## 2 Establishment of the Mathematical Model
According to the principles of acoustics, when the thickness of the air layer is a multiple of \(\lambda/2\) (\(\lambda\) is the wavelength), a standing wave will occur at the middle position of the air layer where the incident and reflected waves superimpose. The standing wave will cause resonance phenomena, and resonance will occur when the frequency of the ultrasonic wave  
Figure 2: Flange fastening device acoustic reflection/transmission diagram. is equal to the natural frequency of the air layer [17]. The expression for the resonance angular frequency is:  
\[\omega=\frac{\pi cn}{h} \tag{5}\]  
The analytical expression for calculating the thickness of the air layer can be obtained after transformation:  
\[h=n\cdot\frac{\lambda}{2}=\frac{nc}{2f_{n}} \tag{6}\]  
where \(n\) is a positive integer representing the order of resonance, and \(f_{n}\) is the resonant frequency of the \(n\)th order. According to Equation (4), it is known that the reflection coefficient function \(R\) is a function of the frequency when the air layer thickness is constant. Assuming that layers 1 and 5 of the medium are Q235 and layers 2 and 4 are air, the relationship between the incident frequency and the reflection coefficient can be calculated under different air layer thickness conditions by looking up the density and sound speed of Q235 and air. By using Formula (4) and calculating the results based on the data in Table 1, the relationship between incident sound wave frequency and reflection coefficient under different air layer thicknesses can be obtained, as shown in Figure 3. The analysis of Formula (4) reveals that when there is a large difference in density between the media, there may be some errors in the calculated results. As the density of the medium gradually increases, the model error decreases and the change in reflection coefficient is independent of density. Therefore, Formula (4) can be used to analyze the resonant frequency of ultrasound in the air layer and to analyze the thickness of the air layer. As shown in Figure 3, when the incident frequency of ultrasound is at a specific set of values, the reflection coefficient of the air layer reaches a minimum value. This means that at this incident frequency, the air layer medium resonates with the ultrasound, resulting in a minimum value of the interface reflection coefficient. Therefore, in practical measurements, the thickness of the air layer can be calculated by extracting the resonant frequency corresponding to the minimum value of the reflection coefficient and combining it with Formula (6). Since the reflection coefficient minimum value corresponding to the first-order resonant frequency is easy to select in the calculation process, it is chosen as the reference value. \begin{table}
\begin{tabular}{c c c} \hline
**Medium** & **Density (kg/m\({}^{3}\))** & **Sound Velocity or Speed of Sound (m/s)** \\ \hline Q235 & 7850 & 5890 \\ Air & 1.29 & 340 \\ Silicone Gel & 1200 & 2300 \\ \hline \end{tabular}
\end{table} 
Table 1: The physical parameters of the medium. Figure 3: Graph of the relationship between incident frequency and reflection coefficient.