [24], which are 0.8 and 1.5 cm, respectively. A uniform (parallel surface), initial posturing configuration was used, and the parameter values (in cm) were as follows:  
\[\xi_{0R1}=0.10,\ \xi_{0R2}=0.10,\ \xi_{BR}=0.005,\ \xi_{0L1}=0.10,\ \xi_{0L2}=0.10,\ \xi_{BL}=0.005.\]  
The vibration frequency of the vocal folds, \(f\), was chosen to be 100 Hz and 500 Hz. The angular frequency, \(\omega\), and the mucosal wave speed, \(c\), were defined as \(2\pi f\) and \(T\pi f\), respectively. For both the right and left medial surfaces, the modal displacement amplitude \(\xi_{m}\) was 0.1 cm, and the inflection point \(z_{m}\) was defined as \(T\times(0.6-0.02\xi_{B})\). In this study, a total of 16 sequential static glottal shapes were extracted for each of the (1,0)-(2,1) and (1,1)-(2,0) sets of glottal wall motions.