# Vibro-Acoustic Performance of a Sandwich Plate with Periodically Inserted Resonators
## 3 Results and Discussion
### Radiation Efficiency of an Infinite LR Sandwich Plate  
It was found from the above analysis that periodic resonators can improve the vibration performance of an LR sandwich plate in the band-gap frequency region. As the sound radiation ultimately depends on vibration, the radiation performance will also be affected significantly. In this subsection, the radiation efficiency of an infinite LR sandwich plate is examined here to reveal the effects of periodic resonators on radiation characteristics. Shown in Figure 8a-c are the structural wave number \(k_{P}\), acoustic wave number \(k_{ac}\), transverse acoustic wave number \(k_{zr}=\text{real}\left(\sqrt{k_{ac}^{2}-k_{p}^{2}}\right)\), and radiation efficiency \(\sigma_{\text{inf}}\) of an infinite homogeneous sandwich plate (HM-SP) in Figure 8a, an infinite LR sandwich plate (LR-SP) with the resonator tuned at \(f_{R}=1000\,\text{Hz}\) in Figure 8b, and an infinite homogeneous sandwich plate with the LR mass added to the core layer (HM-SP2) in Figure 8c. The difference between HM-SP and HM-SP2 is that the mass of resonator is added to the core layer in HM-SP2 in order to examine the effects of increase of weight due to periodic resonators on radiation properties, thus the weight of the structure of HM-SP2 keeps the same as that of the structure of LR-SP. It is well known that, in an infinite structure, sound radiates efficiently only in situations where the phase speed of a structural wave exceeds the speed of sound in  
Figure 6: Comparison of modal contributions for a homogeneous sandwich plate (HM-SP) and a locally resonant sandwich plate (LR-SP) for (**a**) mode (2, 3), (**b**) mode (5, 4), and (**c**) mode (5, 6). (Flex-mode—flexural vibration mode; LR-mode—resonator vibration mode.). Figure 7: Comparison of modal frequencies for a homogeneous sandwich plate (HM-SP) and a locally resonant sandwich plate (LR-SP). the surrounding medium (\(c_{P}>c_{ac}\), \(k_{P}<k_{ac}\)). Thus, \(k_{xr}\) can be considered as an indicator of whether a wave with a specific frequency radiates efficiently or not, that is, a wave radiates efficiently when \(k_{xr}\) exceeds zero and inefficiently when \(k_{xr}\) equals zero. For a homogeneous structure, the frequency that satisfies \(k_{P}=k_{ac}\) is generally called the coincidence frequency, where vibration energy may transform to sound radiation power most efficiently and the corresponding radiation efficiency tends to infinity. As shown in Figure 8a, the coincidence frequency of the HM-SP is \(f_{\rm{CHM}}=3507\,\rm{Hz}\). With the increase in frequency, the radiation efficiency \(\sigma_{\rm{inf}}\) stays constant at zero below \(f_{\rm{CHM}}\), then reaches infinity at \(f_{\rm{CHM}}\), and finally decreases from infinity to unity above \(f_{\rm{CHM}}\). Thus, the effective radiation region begins at 3507 Hz, where vibration energy can be radiated into the acoustic medium efficiently. In Figure 8c for the structure of HM-SP2, the variation trends of wave numbers and radiation efficiency keep the same as that in HM-SP, except that the coincidence frequency changes from 3507 Hz to 4666 Hz because of the increased weight in the core layer. Thus, the effective radiation region for HM-SP2 begins at 4664 Hz. However, the case of the LR sandwich plate differs from that of the homogeneous sandwich plate. As is shown in Figure 8b, the acoustic wave number \(k_{ac}\) is the same as in the homogeneous sandwich plate, while the structural wave number \(k_{P}\) varies significantly near the band-gap region (compare Figure 8a,b)). With the increase in frequency, \(k_{P}\) first increases gradually from zero to infinity over 0-1000 Hz, and then decreases sharply from infinity to zero over 1001-1026 Hz, then remains unchanged at zero over 1027-1097 Hz, and finally increases from 1098 Hz to infinity.