# Vibro-Acoustic Performance of a Sandwich Plate with Periodically Inserted Resonators
## 3 Results and Discussion
### Radiation Efficiency of an Infinite LR Sandwich Plate
As \(k_{xr}\) is calculated from \(k_{ac}\) and \(k_{P}\), it is also significantly affected by the resonators near the band-gap frequency region  
Figure 8: Wave numbers and radiation efficiencies of **(a)** an infinite homogeneous sandwich plate (HM-SP), **(b)** an infinite locally resonant sandwich plate (LR-SP), and **(c)** an infinite homogeneous sandwich plate with the LR mass added to the core layer (HM-SP2). **(d)** The comparison of structural wave numbers among the above three plates. (868-1032 Hz). It is noticed in Figure 8b that the coincidence frequency becomes \(f_{\rm{CLR}}=3378\) Hz, which is 129 Hz lower than that of the structure of HM-SP. Furthermore, because of the variation of \(k_{P}\), \(k_{zr}\) becomes positive from 1018 to 1346 Hz, which corresponds to effective sound radiation. Thus, affected by \(k_{zr}\), two effective radiation regions are formed, namely 1018-1346 Hz and 3378 Hz-\(\infty\). The first one is the resonator-induced effective radiation region, which is located above the band-gap (868-1032 Hz) with a small zone overlapping (1018-1032 Hz). Thus, for waves with frequencies close above the band-gap, acoustic radiation may be enhanced due to the effective radiation region. As the radiation efficiency is strongly related to the structural wave number \(k_{P}\), the structural wave numbers of HM-SP, LR-SP, and HM-SP2 are compared to further examine the radiation properties in Figure 8d. As shown in the figure, in the lower frequencies between 0 Hz to approximately 600 Hz, the structural wave number \(k_{P}\) of LR-SP nearly coincidence with that of HM-SP2. It is noticed that these above two structures have the same weight, thus the increased weight introduced by periodic resonators have some effect on the radiation. In contrast to lower frequencies, in the higher frequencies between approximately 2.5 to 6 kHz, \(k_{P}\) of LR-SP nearly coincides with that of HM-SP, inducing that the weight introduced by periodic resonators have little effect on the radiation properties in high frequencies. It is noticed from the comparison between Figure 8a,b that the effects of periodic resonators on the sound radiation of an infinite sandwich plate mainly contain two terms. The first is the decrease in coincidence frequency from 3507 to 3378 Hz, and the second is the newly generated effective radiation region. Thus, the total effective radiation region is actually enlarged, and the ineffective radiation region in a homogeneous sandwich plate may be changed to an effective radiation region in an LR sandwich plate. Although the vibration in the band-gap is reduced, sound radiation in other frequencies may be enhanced. Therefore, care should be taken when designing a band-gap for noise-reduction purposes as the band-gap design may in fact enlarge the effective radiation region. In fact, the variation of acoustic radiation can be physically explained by the normalized effective mass \(m_{e}\) of an LR sandwich plate, which is expressed as  
\[m_{e}=1+\frac{m_{R}k_{R}}{m_{b}(k_{R}-\omega^{2}m_{R})}, \tag{29}\]  
where \(m_{b}\) is the mass of a unit element of the base sandwich plate. As shown in Figure 8, radiation efficiency is strongly dependent on the structural wave number \(k_{P}\), thus \(m_{e}\) and \(k_{P}\) are shown together in Figure 9 for further analysis. It is observed that, with the increase in frequency, \(m_{e}\) increases from 1.25 to infinity over 0-1000 Hz, then increases from minus infinity to approximately unity (but always less than unity) between 1000 and 6000 Hz. In the lower frequencies, the limit of \(m_{e}\) is 1.25, which is exactly the effective mass of the structure of HM-SP2. In the higher frequencies, \(m_{e}\) tends to unity, which is exactly the effective mass of the structure of HM-SP. This phenomenon is the same as that observed in Figure 8b. It can also be observed that the effective mass is disconnected at 1000 Hz (the resonance frequency of the resonator).