# Vibro-Acoustic Performance of a Sandwich Plate with Periodically Inserted Resonators
## 3 Results and Discussion
### Acoustic Radiation of a Finite LR Sandwich Plate  
As was studied in Section 3.2, the introduction of periodic resonators had significant effects on the vibration response, especially in the band-gap, where vibration response is significantly reduced. This variation in vibration will in turn directly affect the acoustic radiation, owing to the coupling between vibration and acoustic radiation. Thus, it can be inferred that the sound radiation in the band-gap may also be reduced as a result of the great reduction in vibration. However, the sound radiation performance beyond the band-gap cannot be estimated simply from vibration performance, as the sound radiation is also greatly dependent on radiation efficiency, which quantifies how much vibration energy can be transformed to sound radiation energy. Thus, this sub-section examines the radiation efficiency and the final sound radiation power in order to further understand the vibro-acoustic performance of an LR sandwich plate. The structural model is the same as that used in Section 3.2. After the vibration response is determined, the sound radiation power from this finite LR sandwich plate is then determined using a Rayleigh integral in the far field. The corresponding average radiation efficiency is determined from the modal radiation efficiency \(\sigma_{mmpq}\) and modal velocity response \(\hat{V}_{mn}\) using Equation (28). As the radiation surface, structural boundary, and structural size of the homogeneous sandwich plate are the same as those of the LR sandwich plate, the modal radiation efficiencies of both sandwich structures are identical. Therefore, the variation in average radiation efficiency of both sandwich structures is, in effect, due to the variation in the modal velocity response, as shown in Figure 5.  
Shown in Figure 10 is a comparison of the radiation efficiencies of homogeneous (\(\sigma_{\rm HM}\)) and LR (\(\sigma_{\rm LR}\)) sandwich plates. It is noticed that the effects of the periodic resonators on radiation efficiency depend  
Figure 9: Normalized effective mass and structural wave number of a locally resonant sandwich plate. on frequency. In the figure, the whole frequency zone is divided into four parts: zone I (0 Hz\(-f_{11}\)), zone II (\(f_{11}-f_{\rm LR}\)), zone III (\(f_{\rm LR}-f_{\rm LLR}\)), and zone IV (\(f_{\rm LLR}-6000\) Hz), where \(f_{11}=132\) Hz, \(f_{\rm LR}=1000\) Hz, and \(f_{\rm LLR}=3378\) Hz are the fundamental frequency, the resonance frequency of the resonator, and the coincidence frequency of the LR sandwich plate, respectively. Below the fundamental frequency (zone I) and above the coincidence frequency (zone IV), the periodic resonators have little small effect on radiation efficiency and \(\sigma_{\rm LR}\) and \(\sigma_{\rm HM}\) are nearly the same, with \(\sigma_{\rm LR}\) slightly higher than \(\sigma_{\rm HM}\) in zone I and slightly lower in zone IV. Meanwhile, in zone II and zone III, \(\sigma_{\rm LR}\) and \(\sigma_{\rm HM}\) vary significantly and the periodic resonators have a negative effect on radiation efficiency in zone II and a positive effect in zone III. Thus, compared with the homogeneous sandwich plate, the radiation performance of the LR sandwich plate is improved in zone II yet deteriorated in zone III. It is also noticed that, at frequencies close above the band-gap in zone III, \(\sigma_{\rm LR}\) reaches a local maximum of 1.99 at 1158 Hz, which is more than five times that of \(\sigma_{\rm HM}\). In the frequency band close to 1158 Hz, the LR sandwich plate radiates very efficiently. In fact, this frequency region is simply the resonator-induced effective radiation zone in an infinite LR sandwich plate, which can be found by comparing the radiation efficiency of finite and infinite LR sandwich plates, as shown in Figure 11, where the two shaded regions are the effective radiation zones in an infinite LR sandwich plate.