# Numerical Analysis of Vibroacoustic Response of Timber Floor Panels Damped with Porous Materials
## 3 Finite Element Formulation of Sound Radiation of an Elastic Structure with a Poroelastic Layer
### Prediction of Radiated Sound Power  
The acoustic indicators used in this work to evaluate the sound insulation of the studied structure are the radiated sound power \(W_{r}\) and the sound-transmission loss \(TL\). The Rayleigh integral can be used to calculate the pressure \(p\) at any point \(M\) in the external acoustic domain due to the vibration of an elastic-plane structure and radiating in a rigid baffle [28]. This integral equation requires knowledge of the normal velocity \(v_{n}\) at any point \(G\) on the vibrating plate of surface \(\Gamma\). It can be written as Equation (10) [28; 29; 30; 31]:  
\[p(\omega,M)=\rho_{0}\frac{i\omega}{2\pi}\int_{\Gamma}v_{n}(\omega,G)\frac{e^{- ikr}}{r}dS \tag{10}\]  
where \(\rho_{0}\) is the volume density of the external acoustic domain; \(k\) is the wave number (\(k=\omega/c_{0}\)); \(c_{0}\) is the speed of sound in the acoustic domain; and \(r\) is the distance between \(M\) and \(G\). The velocity distribution \(v_{n}(\omega,G)\) can be obtained through finite element simulation.  
Therefore, the radiated power is given by  
\[W_{r}=\frac{1}{2}\;Re\left(\int_{\Gamma}p(G)v_{n}^{*}(G)dS\right) \tag{11}\]  
where \(v_{n}^{*}\) is the complex conjugate of the normal velocity and \(Re\) is the real part of the expression.  
The baffled panel is divided into a grid of \(R\) rectangular elements with equal size whose transverse vibrations are specified in terms of the normal velocities at their center positions. Assuming that the dimensions of the element are small compared with both the structural wavelength and the acoustic wavelength, the total radiated sound power (Equation (11)) can then be expressed as the summation of the powers radiated by each element, so that  
\[W_{r}=\frac{S_{e}}{2}\;Re\left(\stackrel{{\rightarrow}}{{v}}_{n} \stackrel{{\rightarrow}}{{p}}\right) \tag{12}\]  
where the exponent \(H\) denotes the Hermitian transposition; \(\stackrel{{\rightarrow}}{{v}}_{n}\) and \(\stackrel{{\rightarrow}}{{p}}\) are the vectors of the complex amplitudes of the normal velocity and the acoustic pressure in all elements; and is the area of each element. The pressure on each element is generated by the vibration of all elements in the \(\Gamma\) plane. The vector of the acoustic pressure can, therefore, be expressed by the impedance matrix relation  
\[\overrightarrow{p}=[Z]\overrightarrow{v_{n}} \tag{13}\]  
where \([Z]\) is the matrix incorporating the point and transfer acoustic-impedance terms over the grid of elements into which the panel has been subdivided:  
\[Z_{ij}=\frac{i\omega\rho_{0}S_{e}}{2\pi r_{ij}}e^{-ikr_{ij}} \tag{14}\]  
and where \(r_{ij}\) is the distance between the centers of the \(i\)th and \(j\)th elements. Note that, because of reciprocity, the impedance matrix \([Z]\) is symmetric. Substituting Equation (13) into the expression for the total radiated sound power given in Equation (12), we obtain  
\[W_{r}=\frac{S_{e}}{2}\ Re\bigg{(}\overrightarrow{v}_{n}^{H}[Z]\overrightarrow {v}_{n}\bigg{)}=\frac{S_{e}}{4}\ Re\bigg{(}\overrightarrow{v}_{n}^{H}\Big{(} [Z]+[Z]^{H}\Big{)}\overrightarrow{v}_{n}\bigg{)}=\overrightarrow{v}_{n}^{H}[ R]\overrightarrow{v}_{n} \tag{15}\]  
The matrix \([R]\) is defined as the \"radiation resistance matrix\" for the elementary radiators which, for the baffled panel [30; 31], is given by  
\[[R]=\frac{\omega^{2}\rho_{0}S_{e}^{2}}{4\pi c_{0}}\begin{bmatrix}1&\frac{\sin (kr_{12})}{kr_{12}}&\dots&\frac{\sin(kr_{1R})}{kr_{1R}}\\ \frac{\sin(kr_{21})}{kr_{21}}&1&\dots&\frac{\sin(kr_{2R})}{kr_{2R}}\\ \vdots&\vdots&&\ddots&\vdots\\ \frac{\sin(kr_{1R})}{kr_{1R}}&\frac{\sin(kr_{2R})}{kr_{2R}}&\dots&1\end{bmatrix} \tag{16}\]  
The normal incidence sound transmission of the double-wall sandwich panels is evaluated using the following formula:  
\[TL=10\text{log}\Big{|}\frac{W_{i}}{W_{r}}\Big{|} \tag{17}\]  
where \(W_{i}\) is the incident acoustic power.