# Vibro-Acoustic Performance of a Sandwich Plate with Periodically Inserted Resonators
## 2 Formulations
### Band-Gap of, and Sound Radiation from, an Infinite Periodic Structure
#### 2.2.2 Sound Radiation  
In an infinite LR sandwich plate with wave vector \(\mathbf{k}=\left(k_{x},k_{y}\right)\), the displacement functions can be assumed as  
\[\left[ \begin{array}{cccc}u&v&w\end{array} \right]=\left[ \begin{array}{cccc} U&V&W\end{array} \right]e^{-j\mathbf{k}x}e^{-j\mathbf{k}y}, \tag{11}\]where \(k_{x}=k_{p}\cos\varphi\), \(k_{y}=k_{p}\sin\varphi\), and \(k_{p}=\sqrt{k_{x}^{2}+k_{y}^{2}}\) is the structural wave number. The situation of free vibration is considered here to determine the relationship between \(k_{p}\) and frequency. Substitution of Equation (11) into Equations (5) and (6) yields  
\[\left[ \begin{array}{cccc}a_{11}k_{p}^{2}+a_{12}&a_{13}k_{p}^{2}&a_{14}k_{p}& 0\\ a_{21}k_{p}^{2}&a_{22}k_{p}^{2}+a_{23}&a_{24}k_{p}&0\\ a_{31}k_{p}&a_{32}k_{p}&a_{33}k_{p}^{4}+a_{34}k_{p}^{2}+a_{35}&a_{36}\\ 0&0&a_{41}&a_{42}\end{array} \right]\left[ \begin{array}{c}U\\ V\\ W\\ w_{R0}\end{array} \right]=0, \tag{12}\]  
where the coefficients \(a_{ij}\) are expressed as a function of frequency, which are defined in Equations (A28)-(A43) in Appendix A. By calculating the matrix determinant of Equation (12), the dispersion equation can be obtained as  
\[k_{p}^{8}+\varepsilon_{6}k_{p}^{6}+\varepsilon_{4}k_{p}^{4}+\varepsilon_{2}k_ {p}^{2}+\varepsilon_{0}=0, \tag{13}\]  
where \(\varepsilon_{0}\), \(\varepsilon_{2}\), \(\varepsilon_{4}\), and \(\varepsilon_{6}\) can be calculated with Equations (A44)-(A48) defined in Appendix A. As can be seen in Equation (13), each frequency corresponds to four pairs of conjugate wave numbers, with two of them related to \(x\)-wise and y-wise in-plane waves and the other two related to \(z\)-wise flexural waves. The far-field sound radiation is strongly related to flexural waves instead of in-plane waves. Further study shows that, for the flexural waves, only one pair corresponds to the propagating waves and the other corresponds to the evanescent (near-field) waves. The waves considered in the theoretical model are in sinusoidal form distributed in the whole infinite structure as shown in Equation (11) instead of point-excited situation, in order to minimize the effect of near-field effect. For the sinusoidal wave form, the sound pressure generated by evanescent wave attenuates rapidly from near surface to far-away sound field, and the sound pressure only exists near the plate surface. In addition, the sound pressure is out of phase with the plate velocity, resulting that there's little sound power radiated from evanescent wave. For the finite sandwich structure in the next subsection, the present model is restricted to lightly damped situation and light-fluid acoustic medium so that the effect of near-field effect can be reasonably ignored. In order to make the theoretical model not too complex, the effects of evanescent flexural waves are ignored in the following analysis. It is well known that the Helmholtz wave equation in a Cartesian coordinate system is expressed as  
\[\frac{\partial^{2}p}{\partial^{2}x}+\frac{\partial^{2}p}{\partial^{2}y}+\frac {\partial^{2}p}{\partial^{2}z}-k_{ac}^{2}p=0, \tag{14}\]  
where \(k_{ac}=\omega/c\) is the acoustic wave number and \(c\) is sound velocity in the surrounding acoustic medium. The sound pressure solution of Equation (14) has the form of \(p=p_{0}e^{-jk_{x}}e^{-jk_{y}}e^{-jk_{z}}\), substitution of which into Equation (14) yields \(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=k_{ac}^{2}\). Thus, the transverse acoustic wave number can be expressed as \(k_{z}=\pm\sqrt{k_{ac}^{2}-k_{p}^{2}}\). The continuity condition for normal velocity at the interface between the plate surface and acoustic medium states that  
\[\hat{v}=-\frac{1}{j\omega\rho_{0}}\left.\frac{\partial p}{\partial z}\right|_{ z=0}, \tag{15}\]  
where \(\rho_{0}\) is the density of the acoustic medium and \(\hat{v}=-j\omega w\) is the normal velocity of the plate.