## 3 Examples and Discussion
### Acoustic Radiation Property of the Finite LR Plate
It can be seen that, besides the range where the infinite LR plate radiates sound effectively above the critical frequency of about 4 kHz, the acoustic wave number also exceeds the structural wave number in the two additional bands of 201-215 Hz and 300-320 Hz. The two bands coincide well with the band gaps, which means that the acoustic radiation from the infinite LR plate is effective in the band-gap frequency ranges. Therefore, the acoustic radiation from the finite LR plate of identical resonator parameters can reach a high effectiveness in the band-gap frequency ranges, as expressly shown in Figure 11b. It can be seen that the radiation efficiencies of the finite FFFF, SSSS, and CCCC LR plates are sharply increased in two band-gap frequency ranges. Furthermore, by comparing the results of three different LR plates, the results behave almost the same and closer to unity in the band-gap frequency ranges. However, at most frequencies in 0-500 Hz, the greater the boundary conditions are, the higher the radiation efficiency of the LR plate is in this case. Figure 12a-c shows the acoustic power radiated from the finite ML and LR plates with FFFF, SSSS, and CCCC boundary conditions. It can be observed from the comparison between Figures 12a and 12b (or Figure 12c) that the acoustic behavior behaves quite different when no constraints are applied to the plate. It has been explained in reference [29] that the acoustic power in the low-frequency range radiated from the FFFF bare plate is mainly dominated by three rigid body modes. The rigid body motion and rotation of the FFFF plate make the first few flexural modes inefficiently radiating ones, and then the acoustic power at these modal frequencies is not highlighted, unlike the results of the finite plates with higher constraints. As shown in Figure 12a, the acoustic power radiated from the ML plate is almost constant below 200 Hz in this case. For the LR plate, the radiated power is approximate to that from the ML plate in 0-100 Hz and is decreased below the lower frequency of the 1 st band gap, while it turns out to be increased at a certain frequency within the band gap. A similar phenomenon is found around the 2 nd band-gap frequency range. Figure 11: (**a**) The structural and acoustic wave numbers of the infinite locally resonant (LR) plate, and (**b**) the radiation efficiencies of the finite LR plates with undamped resonators (\(\eta_{R1,2}~{}=~{}0\)), with fully free (FFFF) boundary conditions, fully simply supported (SSSS) boundary conditions, and fully clamped (CCCC) boundary conditions, together with the wave numbers for comparison. For the LR plates with greater constraints, such as the SSSS and CCCC boundary conditions considered in this paper, it is expected that the acoustic power is decreased in the band-gap frequency ranges, as shown in Figure 12b,c. It can be concluded that, although the radiation efficiency of the LR plate with greater constraints, is increased in the band-gap frequency ranges, due to the good performance of the vibration attenuation the acoustic power is reduced much in the same bands. Moreover, when considering the damping characteristic of the resonators, the acoustic power curve becomes smooth from below the lower frequency of the 1 st band gap. Although the degree of acoustic power reduction in the band-gap frequency ranges is weakened, the attenuation band of acoustic radiation is broadened. It can be seen that the application of the periodic resonators on the FFFF base plate cannot be a good noise-control method in this case, especially in and above the band gap frequency ranges. In fact, it can be understood from the vibration profile shown in Figure 13a that the vigorously vibrating FFFF ML plate at the two frequencies may not radiate much sound power due to radiation cancellation from the uniform distribution of anti-phase regions.