## 2 Basic Theories
### Acoustic-Structural Coupling Modal Theory  
Acoustic-structure coupling is a multi-physics analysis system that examines the interactions between solid structures and acoustic waves in air. This research method interprets the coupling process by solving the acoustic wave equation for the system under defined boundary conditions through classical acoustic analytical techniques. The diagram of the acoustic-structure coupling system is shown in Figure 3. The complete coupling system consists of a structural domain (\(\Omega_{s}\)), an acoustic cavity domain (\(\Omega_{f}\)), and a coupling interface (\(\Omega_{sf}\)). The boundary conditions of domains include the pressure boundary condition (\(\Gamma_{f}\)) and displacement boundary condition (\(\Gamma_{u}\)) [19]. The structure and acoustic waves interact at the coupling surface between the solid structure and the acoustic cavity of the muffler. To ensure proper coupling, continuity of sound pressure and displacement must be maintained on the coupling surface. Therefore, the coupling process must satisfy the following conditions:  
\[u_{i}\mathbf{n}_{i}\mid_{\Omega_{sf}}=u_{s}\mathbf{n}_{s}\mid_{\Omega_{sf}} \tag{13}\]  
where \(u_{i}\) is the displacement of the acoustic cavity domain at the coupling boundary, \(u_{s}\) is the displacement of the structural domain at the coupling boundary, \(\mathbf{n}_{i}\) is the normal vector of the acoustic cavity surface at the coupling boundary, and \(\mathbf{n}_{s}\) is the normal vector of the structural surface, with the relationship \(\mathbf{n}_{i}=-\mathbf{n}_{s}\). Figure 3: Diagram of the acoustic-structure coupling system. Equation (21) is substituted into Equations (19) and (20). Consequently, Equations (19) and (20) can be simplified as follows:  
\[F_{is}=RP \tag{22}\]  
\[F_{si}=-\rho R^{T}\ddot{u} \tag{23}\]  
The finite element equation of the acoustic-structure coupling system can be obtained by combining Equations (17)-(23):  
\[ \begin{bmatrix}M_{s}&0\\ \rho R^{T}&M_{i}\end{bmatrix}\begin{Bmatrix}\ddot{U}\\ \ddot{p}\end{Bmatrix} + \begin{bmatrix}C_{s}&0\\ 0&C_{a}\end{bmatrix}\begin{Bmatrix}\dot{U}\\ \dot{p}\end{Bmatrix} + \begin{bmatrix}K_{s}&-R\\ 0&K_{i}\end{bmatrix}\begin{Bmatrix}U\\ P\end{Bmatrix} = \begin{Bmatrix}F_{s}\\ F_{i}\end{Bmatrix}  \tag{24}\]  
Assuming that both displacement and acoustic waves are harmonic functions, Equation (24) can be rewritten as a formula without the differential term [21]:  
\[ \begin{bmatrix}K_{s}-\omega^{2}M_{s}+j\omega C_{s}&-R\\ \rho\omega^{2}R^{T}&K_{i}-\omega^{2}M_{i}+j\omega C_{i}\end{bmatrix}\begin{Bmatrix} U\\ P\end{Bmatrix} = \begin{Bmatrix}F_{s}\\ F_{i}\end{Bmatrix} \] (  
mitted sound power level at the outlet of the muffler (measured in dB). The formula for calculating transmission loss is  
\[L_{TL}=L_{Win}-L_{Wout}=10\log 10\frac{W_{in}}{W_{out}}=20\log 10\left|\frac{p_{in} }{p_{out}}\right| \tag{31}\]  
From Equation (31), assuming total silence at the airflow outlet of the muffler, the transmission loss is the difference between the incident sound power level and the transmitted sound power level of the muffler [22]. The amplitude of the function reaches its maximum at the midpoint of the source field and is zero at both ends. The subscripts '1' and '2' represent the forward and backward traveling waves, respectively. \(L_{Win}\) refers to the incident sound power level of the muffler, while \(L_{Wout}\) refers to the transmitted sound power level. \(W_{in}\) represents the incident sound power, and \(W_{out}\) represents the transmitted sound power. \(p_{in}\) refers to the incident sound pressure, and \(p_{out}\) refers to the transmitted sound pressure (Figure 3). \(p_{in}\) is expressed as the sum of \(p_{in1}\) and \(p_{in2}\).