# Analysis of the Effect of Temperature on the Sound Transmission Loss of a Curved Plate
## 2 Theoretical Model
### Sound Transmission Loss  
According to the fluctuation theory, the velocity potential functions of the incident and radiated sound fields can be expressed as follows:  
\[\Phi_{1}(\theta,L,R;t)=le^{-j(k_{\theta}\theta+k_{L}L+k_{R}(R-R_{0})-\omega t )}+\beta e^{-j(k_{\theta}\theta+k_{L}L-k_{R}(R-R_{0})-\omega t)}, \tag{18}\]  
\[\Phi_{2}(\theta,L,R;t)=\varepsilon e^{-j(k_{\theta}\theta+k_{L}L+k_{R}(R-R_{ 0})-\omega t)}. \tag{19}\]  
The wave numbers in each direction are \(k_{\theta}=k_{0}sin\sigma cos\xi\), \(k_{L}=k_{0}sin\sigma sin\xi\), and \(k_{R}=k_{0}cos\sigma\), where \(k_{0}=\omega/c\). Because the boundary conditions are simply supported, at the plate's outer edge, both the bending moment and displacement are zero. Therefore, the mathematical description of the curved plate is as follows:  
\[\theta=0,\theta_{0}:\,w=0,\,\,\,\frac{\partial^{2}w}{\partial\theta^{2}}=0, \tag{20}\]  
\[L=0,L_{0}:\,w=0,\,\,\,\frac{\partial^{2}w}{\partial L^{2}}=0. \tag{21}\]  
According to Equations (20) and (21), the deflection (\(w\)) can be expressed as a double trigonometric expression, as follows:  
\[w(\theta,L,t)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}A_{mn}\varphi_{mn}e^{i \omega t}, \tag{22}\]  
where \(m\) and \(n\) represent the number of waves in the circumferential and axial directions, respectively, and \(A_{mn}\) is the modal array coefficient. \(\varphi_{mn}\) is the modal function, whose expression is as follows:  
\[\varphi_{mn}=\sin\frac{m\pi\theta}{\theta_{0}}\sin\frac{n\pi L}{L_{0}}. \tag{23}\]According to the deflection equation, Equation (22), the velocity potential function of the sound pressure can be expressed as follows:  
\[\Phi_{1}(\theta,L,R;t)=\sum_{m,n}I_{mn}\varphi_{mn}e^{-j(k_{R}(R-R_{0})-\omega t) }+\sum_{m,n}\beta_{mn}\varphi_{mn}e^{-j(-k_{R}(R-R_{0})-\omega t)}, \tag{24}\]  
\[\Phi_{2}(\theta,L,R;t)=\sum_{m,n}\varepsilon_{mn}\varphi_{mn}e^{-j(k_{R}(R-R_ {0})-\omega t)}, \tag{25}\]  
where \(I_{mn}\) and \(\beta_{mn}\) are the incident and reflected wave amplitudes, respectively, and \(\varepsilon_{mn}\) is the radiation wave amplitude. \(I_{mn}\), \(\beta_{mn}\), and \(\varepsilon_{mn}\) are related to the original coefficients as follows:  
\[\lambda_{mn}=\frac{4}{\theta L}\int_{0}^{\theta}\int_{0}^{L}\lambda e^{-j(k_{ \theta}\theta+k_{L}L)}\sin\frac{m\pi\theta}{\theta_{0}}\sin\frac{n\pi L}{L_{0 }}d\theta dL, \tag{26}\]  
where \(\lambda\) refers to the coefficients \(I\), \(\beta\), and \(\varepsilon\) in general. At the air-and-plate intersection, the following equation is obtained according to the condition of velocity continuity. \[R=R_{0}+h:\qquad-\frac{\partial\Phi_{1}}{\partial R}=j\omega w\ \, \tag{27}\]  
\[\Phi_{2}(\theta,L,R;t)=\varepsilon e^{-j(k_{\theta}\theta+k_{L}L+k_{R}(R-R_{ 0})-\omega t)}. \tag{19}\]0  
Substituting Equations (24) and (25) into Equations (27) and (28), respectively, we obtain the following:  
\[\Phi_{2}(\theta,L,R;t)=\varepsilon e^{-j(k_{\theta}\theta+k_{L}L+k_{R}(R-R_{ 0})-\omega t)}. \tag{19}\]1  
\[\Phi_{2}(\theta,L,R;t)=\varepsilon e^{-j(k_{\theta}\theta+k_{L}L+k_{R}(R-R_{ 0})-\omega t)}. \tag{19}\]2  
The conjunction of Equations (17) and (30) gives  
\[ \begin{array}{l}A_{mn}D\
abla^{8}\varphi(\theta,L)+A_{mn} Eh\
abla_{1}^{4}\varphi(\theta,L)-\omega^{2}\rho hA_{mn}\
abla^{4}\varphi( \theta,L)=\\ j\omega\
abla^{4}\varphi(\theta,L)\left[\rho_{1}I_{mn}e^{-jk_{1R}h}+\rho_{1} \beta_{mn}e^{jk_{1R}h}-\rho_{2}\varepsilon_{mn}\right]\end{array}