## 5 Conclusions  
Based on the consideration of the resonance frequency theory of the air layer in the incident ultrasonic wave, this paper solved the resonance conditions of different thicknesses of air layers at different frequencies using both mathematical and finite element models. The measurement problem of the air layer thickness between the flange contact interface was solved, and the following conclusions were drawn. 1. Under the condition of the vertical incidence of sound waves, by using the displacement and stress equations of longitudinal waves in solids and the reflection conditions of elastic waves on non-homogeneous interfaces, the expressions of air layer thickness in the flange model containing an air layer were obtained. Based on the boundary conditions of elastic waves, the ultrasound resonance frequency under different air layer thickness conditions was calculated. 2. Based on considering the resonance frequency theory of ultrasonic waves generated by the air layer, this paper derived the expression of the air layer thickness in the flange model with the help of displacement and stress equations of longitudinal waves in solids and the reflection condition of elastic waves at the interface of dissimilar media under the condition of vertical ultrasonic wave incidence. Based on the boundary conditions of elastic waves, the resonant frequencies of ultrasonic waves were calculated for different air layer thicknesses. Additionally, a finite element simulation model of the flange contact surface with an air layer was established. By simulating the pressure sound field distribution of reflection and transmission at different incident angles and frequencies, the angle and frequency spectra of ultrasonic reflection coefficients for different air layer thicknesses were extracted. Comparing the finite element results with the mathematical model results showed high consistency. The reliability of using the ultrasonic reflection coefficient method to evaluate the interface contact quality by measuring the air layer thickness was confirmed. 3. Based on the vertical incidence model, a multi-angle incidence air layer thickness measurement model was derived and constructed. The relationship between air layer thickness and incidence angle under multiple incidence angles was obtained. The influence of the ultrasonic incidence angle and air layer thickness on the resonance frequency of the air layer was analyzed in detail. The results showed that when the incidence angle was constant, the resonance frequency of the air layer gradually decreased as the air layer thickness increased in a power function form. When the air layer thickness was constant, the resonance frequency of the air layer gradually increased with increasing incidence angle. The thickness of the air layer between the flanges can be analyzed by the resonance frequency of the air layer at different incidence angles to infer the fitting degree of the flange contact surface and prevent leakage. \begin{table}
\begin{tabular}{c c} \hline
**Figure** & **Error Value** \\ \hline Figure 4 & 0.01056 \\ \hline  & 0.03343 \\  & 0.03101 \\  & 0.04741 \\ Figure 6 & 0.04303 \\  & 0.02933 \\  & 0.04291 \\  & 0.04898 \\ \hline Figure 10 & 0.00361 \\ \hline Figure 11 & 0.00138 \\ \hline \end{tabular}
\end{table} 
Table 6: RMS error value. The above research results have led to the development of a multi-angle ultrasonic flange air gap thickness detection method which addresses the limitations of probe incidence angles in the detection of certain flange structures, ensures the operability of flange structure detection in any environment, and improves the accuracy issues caused by incidence angle limitations. This can provide technical and theoretical support for practical industrial applications. Conceptualization, F.S. and H.Z. ; methodology, B.S. ; software, B.S. ; validation, F.S., H.Z. and B.S. ; formal analysis, H.Z. ; investigation, H.Z. ; resources, F.S. ; data curation, F.S. ; writing--original draft preparation, H.Z. ; writing--review and editing, B.S.