## 3 Validation and Analysis  
In the process of establishing the theoretical model, some constants were defined and the high-order terms of the equations were neglected. In order to validate the model precision, the sound transmission loss (STL) comparison of the analytical models and numerical models and experiments was carried out. Three WU cases were carried out using different structural parameters, as shown in Table 2.  
Impedance-tube experiments based on the transfer function method were carried out using a test system manufactured by BSWA and the experimental principle is shown in \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Bulk viscosity/(Pa-s) & \(1.11\times 10^{-5}\) \\ Dynamic viscosity/(Pa-s) & \(1.85\times 10^{-5}\) \\ Density/(kg\(\cdot\)m\({}^{-3}\)) & \(1.21\) \\ Thermal conductivity/(W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\)) & \(2.63\times 10^{-2}\) \\ Heat capacity at constant & \(1005.70\) \\ pressure/(J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\)) & Ratio of specific heats & \(1.40\) \\ Speed of sound/(m\(\cdot\)s\({}^{-1}\)) & \(343.00\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of air. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameters** & \(d\) **(mm)** & \(t\) **(mm)** & \(a\) **(mm)** & \(w\) **(mm)** & \(I_{1}\) **(mm)** & \(I_{2}\) **(mm)** \\ \hline Case 1 & 12.0 & 2.0 & 30.0 & 12.0 & 208 & 56 \\ \hline Case 2 & 12.0 & 2.0 & 25.0 & 12.0 & 193 & 50 \\ \hline Case 3 & 12.0 & 2.0 & 30.0 & 12.0 & 223 & 44 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Structure parameters. Figure 2: Boundary conditions and finite element meshes. (**a**) WU and (**b**) LC. Figure 3a. The thickness of the experimental specimen was \(d\) = 12 mm, the distance between microphones \(m_{1}\) and \(m_{2}\) was \(s_{1}\), and the distance between microphones \(m_{3}\) and \(m_{4}\) was \(s_{2}\). When \(s_{1}\) = \(s_{2}\) = 0.3 m, \(l_{1}\) = 0.05 m, and \(l_{2}\) = 0.15 m, the effective test range of the impedance tube was 63-550 Hz. The experimental physical diagram is shown in Figure 3b, and the modified impedance loading \(Z_{i}\) on the end was realized by adjusting the end cap. The LC specimen was fabricated using fused deposition modeling (FDM) technology with filament diameter of 1.75 mm, layer height of 0.1 mm, and nozzle diameter of 0.2 mm, as shown in Figure 3c. In the experiments, the STL was calculated as [35]  
\[\left\{ \begin{array}{l}\mathrm{STL}=20\log_{10}\left|\frac{1}{\tau}\right|\\ \tau=\frac{T_{11}+\frac{T_{22}}{L_{0}}+T_{21}Z_{0}+T_{22}}{2e^{\mathrm{i}d}} \end{array} \right. \tag{14}\]  
where, \(\tau\) is the transmission coefficient from the experiment, \(\mathbf{T}\) is the transfer matrix calculated from the measurements, and \(Z_{0}\) is the characteristic impedance of air. The comparison results are shown in Figure 4. In Case 1, for example, the results of the analytical and numerical model were in good agreement. The STL peak of the analytical model was located at 378.5 Hz, and that of the numerical model was located at 377.5 Hz, with an error of no more than 0.26%. The accuracy of the analytical model was clarified. There was a slight error between the experimental and the theoretical results. On the one hand, this was due to the processing accuracy during the manufacturing process of the specimens, and on the other hand, the fact that in the theoretical models, the rigid walls were defined, but it was difficult to realize the ideal state in the experiments. Cases 2 and 3 showed a certain frequency shift compared to case 1 due to the variation in structural parameters, but comparison of the theoretical and experimental results showed that they were in good agreement. Figure 4: STL comparison of analytical and numerical models. Figure 3: Impedance-tube experiment. (**a**) Experimental principle of transfer function method, (**b**) physical diagram of the experiment, and (**c**) samples.