# Vibro-Acoustic Performance of a Sandwich Plate with Periodically Inserted Resonators
## 2 Formulations
### Vibro-Acoustic Formulation of a Finite Periodic Structure
Then the modal frequencies and modal shapes can be obtained by setting \(\mathbf{F}\) as zero, and the modal coefficient \(\overline{\mathbf{\Phi}}\) can be determined by setting \(\mathbf{F}\) as nonzero. After \(\overline{\mathbf{\Phi}}\) is known, the displacement response of the LR sandwich plate can finally be determined using Equation (17), from which the transverse velocity can be obtained as \(\vartheta(x,y)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\hat{V}_{mn }\sin(k_{n}x)\sin(k_{n}y)\), where \(\hat{V}_{mn}=-j\omega\overline{W}_{mn}\). The spatially averaged mean square velocity can be calculated as  
\[\left\langle\overline{\theta^{2}}\right\rangle=\sum\limits_{m=1}^{M}\sum \limits_{n=1}^{N}\frac{1}{8}\hat{V}_{nm}\hat{V}_{mn}^{*}, \tag{20}\]  
where the superscript asterisk denotes the complex conjugate of a variable. According to the Rayleigh integral, the acoustic pressure \(p\) at the far-field observation point \((r,\theta,\varphi)\) can be written in terms of the plate-surface velocity \(\hat{v}(x,y)\)  
\[p(r,\theta,\varphi)=\frac{jk_{ac}\varphi_{0}}{2\pi}\int\limits_{0}^{a}\int \limits_{0}^{b}\frac{\vartheta(x,y)e^{-jkr}}{r}dydx. \tag{21}\]  
Substitution of Equation \(\vartheta(x,y)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\hat{V}_{ mn}\sin(k_{m}x)\sin(k_{n}y)\) into Equation (21) gives  
\[p(r,\theta,\varphi)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\hat{V }_{mn}T_{mn}(r,\theta,\varphi). \tag{22}\]  
Here  
\[T_{mn}(r,\theta,\varphi)=jk_{ac}\varphi_{0}c\frac{e^{-jkr}}{2\pi r}\frac{ab}{ mn\pi^{2}}\frac{\left(-1\right)^{m}e^{-jk_{x}}-1}{\left(\tau_{x}/m\right)^{2} -1}\frac{\left(-1\right)^{n}e^{-jk_{y}}-1}{\left(\tau_{y}/n\pi\right)^{2}-1}, \tag{23}\]  
where \(\tau_{x}=ka\sin\theta\cos\varphi\) and \(\tau_{y}=kb\sin\theta\sin\varphi\). Thus, the total acoustic power radiated from the LR sandwich plate can be obtained by integrating sound intensity over a far-field hemisphere with radius \(r\)  
\[\Pi=\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi/2}\frac{\left|p(r,\theta, \varphi)\right|^{2}}{2\rho_{0}c}r^{2}\sin\theta d\theta d\varphi. \tag{24}\]  
Substituting Equation (22) into Equation (24) gives  
\[\Pi=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\sum\limits_{p=1}^{ \infty}\sum\limits_{q=1}^{\infty}\Pi_{mmpq}, \tag{25}\]  
where \(\Pi_{mmpq}\) is the acoustic power radiated by mode \((m,n)\) due to the vibration of mode \((p,q)\), and it is given by  
\[\Pi_{mmpq}=\hat{V}_{mn}\hat{V}_{pq}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi /2}\frac{T_{mn}(r,\theta,\varphi)T_{pq}^{*}(r,\theta,\varphi)}{2\rho_{0}c}r^{ 2}\sin\theta d\theta d\varphi. \tag{26}\]Thus, the modal radiation efficiency can be expressed as  
\[\sigma_{mmpq}=\frac{\Pi_{mmpq}}{\frac{1}{8}\rho_{0}cab\hat{V}_{mn}\hat{V}_{pq}^{*} }=\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi/2}\frac{4}{\rho_{0}^{2}c^{2}ab}T_{ mn}(r,\theta,\varphi)T_{pq}^{*}(r,\theta,\varphi)r^{2}\sin\theta d\theta d\varphi, \tag{27}\]  
and the total average radiation efficiency of the LR sandwich plate is given by  
\[\sigma=\frac{\Pi}{\frac{1}{2}\rho_{0}cab\left(\widehat{\phi}^{2}\right)}=\frac {\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\sum\limits_{p=1}^{ \infty}\sum\limits_{q=1}^{\infty}\hat{V}_{mn}\hat{V}_{pq}^{*}\sigma_{mmpq}}{ \sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\hat{V}_{mn}\hat{V}_{mn}^{* }}. \tag{28}\]  
From Equation (28), when the modal vibration coefficient and the modal radiation efficiency are obtained, the sound radiation power can be finally determined to illustrate the sound radiation performance caused by the coupling effect between resonators and a sandwich plate.