## 4 Comparative Analysis of the Mathematical Model and Finite Element Model Results
### Comparison of Results from the Multi-Angle Incident of Sound Waves
Figure 14: Fitting curve for the exponent term of the first-order resonant equation. Figure 13: Relationship curve between curve coefficients and incidence angle and its fitting. The fitting equation for the exponential term is:  
\[ \begin{array}{l}i=8.3333\times 10^{-11}\theta^{5}-2.3106\times 10^{-8}\theta^{4}+2.0 719\times 10^{-6}\theta^{3}\\ -5.5416\times 10^{-5}\theta^{2}-8.3530\times 10^{-4}\theta-0.9661\end{array}  \tag{14}\]  
The first harmonic resonance equation under multi-angle incidence can be obtained from Equations (13) and (14):  
\[f(\theta)=(0.0018\theta+0.1688)h^{i} \tag{15}\]  
where \(f(\theta)\) is the first-order resonant frequency equation. To validate the reliability of the curve coefficient equation, numerical calculations and finite element simulations were conducted to obtain the first-order resonant frequency under four different operating conditions: air layer thickness of 0.03 mm and 0.05 mm, and incident angles of 15\({}^{\circ}\) and 45\({}^{\circ}\). The results are shown in Table 5 for comparison and analysis. Based on Table 5, the first-order resonant frequencies obtained by fitting function and finite element simulation calculations were in good agreement for different operating conditions, with a minimum difference of 0.08 MHz and a maximum difference of 0.14 MHz. Therefore, the fitting function of the first-order resonant frequency can be applied to measure the thickness of the air layer under different incident angles, which indirectly characterizes the actual contact state between the flange and the silicone gasket. When conducting curve fitting, the number of fitting points is limited and relatively small. Therefore, there exists an error between the values of the fitting curve and those of the fitting points. To reduce the interference caused by errors when calculating the reflection coefficient, an error analysis of the fitting curve is necessary. Considering the characteristics between the discrete points and the fitting curve, the RMS error analysis method was chosen, as shown in Equation (16). \[X_{\text{RMSE}}=\sqrt{\frac{\sum\limits_{i=1}^{N}(X_{\text{obs},i}-X_{\text{ model},i})}{N}} \tag{16}\]  
where \(X_{\text{RMSE}}\) is the RMS error value, \(X_{\text{obs},i}\) is the observed value, \(X_{\text{model},i}\) is the true value, and \(N\) is the number of observations. The errors calculated according to Equation (16) are shown in Table 6. It can be concluded from the data in Table 6 that the fitting between the discrete points and the fitted curve was good, and the error can be ignored when calculating the thickness of the air layer through the fitted curve. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type of** & **Angle of Incidence** & \multicolumn{2}{c}{**Air Layer Thickness (\({}^{\circ}\))**} \\ \cline{3-4}
**Computation** & **(mm)** & **0.03** & **0.05** \\ \hline Mathematical & 15 & 6.20 & 3.75 \\ calculation & 45 & 8.52 & 5.09 \\ \hline Finite element & 15 & 6.11 & 3.61 \\ analysis & 45 & 8.41 & 5.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Resonant frequencies obtained from numerical and finite element simulations.