# Effects of Vertical Glottal Duct Length on Intraglottal Pressures in the Convergent Glottis
## 2 Materials and Methods  
### Glottal Geometries  
Figure 1 shows the schematic of the design of the vocal folds and glottal duct for the 0.308 cm vertical glottal duct length with a \(-5^{\circ}\) convergent glottal half-angle. Because the model was symmetric about the X axis, only half of the model is shown, and thus the actual included angle for the two-sided glottis is a convergent \(-10^{\circ}\) for this schematic. Here, the vertical glottal duct length is defined as the length between the glottal entrance and the glottal exit, using the geometry based on the M5 model [1]. Previous studies have reported that the vertical glottal duct length values range from approximately 0.1 to 0.5 cm [8; 34; 35]. The intermediate value of 0.30 cm vertical glottal duct length has been used in numerous physical and mathematical speech production models [1; 2; 5; 36; 37]. In this study, the chosen lengths were 0.108 cm, 0.308 cm, 0.608 cm, and 0.908 cm, which represent lengths expected for relatively high to low \(f_{o}\) during speech. Because the glottal exit radius was kept constant at 0.108 cm, the 0.108 cm glottal duct length cases were conditions of zero glottal angle over zero glottal length (see Figure 2a). Three typical convergent glottal full angles (\(-5^{\circ}\), \(-10^{\circ}\), and \(-20^{\circ}\)), three transglottal pressures (500 Pa, 1000 Pa, and 1500 Pa), and three glottal minimal diameters (0.01 cm, 0.04 cm, and 0.16 cm) were selected and used for each of four vertical glottal duct lengths (0.108 cm, 0.308 cm, 0.608 cm, and 0.908 cm), for a total of 30 geometries and 90 cases. The computational model was two-dimensional (rectangular in the anterior-posterior direction or transverse plane). Constant parameters for each case were the inferior vocal fold surface angle (\(50^{\circ}\) relative to the tracheal center line), the superior vocal fold surface angle (\(90^{\circ}\) relative to the vertical center line), the radius of the vocal fold at glottal entrance R\({}_{\text{in}}\) (0.15 cm), the radius of the vocal fold at glottal exit R\({}_{\text{out}}\) (0.108 cm), the anterior-posterior glottal length (1.2 cm), the upstream (tracheal) inlet computational domain length, and the downstream outlet computational domain length. The anterior-posterior length of 1.2 cm was included for all calculations relying on volume airflow measures. Because the glottal shape was convergent, the minimal glottal diameter was located just prior to the rounded vocal fold shaping near the glottal exit.  
Figure 1: The outline of the glottal configuration for a 0.308 cm vertical (axial) glottal duct length and 0.01 cm minimal glottal diameter for a \(-10^{\circ}\) (convergent) glottal angle. Distances are in centimeters. R\({}_{\text{in}}\) = 0.15 cm; R\({}_{\text{out}}\) = 0.108 cm.  
Figure 2: _Cont._  
Figure 2: _Cont_.  
Figure 2: _Cont._  
Figure 2: (Color online) Schematic of the glottal configurations and the corresponding pressure distributions for 0.108 cm (for zero glottal angle), 0.308 cm, 0.608 cm, and 0.908 cm vertical glottal duct length for \(-5^{\circ}\), \(-10^{\circ}\), and \(-20^{\circ}\) convergent glottal  
angles, 0.01 cm, 0.04 cm, and 0.16 cm glottal minimal diameters, and 500, 1000, and 1500 Pa transglottal pressures, with zero (atmospheric) pressure being the outlet pressure. The lossless pressure distributions are also shown, designated with a \"B\" (for Bernoulli). The upper portion of each small figure shows the outlines of the glottal configurations corresponding to each duct length, and the lower portions show the pressure distributions. Refer to Table 2 for the conditions that correspond to the specific figures.