# Using a Lossy Electrical Transmission Line Model for Optimizing Straw Phonation Configurations
## 2 Materials and Methods
Moreover, the propagation constant, a complex number of the form \(\gamma=\alpha+j\beta\) where \(\alpha\) is the attenuation constant and \(\beta\) is the phase constant, further provides information about the process of phonation which CFD cannot. The attenuation constant describes the reduction of the amplitude of the wave is it travels down the line while the phase constant shows the phase difference between the voltage and current. As with [PERSON] et al., the electrical circuit matrices are left multiplied to effectively iterate through the vocal tract backwards, from the mouth (or straw) opening to the glottis. The effect of the transmission line on the impedance of each tubelet in the direction of the wave propagation is given by  
\[Z_{in}(z^{\prime})=Z_{0}\frac{Z_{L}+Z_{0}\tanh(\gamma z^{\prime})}{Z_{0}+Z_{L} \tanh(\gamma z^{\prime})} \tag{5}\]  
where \(z^{\prime}\) is the length along the transmission line element. Since each of the tubelets have the same length and the tubelets are connected in series so the input impedance of one tubelet is the load impedance of the next tubelet closer to the glottis, input impedance of the tubelet element \(i+1\) is  
\[Z_{i+1}=Z_{0,i}\frac{Z_{i}+Z_{0,i}\tanh(\gamma_{i}\ell_{i})}{Z_{0,i}+Z_{i} \tanh(\gamma_{i}\ell_{i})} \tag{6}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Definition** & **Value** \\ \hline \(\rho\) & density of air & \(1.14\times 10^{-3}\) g/cm\({}^{3}\) \\ c & sound velocity & \(3.5\times 10^{4}\) cm/s \\ \(\mu\) & viscosity coefficient & \(1.86\times 10^{-4}\) dyne-s/cm\({}^{2}\) \\ \(\lambda\) & coefficient of heat conduction & \(5.5\times 10^{-5}\) cal/g-s-deg \\ \(\eta\) & adiabatic constant & 1.4 \\ \(C_{p}\) & specific heat of air at constant pressure & \(0.24\) cal/g-deg \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values for physical parameters inside the human vocal tract. Radiation impedance is the impedance created from the atmosphere as sound exits the mouth (or straw). In this way, radiation impedance acts as the load which the sound wave is transmitted to. Most research, including [PERSON] et al. and [PERSON], agree that radiation impedance is best modeled as a vibrating piston in an infinite baffle. The vibrating piston produces a sinusoidal pressure wave like the oscillation of the vocal folds which is reflected off the baffle. However, since the walls of the baffle are infinitely far away from the emission of the sound, the reflection becomes nothing, and the equation devolves to  
\[Z_{L}=\frac{j\omega RL}{R+j\omega L}=\frac{\omega^{2}RL^{2}+j\omega R^{2}L}{R^{ 2}+\omega^{2}L^{2}} \tag{7}\]  
where \(R\) and \(L\) are the effective resistance and inductance of the equivalent circuit for radiation impedance, respectively. Since the end of the acoustic transmission line is open, there is no compression of the air or losses due to heat conduction with the tubelet walls, and as such there is no term for capacitance or shunt conductance. For more information on the formulae for calculation radiation impedance, the reader is directed to the paper from [PERSON] et al. Importantly, radiation impedance is the ratio of the pressure to volume velocity at the mouth or straw opening such that \(Z_{L}=P_{rad}/U_{rad}\), which only depends on the frequency and geometry of the mouth or straw opening. By substituting \(Z_{L}U_{rad}\) into the final equation for \(P_{rad}\), a final expression can be found relating the input impedance to the radiation impedance and the product of tubelet matrices. Therefore, the input impedance does not depend on the initial conditions of subglottal pressure or volume velocity. As a result, the use of the chain matrix and transmission line models require significantly fewer simulations and calculations than comparable CFD.