## 3 Examples and Discussion
### Acoustic Radiation Property of the Finite LR Plate  
As derived in Sections 2.1 and 2.2, the theory in this paper can calculate the radiation efficiencies of the finite LR plate with various boundary conditions. Figure 10a-c gives the radiation efficiencies of the FFFF, SSSS, and CCCC plates, including the results of the ML plate, the LR plates with undamped resonators, and the LR plate with damped resonators of \(\eta_{R1,2}~{}=~{}0.05\). It is not difficult to observe that more peaks and dips appear in the curves of the RMSV and the radiation efficiency of the FFFF plates in 0-500 Hz, than in those in the case of greater constraints, such as the SSSS and CCCC boundary conditions simulated in this paper. Moreover, the modal frequencies shift to higher frequencies when more constraints are applied to the plate boundaries; for instance, the 1 st modal frequencies of the LR plates with FFFF, SSSS, and CCCC boundary conditions are respectively 22 Hz, 36 Hz, and 69 Hz in this paper. As for the radiation efficiency, it can be easily seen from Figure 10a-c that, in contrast to the vibration attenuation in the two band-gap frequency ranges, the radiation efficiency has been obviously increased in both bands compared with the results of the ML plates. When the resonators are damped,  
Figure 9: The root-mean-square velocities (RMSVs) averaged over the whole surface of the mass-loaded (ML) plate and the locally resonant (LR) plate with the resonator damping loss factors of (**a**) \(\eta_{R1}\) varied by 0, 0.05, 0.1, and \(\eta_{R2}\) kept as 0, (**b**) \(\eta_{R1}\) kept as 0 and \(\eta_{R2}\) varied by 0, 0.05, and 0.1, and (**c**) \(\eta_{R1}\) and \(\eta_{R2}\) simultaneously varied by 0, 0.05, 0.1, and 0.2.  
the curves become smoother than those in the undamped case from below the lower frequency of the 1 st band gap, while the acoustic radiation is still high in the band-gap frequency ranges in each case. It is known that the acoustic radiation from the infinite bare plate is effective only above the critical frequency [31] where the acoustic wave number exceeds the structural wave number. For its finite counterpart, though its acoustic power can effectively radiate in the whole frequency range, the radiation efficiency reaches the highest at the critical frequency and tends to unity above the critical frequency. Based on this relationship, the structural and acoustic wave numbers of the infinite LR plate are then calculated. In an infinite LR plate with the wave vector \(\mathbf{k}~{}=~{}\left(k_{x},k_{y}\right)\), the displacement function can be assumed as:  
\[w(x,y)~{}=~{}We^{-j_{x}x}e^{-j_{y}y} \tag{42}\]  
The substitution of Equation (42) into the governing equation of the infinite LR plate:  
\[\left(D\
abla^{4}-\omega^{2}\rho_{p}h_{p}\right)w(\mathbf{r})~{}=~{}\sum_{i~{}=~{} 1}^{2}\sum_{s~{}=~{}1}^{S}\sum_{t~{}=~{}1}^{T}F_{ist}\delta(x-x_{is})\delta(y- y_{t}), \tag{43}\]  
Yields:  
\[\left[ \begin{array}{cc}Dk_{p}^{4}-\omega^{2}\rho_{p}h_{p}&-\frac{\omega^{2 }m_{R1}}{2t_{L}^{2}}&-\frac{\omega^{2}m_{R2}}{2t_{L}^{2}}e^{j_{x}a_{L}}\\ -k_{R1}&k_{R1}-\omega^{2}m_{R1}&0\\ -k_{R2}e^{-j_{x}a_{L}}&0&k_{R2}-\omega^{2}m_{R2}\end{array} \right]~{}=~{}\mathbf{0}, \tag{44}\]  
Figure 10: The radiation efficiencies of the mass-loaded (ML) plate, the locally resonant (LR) plate with undamped resonators (\(\eta_{R1,2}~{}=~{}0\)), and the LR plate with damped resonators (\(\eta_{R1,2}~{}=~{}0.05\) ), with (**a**) fully free boundary conditions, (**b**) fully simply supported boundary conditions, and (**c**) fully clamped boundary conditions. where \(k_{P}~{}=~{}\sqrt{k_{x}^{2}+k_{y}^{2}}\) is the structural wave number. To obtain the non-zero displacement responses of the base plate and the resonators, the matrix determinant of Equation (44) should equal zero, and then the structural and acoustic wave numbers can be obtained. The structural and acoustic wave numbers of the infinite LR plate are compared in 0-500 Hz and 3-6 kHz, where the effective radiation frequency ranges are shaded, as shown in Figure 11a.