## 3 Results and Discussion  
### Predicted Sound Absorption Coefficients  
In this study, we made the solid phase vibrate violently due to incident sound waves to induce relative velocity and viscous damping at the boundary between the fluid and solid phases as a sound absorption mechanism without sound wave penetration, which is unlike the conventional method shown in Section 2.2. The main result of this verification is the goal of this study. The normal-incidence sound absorption coefficient was calculated using the model shown in Section 2.1 to verify the abovementioned concept. The Kelvin cell model in Figure 6 was used to represent the microstructure of the porous sound-absorbing material. Its unit cell (Figure 6b) consisted of a rectangle and a hexagon with bar-like ligaments. Figure 5: Simple thermal conductivity measurement apparatus. Given the same lengths \(w\) of the three sides of the unit cell (\(w_{x}=w_{y}=w_{z}=w\)), the sizes of the square and hexagonal vacancies were approximately \(1/3w\) and \(2/3w\), respectively. The lengths \(w\) of the three sides of the unit cell were varied from 1 to 10 \(\upmu\)m (the average pore size was approximately 0.5-5.2 \(\upmu\)m), which are almost impenetrable to sound waves. The material of the ligaments was glass, and the material properties of the joints between ligaments were set, as shown in Table 1.  
Figure 7 shows the spectra of the predicted sound absorption coefficients. Table 2 shows the average sound absorption coefficients in the 1/3-octave band from 500 to 3150 Hz and from 1000 to 5000 Hz, commonly required for sound-absorbing materials in the automotive field. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Young’s Modulus** & **Poisson’s Ratio** & **Density** & **Loss Factor** \\  & **(GPa)** & **Poisson’s Ratio** & **(kg/m\({}^{3}\))** & **Loss Factor** \\ \hline Glass & 71.6 & 0.23 & 2200 & 0.002 \\ Rubber 1 & 0.1 & 0.35 & 1100 & 0.3 \\ Rubber 2 & 0.1 & 0.35 & 1100 & 0.1 \\ Rubber 3 & 0.1 & 0.35 & 1100 & 0.5 \\ Rubber 4 & 0.01 & 0.35 & 1100 & 0.3 \\ Rubber 5 & 1 & 0.35 & 1100 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of ligaments and joints in calculation model. Figure 6: Calculation model of the microstructure of simulated porous material: (**a**) periodic structure and (**b**) unit cell. be due to (1) an increase in viscoelastic damping caused by the distortion of the rubber at the joint or (2) an increase in viscous damping caused by an increase in the resonance amplitude of the ligament due to a smaller ratio of the spring constant at the joint to the ligament weight. The results were compared by varying only the loss factor or Young's modulus at the joint to determine the cause. First, the sound absorption coefficient spectra of rubber 1 (Figure 7b), rubber 2 (Figure 7c), and rubber 3 (Figure 7d), which differ only in loss factor, were compared. The results show that a lower loss factor at the joint makes the peak of the sound absorption coefficient more intense. Given a larger contribution of viscoelastic damping caused by the distortion of the rubber at the joint, a higher loss factor at the joint improves the peak value due to the higher thermal conversion effect caused by the distortion. On the contrary, given a larger contribution of viscous damping caused by the larger amplitude of the ligament resonance, a lower loss factor at the joint enhances the peak value because the resonance is not suppressed. Therefore, the contribution of viscous damping to the sound absorption phenomenon is higher than that of viscoelastic damping. The average sound absorption coefficients of rubber 1, rubber 2, and rubber 3 in the 1/3-octave band were also compared (Table 2). The results show that the average sound absorption coefficient is slightly higher at higher loss factors at the joint.