## 3 Analysis of Results and Mechanisms  
### Very Low-Frequency Wideband Characteristic Dispersion Relation Curve  
In this section, the dispersion relation curves of the AREM and WREM (\(a\) = 5.00 mm, \(b\) = 3.40 mm, \(c\) = 0.04 mm, and \(l\) = 2.00 mm) were obtained by applying the finite element approach. The simulations in this paper were performed using COMSOL Multiphysics 5.6 software. The model material is silicone rubber and the material parameters are [PERSON]'s modulus \(E=1.199\times 10^{5}\) Pa, density \(\rho=1300\) kg/m\({}^{3}\), and [PERSON]'s ratio \(v=0.49998\)[21]. The transverse wave velocity in silicone rubber is \(c_{t}=5.6\) m\(\cdot\)s\({}^{-1}\). Figure 2 illustrates the dispersion relation curve of the AREM, with the imaginary part of the dispersion relation curve (used to describe the wave attenuation performance within the BG) on the left and the real part of the dispersion relation curve on the right. There are five dispersion curves of the AREM in the range below 1300 Hz, and one flexural BG and two longitudinal BGs are generated. The start and cutoff frequencies of the first flexural BG are 115.4 Hz and 1213.3 Hz, respectively, with a center frequency w\({}_{\rm g}\) of 664.4 Hz and a normalized bandwidth of \(\Delta\)w/w\({}_{\rm g}=1.653\) (where \(\Delta\)w is the bandwidth of the BG). The start and cutoff frequencies of the first longitudinal BG are 0 Hz and 70.0 Hz, respectively, and the center frequency is 35.0 Hz. The normalized bandwidth is \(\Delta\)w/w\({}_{\rm g}=2.0\). The start and cutoff frequencies of the second longitudinal BG are 316.6 Hz and 1228.9 Hz, respectively, and the center frequency is 772.8 Hz. The normalized bandwidth is \(\Delta\)w/w\({}_{\rm g}=1.181\). For comparison, the dispersion relation curves for the imaginary and real parts of the WREM were calculated as shown in Figure 3. The WREM has six dispersion curves in the range below 1300 Hz, producing three flexural BGs and two longitudinal BGs. The start and cutoff frequencies of the first flexural BG are 42.7 Hz and 193.8 Hz, respectively,  
Figure 2: The dispersion relation curve of BG characteristics of AREM. The brown scatter represents the imaginary part of the reduced wave vector and the green and blue lines represent the real part of the reduced wave vector, where the green line indicates the flexural vibration mode, and the blue line represents the longitudinal vibration mode. and the center frequency \(w_{g}\) is 118.3 Hz, with the normalized bandwidth \(\Delta w/w_{g}=1.277\). The start and cutoff frequencies of the second flexural BG are 398.6 Hz and 1129.7 Hz, respectively, and the center frequency \(w_{g}\) is 750.6 Hz, with a normalized bandwidth of \(\Delta w/w_{g}=0.957\). The start and cutoff frequencies of the third flexural BG are 1197.3 Hz and 1235.8 Hz, respectively, with a center frequency \(w_{g}\) of 1216.6 Hz and a normalized bandwidth of \(\Delta w/w_{g}=0.032\). The start and cutoff frequencies of the first longitudinal BG are 0 Hz and 71.6 Hz, respectively, and the center frequency \(w_{g}\) is 35.8 Hz. The normalized bandwidth is \(\Delta w/w_{g}=2.0\). The start and cutoff frequencies of the second longitudinal BG are 447.4 Hz and 1101.7 Hz, respectively, and the center frequency \(w_{g}\) is 774.6 Hz. The normalized bandwidth is \(\Delta w/w_{g}=0.845\). By comparing Figures 2 and 3, it is observed that the flexural BG of the AREM is one large forbidden band in the range of 0-1300 Hz, while the flexural BG of the WREM is three forbidden bands, and the total bandwidth of the flexural BG of the AREM is better than that of the WREM. The proposed AREM structure is isometrically scaled up by a factor of 46 in this paper and compared with the existing typical structures. With the same material and similar lattice constants, the AREM structure has an ultra-wide BG. As shown in Table 1, the forbidden band share of the AREM is also better than the other two structures in the range below 5000 Hz.