## 4 Comparative Analysis of the Mathematical Model and Finite Element Model Results  
### Comparison of Results from the Vertical Incident of Sound Waves  
In order to verify the accuracy of the model, it is necessary to compare the numerical results with those obtained from finite element simulations. Using the first-order resonance theory, the thickness of the air layer at different resonance frequencies was calculated and compared with the simulated results of the vertical incidence measurement model. The theoretical thickness of the air layer obtained by mathematical methods was obtained and is shown in Table 2.  
The data in Table 2 represent a comparison between the results of the mathematical model and the finite element simulation model at the same frequency. The maximum deviation of the air layer thickness was 3.6 \(\upmu\)m, with a deviation of 3.17%, and the minimum deviation was 0.3 \(\upmu\)m, with a deviation of 0.3%, which is acceptable. The calculation nodes of the air layer thickness in the finite element simulation and the image of the first-order resonance theory curve for when the sound field was vertically incident are shown in Figure 10.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Air Layer** & **Reflectance** & **Resonance** & **Theoretical** & **Deviation of Air** & **Percentage** & **Deviation of Air** \\
**Thickness** & **Coefficient** & **Frequency** & **Thickness** & **Layer Thickness** & **Layer Thickness** \\
**Q/\(\upmu\)m** & **R** & **f/MHz** & **Q/\(\upmu\)m** & **\(\upmu\)m** & **\%** \\ \hline
30 & 0.013 & 5.61 & 30.6 & 0.6 & 2.0 \\
50 & 0.067 & 3.41 & 50.2 & 0.2 & 0.4 \\
70 & 0.024 & 2.41 & 71.1 & 1.1 & 1.5 \\
90 & 0.114 & 1.91 & 89.7 & 0.3 & 0.3 \\
110 & 0.049 & 1.51 & 113.6 & 3.6 & 3.2 \\ \hline \end{tabular}
\end{table}
Table 2: Data table for the extreme points of the reflection coefficient.  
Figure 10: The relationship curve between air layer thickness and resonance frequency.  
As shown in Figure 10, when the sound waves were vertically incident the air layer thickness nodes of the finite element model coincided with the resonance curve. This indicated that the vertical incidence model could be used to measure the air layer thickness in flange fastening devices. The first-order resonance frequency decreased as the air layer thickness between the flange cover and the silicone gasket increased. Moreover, when the air layer thickness tended to infinity, the first-order resonance frequency tended to infinity, and the two exhibited a power-law function relationship.  
Based on the comparison between the results of the mathematical model and the simulation results of the finite element model, it was found that there was a good agreement between the finite element solution and the numerical calculation solution, which indicates that this model can be used to calculate the thickness of the air layer between flanges when ultrasonic waves are vertically incident. However, during the inspection process, due to the special nature of the structure of the flange fastening device and the service environment, the sound field cannot be vertically incident. Therefore, a multi-angle finite element measurement model was established to investigate the influence of air layer thickness on the first-order resonance frequency under different incidence angles.