## 2 Physics-Driven Models of Sound Transmission Loss
### Methodology of the STL Analyses
#### 2.1.3 Modal Summation  
An analytical solution of the fluid-structure iteration problem of a simple supported plate can also be obtained by modeling the plate displacement \(w\) in terms of a modal summation [29; 30]:  
\[w(x,y,z)=\sum_{m,n}\alpha_{m,n}\varphi_{m,n}(x,y), \tag{23}\]  
where \(\sum_{m,n}\) is the short format for \(\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\), the subindices \(m\) and \(n\) indicate the mode index, \(\alpha_{m,n}\) is the coefficient of contribution of each mode for the displacement field, and \(\varphi_{m,n}(x,y)\) (Equation (24)) is the modal function that satisfies the simple supported boundary conditions (Equation (25)).  
\[\varphi_{m,n}(x,y)=\sin\Bigl{(}\frac{m\pi x}{a}\Bigr{)}\sin\Bigl{(}\frac{n\pi y }{b}\Bigr{)}. \tag{24}\]  
\[w=\frac{\partial^{2}w}{\partial x^{2}}+\
u\frac{\partial^{2}w}{ \partial y^{2}}=0,\ \ \ \text{for}\ x=\pm a/2, \tag{25}\] \[w=\frac{\partial^{2}w}{\partial y^{2}}+\
u\frac{\partial^{2}w}{ \partial x^{2}}=0,\ \ \ \text{for}\ y=\pm b/2.\]  
Using the 2D Fourier Transform, each pressure field \(P_{i}\) can also be described as a modal summation:  
\[P_{i}(x,y,z)=\sum_{m,n}p_{i_{m},n}\varphi_{m,n}(x,y)e^{\pm k_{z}z}, \tag{26}\]  
where the signal of the exponential function depends on the propagation direction of the plane wave in the field, and in which the coefficient of contribution of the pressure field \(p_{i_{m},n}\) reads as:  
\[p_{i_{m},n}=\frac{4}{ab}\int_{0}^{b}\int_{0}^{a}p_{i}e^{-i(k_{x}x+k_{y}y)} \varphi_{m,n}(x,y)dxdy. \tag{27}\]  
Substituting Equations (23), (26) and (27) into Equation (14) and applying the weighted residuals method:  
\[\int_{0}^{b}\int_{0}^{a}\Biggl{[}D\
abla^{4}\sum_{m,n}\alpha_{m,n}\varphi_{m, n}-\omega^{2}m\sum_{m,n}\alpha_{m,n}\varphi_{m,n}-\sum_{m,n}2\Bigl{(}p_{1_{m},n}-p _{T_{m},n}\Bigr{)}\varphi_{m,n}\Biggr{]}\varphi_{m,n}dxdy=0. \tag{28}\]  
As the modal functions have orthogonal properties \(\{\varphi_{j,k}\}^{T}M\{\varphi_{h,g}\}=0\) for \(j\
eq h\) or \(k\
eq g\), Equation (28) reads as  
\[\alpha_{m,n}D\frac{\int_{0}^{b}\int_{0}^{a}\bigl{[}\
abla^{4}\varphi_{m,n} \bigr{]}\varphi_{m,n}dxdy}{\int_{0}^{b}\int_{0}^{a}\varphi_{m,n}dxdy}-\alpha_ {m,n}\omega^{2}m-2\Bigl{(}p_{1_{m},n}-p_{T_{m},n}\Bigr{)}=0 \tag{29}\]  
if inter-modal coupling is neglected. The integrals on the first term can be analytically evaluated, and the relation from Equation (8) is used, resulting in:  
\[\biggl{[}\omega_{m,n}^{2}-\omega^{2}+2\frac{i\omega\rho_{0}c_{0}}{m\cos\theta }\biggr{]}\alpha_{m,n}=\frac{2}{m}p_{1_{m},n}, \tag{30}\]  
where \(\omega_{m,n}\) are the natural frequencies of the simply supported plate, and \(p_{1_{m},n}\) can be analytically evaluated by solving the integrals from Equation (27). Finally, the solution of the system of equation in Equation (30) provides the coefficients of modal participation \(\alpha_{m,n}\)and, consequently, the displacement \(w\) (Equation (23)). The coefficients \(p_{Tm,n}\) are evaluated with Equation (9), and the acoustic transparency is obtained by:  
\[\tau(\theta,\phi)=\frac{\sum_{m,n}|p_{Tm,n}|^{2}}{\sum_{m,n}|p_{1m,n}|^{2}}. \tag{31}\]