## 3 Results and Discussion
The porosity of the specimens was calculated using Equation (5) and the estimated intrinsic parameters (\(\alpha_{\infty}\), \(\Lambda\), and \(\Lambda^{\prime}\)) from the inverse acoustic characterization performed on specimens with thickness h = 70 mm, for all considered densities, are given in Table 2. The same values were adopted for the other thicknesses with the same density. Comparisons between the experimental results using the transfer matrix method and the finite element method are shown in Figures 7-9. The predicted results compared well with the experimental results for samples with thickness h = 7 cm. Although the optimization was performed for frequencies above 650 Hz, there was a reasonable match even for some of the lower frequencies, which was a good indicator of the quality of the obtained parameters [28]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Density [kg/m\({}^{3}\)]**} & \multirow{2}{*}{**Porosity \(\phi\)**} & \multirow{2}{*}{**Tortuosity \(\alpha_{\infty}\)**} & **Viscous** & **Thermal** \\  & & & **Characteristic** & **Characteristic** \\  & & & **Length \(\Lambda\) [\(\mu\)m]** & **Length \(\Lambda^{\prime}\) [\(\mu\)m]** \\ \hline
40 \(\pm\) 10\% & 0.97 & 1.06 & 111 & 386 \\
55 \(\pm\) 10\% & 0.95 & 1.13 & 88 & 238 \\
80 \(\pm\) 10\% & 0.93 & 1.67 & 156 & 156 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Estimated intrinsic parameters from inverse acoustic characterization. Figure 7: Comparisons between experimental results, TMM and FEM for RPU foams with \(\rho\) = 40 kg/m\({}^{3}\). The prediction for samples with thickness h = 45 mm and h = 95 mm was more successful for some cases than others. The prediction performance may be evaluated through the lowest value of the function F, which was estimated using Equation (6) for \(N=721\) frequencies in the frequency region from 650 to 2000 Hz. For the RPUF with density \(\rho=40\) kg/m\({}^{3}\), the Fmin values were 1.93, 0.26, and 0.24 for thicknesses h = 4.5 cm, h = 7.0 cm, and h = 9.5 cm, respectfully. For density \(\rho=55\) kg/m\({}^{3}\), the Fmin values were 1.62, 0.23, and 0.43 for thicknesses h = 4.5 cm, h = 7.0 cm, and h = 9.5 cm, respectfully. For density \(\rho=80\) kg/m\({}^{3}\), the Fmin values were 0.54, 0.19, and 0.99 for thicknesses h = 4.5 cm, h = 7.0 cm, and h = 9.5 cm, respectfully. According to the results, the highest error occurred for the thickness h = 4.5 cm and densities \(\rho=40\) kg/m\({}^{3}\) and \(\rho=60\) kg/m\({}^{3}\) and for the thickness h = 9.5 cm and density \(\rho=80\) kg/m\({}^{3}\). For all the considered samples with thickness h = 4.5 cm, the prediction mainly underestimated the sound absorption coefficients. For thickness h = 9.5 cm and density \(\rho=40\) kg/m\({}^{3}\), the estimated sound absorption coefficients corresponded fairly well to the measured values in the optimization region. For thickness h = 9.5 cm and density \(\rho=55\) kg/m\({}^{3}\), they were overestimated, and for \(\rho=80\) kg/m\({}^{3}\), the behavior was somewhat shifted. Some possible reasons for the discrepancies include the variation of 10% between the densities of the samples and hence other parameters, the inhomogeneity of the rebonded polyurethane foam, and some resonance effects. Furthermore, TMM and FEM were used to predict the sound absorption coefficients for a wider range of thicknesses. The sound absorption coefficients for the thicknesses h = 2 cm,  
Figure 8: Comparisons between experimental results, TMM and FEM for RPU foams with \(\rho=55\) kg/m\({}^{3}\). Figure 9: Comparisons between experimental results, TMM and FEM for RPU foams with \(\rho=80\) kg/m\({}^{3}\). h = 4 cm, h = 6 cm, h = 8 cm, and h = 10 cm are shown in Figure 10. The results obtained by TMM and FEM matched. As expected, the thickness affected the sound absorption efficiency in the low to mid frequencies.