## 2 Theoretical Modeling
### Computation of Sound Radiation Fields  
The Rayleigh Integral gives the acoustic pressure for any point in the half-space \(\big{(}x_{p},y_{p},z_{p}\big{)}\) as shown in Figure 2.  
\[p\big{(}x_{p},y_{p},z_{p}\big{)}=\frac{j\omega\rho_{0}}{2\pi}\int_{S}V(x,y) \cdot\frac{e^{-j\kappa R}}{R}dS \tag{29}\]where \(R=|x-x_{s}|=\sqrt{\left(x_{p}-x\right)^{2}+\left(y_{p}-y\right)^{2}+z_{p}^{2}}\) is the distance between the source point \(|x_{s}|=\left(x,y,0\right)\) on the plate surface to the observation point \(|x|=\left(x_{p},y_{p},z_{p}\right)\) with \(z_{p}\geq 0\) The density of the acoustic medium, air is given by \(\rho_{0}\), \(\kappa\) is the acoustic wavenumber, \(c_{0}\) is the speed of sound in acoustic medium air, and \(S\) is the plate's radiating surface. Each element can also be considered a compact acoustic source, resembling a monopole source. \(S_{e}=dx*dy\), gives the elemental area, where \(dx\) and \(dy\) are the element sizes in the x and y directions, respectively. Every elemental radiator has a constant velocity [32; 52] because the characteristics length of each elemental radiator is smaller than acoustic and structural wavelengths. Thus, Equation (29) can be expressed in discretized form as  
\[p\left(x_{p},y_{p},z_{p}\right)=\frac{j\omega\rho_{0}}{2\pi}\sum_{s}V_{M,N} \frac{e^{-j\kappa R}}{R}dxdy \tag{30}\]  
For a meshed P-FGM plate, Equation (30) can be expressed as an impedance matrix  
\[\overset{\rightarrow}{p}\approx\mathbf{Z}_{p}\overset{\rightarrow}{\mathbf{V}} \tag{31}\]  
Writing Equations (30) and (31) in terms of impedance matrix \(\mathbf{Z}_{p}\) as  
\[\left(\mathbf{Z}_{p}\right)_{il}=\frac{j\omega\rho_{0}S_{e}}{2\pi}\frac{e^{-j\kappa R _{il}}}{R_{il}};\ \ \ \text{for}\ i,l=M,N \tag{32}\]  
where \(S_{e}\) is the \(i^{th}\) elemental radiator area located at source point \((x,y,0)\) and \(R_{il}\) denotes the distance from \(i^{th}\) to \(l^{th}\) element located at observation point \(\left(x_{p},y_{p},0\right)\). For [PERSON]'s function integrand [10; 32], the reciprocity principle holds. \[\left(\frac{e^{-j\kappa R_{il}}}{R_{il}}\right)\text{ \emph{i.e.} }G(x_{s}|x)=G(x|x_{s})\]  
The radiation resistance matrix \(\mathbf{R_{mat}}\) is defined per relation \(\mathbf{R_{mat}}=\frac{S_{e}}{2}Re(\mathbf{Z}_{p})\) as given by [35], which converts to  
\[\mathbf{R_{mat}}=\frac{\omega_{mn}^{2}\rho_{0}S_{e}^{2}}{4\pi c} \begin{bmatrix} 1&\frac{sin\kappa R_{12}}{\kappa R_{12}}&\dots&\frac{sin\kappa R_{1N}}{\kappa R _{1N}}\\ \frac{sin\kappa R_{21}}{\kappa R_{21}}&1&\dots&\vdots\\ &\vdots&\ddots&\vdots\\ \frac{sin\kappa R_{1N}}{\kappa R_{1N}}&\dots&\dots&1\end{bmatrix}  \tag{33}\]  
Figure 2: Schematic diagram of elemental radiators. Because of reciprocity, \(R_{mat}\) is symmetric, real, and positive definite. Equation (33)'s matrix elements are determined by the acoustic medium's properties, the natural frequencies and geometry of the P-FGM plate. As a result, sound power from each of the elemental radiators \(\overline{W_{e}}\) is given as  
\[\overline{W_{e}}=\frac{S_{e}}{2}Re\{(\overset{\rightarrow}{V^{*}})\mathbf{Z_{p}} \overset{\rightarrow}{\mathbf{V}}\}=\overset{\rightarrow}{\mathbf{V}^{*}}R_{mat} \overset{\rightarrow}{\mathbf{V}} \tag{34}\]  
where \(\overset{\rightarrow}{\mathbf{V}}\) is the normal surface velocity vectors of each elemental radiator and * denotes the complex conjugate.