23.2. The damping properties of the vocal folds

The damping of the oscillation is caused by the viscous resistance of the folds and the larynx tissues as well as by the "adhesiveness" effect of the soft, moist contacting surfaces during the contact of the folds. According to Ishizaka and Flanagan (1972:188), the viscous losses are expressed by the damping ratios zeta_i, although the coupling between oscillators is disregarded:

(30)                

where ki represents the linear components of the stiffness of the springs si given in eq. (26). Again, the coefficients are different during the closed and open glottis phases. For the no-contact phase the damping ratios are equal to zeta1=0.1 and zeta2=0.6. During the closed glottis phase, the damping approaches a critical level and this is expressed by the increase of the coefficients to zeta1=1.1 and zeta2=1.6.

23.3. The aerodynamic forces acting on the vocal folds.

The pressure distribution along the glottis is considered under the following assumptions:

• the dimensions of the glottis are small compared to the wavelength of the produced sound
• the velocity of the glottal flow is much higher than the velocity of the vocal folds motion, which is equivalent to the assumption of quasi-stationarity of the glottal flow
• the pressure drop on the inlet of the glottis (along the contraction of the length lc) is equal to the Bernoulli pressure (, where rho is the air density and ug1 is the particle velocity on the lower edge of the fold m1) which is increased by the empirical coefficient k approx. = 0.37 (van den Berg et al., 1957). The increase amounts to equal to 0.69rhoU_g²/A_g1²

The pressure drop above the glottis is then linear with the exception of the transition point between the masses m1 and m2. At this transition point the volume flow is constant, but the air particle velocity has to vary due to an abrupt change in the glottis area (as shown in Fig.40). This pressure change is equal to the change in the kinetic energy per unit of the volume of the air:

(31)                  

To complete the boundary conditions of the pressure computation, it is necessary to describe the extend of air expansion in the vocal tract. At the expansion of the length le the pressure approaches the pressure of the surrounding air. The pressure at the glottis output is then estimated as (Ishizaka & Flanagan, 1972:1240):

(32)                  

where N is the area ratio of Ag2 to A1, with A1 denoting the inlet area of the vocal tract. The value of 2N(1-N) typically changes between 0.05 and 0.40 and influences the acoustic interaction between the excitation source and the vocal tract.

The above equation completes the boundary conditions necessary for the computation of the pressures along the glottis and so the following set of equations can be formulated

(33)                

The pressure indices are given according to Fig.40. Eq. (33) includes not only static, but also dynamic pressure changes caused by the inertance of the vibrating air column (the instances of the time derivative of Ug).

The pressure drop at the junction of both masses is an artifact introduced by the two-mass model. A better approximation of the pressure distribution in the glottis is provided by the body-cover theory of the vocal folds (Titze, 1988). In the body-cover model, the surface wave propagates ahead of the vocal folds, gathering energy from the airflow and synchronizing the non-symmetrical vocal fold movement and the pressure changes. However, the mathematical description of the surface wave propagation is complicated (Titze, 1988) and leads to the same glottis shapes predicted by the two-mass model.