## 4 Comparative Analysis of the Mathematical Model and Finite Element Model Results
### Comparison of Results from the Multi-Angle Incident of Sound Waves  
When the incidence angle of sound waves was non-perpendicular, the first-order resonance frequency of the air layer at different incidence angles and different thicknesses was calculated through COMSOL finite element simulation. The results are shown in Table 3.  
The first-order resonant theory model only provides the relationship between the thickness of the medium and the resonant frequency for vertical incidence, and the relationship between the thickness of the medium and the resonant frequency for inclined incidence is unknown. Figure 11 shows the resonant frequency curves for different incidence angles. It can be observed that the resonant frequency under the same air layer thickness followed a power law function, which was consistent with the trend of resonant frequency under vertical incidence. The resonant frequency under different incidence angles can be fitted and used as a resonant model for multi-angle conditions. The fitting resonant theory model curves for different incidence angles are shown in Figure 12.  
Based on Figure 12, it can be seen that the fitted curve shows an increasing trend of the first resonant frequency with the increasing incidence angle of the sound wave at a constant thickness of the air layer. Therefore, it is recommended to use a low incidence angle for measurement in practical applications to reduce the attenuation of high-frequency sound waves and improve the detection efficiency. When the air layer thickness is relatively thin, using a wide-band ultrasound probe can collect more frequency-domain information, thus making the measurement results more accurate. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Air Layer Thickness**} & \multicolumn{8}{c}{**Angle of Incidence (\({}^{\circ}\))**} \\ \cline{2-7}  & **10** & **20** & **30** & **40** & **50** & **60** & **70** \\ \hline
0.03 & 5.81 & 6.41 & 7.31 & 8.11 & 8.81 & 9.31 & 9.61 \\
0.04 & 4  
By fitting the simulated power function curves using MATLAB, the relationship between the first harmonic resonance frequency and the air layer thickness at different angles can be obtained as follows:  
\[f(\theta)=kh^{i} \tag{12}\]  
where \(\theta\) is the incidence angle of the sound wave, \(k\) is the coefficient of the fitting curve equation, \(h\) is the thickness of the air layer, and \(i\) is the exponent term of the thickness of the air layer. The coefficients of Equation (12) obtained by fitting are shown in Table 4.  
Figure 11: Resonance frequency points at multiple incidence angles. Figure 12: Resonance theoretical model curves for multi-angle incidence. From Equation (12) and Table 4, it can be seen that the coefficients in Equation (12) increase with increasing incidence angle. Therefore, a function can be established to relate the coefficient \(k\) to the incidence angle \(\theta\), and the reflection coefficient under different incidence angles can be analyzed using the newly established fitting function. The graph of the fitting function and the coefficient \(k\) is shown in Figure 13.  
From the graph, it can be observed that the coefficient \(k\) follows a linear distribution. A linear function relationship can be established between the coefficient \(k\) and the incident angle \(\theta\), as shown in Equation (13). \[k=0.0018\theta+0.1688 \tag{13}\]  
According to Table 4, the exponential term in Equation (12) varies as a polynomial. The exponent \(i\) and its fitting curve are shown in Figure 14. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Angle of Incidence** & **0** & **10** & **20** & **30** & **40** & **50** & **60** & **70** \\ \hline Coefficient \(k\) & 0.1715 & 0.1877 & 0.1993 & 0.2182 & 0.2392 & 0.2609 & 0.2776 & 0.2891 \\ Exponent \(i\) & \(-1\) & \(-0.978\) & \(-0.992\) & \(-1.001\) & \(-1.007\) & \(-1.005\) & \(-1.003\) & \(-1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Coefficients of the fitted curve equation.