## 3 Numerical Examples
### The Advantages of the Proposed MRRM as Compared to the Transfer Matrix Method (TMM)  
Consider the bi-coupled periodic beams with one Binary-II as the unit cell and three Binary-IIs as the super cell. The cross-sectional area and the cross-sectional inertia moment of all constituent beams are \(A=2\times 10^{-4}\) m\({}^{2}\) and \(1/6\times 10^{-8}\) m\({}^{4}\) in the respective unit/super cell of the above two exemplified periodic beams. The phase and attenuation constant spectra of these two periodic beams are computed by the proposed MRRM and the well-known TMM. These band structures of the periodic beam with one Binary-II as the unit cell are given and compared in Figure 3 within 0-700 Hz, while those of the periodic beam with three Binary-IIs as the super cell are provided and contrasted in Figure 4 within 0-100 Hz. Figure 4: Comparison of the band structures obtained by the method of reverberation-ray matrix (MRRM) and transfer matrix method (TMM) for a bi-coupled periodic beam consisting of one Binary-II as the unit cell. Figure 3: The first several phase constant spectra and the boundary frequencies of band gaps of characteristic bending waves in the bi-coupled periodic beam with Binary-I unit cell. EBT: Euler–Bernoulli beam theory; TBI: Timoshenko beam theory; MRRM: Method of reverberation-ray matrix; TMM: Transfer matrix method; DQM: Differential quadrature method; BG: Band gap. Figures 4 and 5 indicates that the classical TMM formulation is ill-conditioned, as the frequency is above 380 Hz and 40 Hz for the exemplified periodic beam with one Binary-II as the unit cell and the periodic beam with three Binary-IIs as the super cell under consideration, respectively. The comparison between Figures 4 and 5 demonstrate that the increasing of transfer path will reduce the critical frequency above which the TMM will be ill-conditioned. These findings illustrate the possible ill-condition problem of the TMM in cases of relatively high frequency and long transfer path. However, our proposed MRRM formulation is always well-conditioned for both exemplified periodic beams within the respectively considered frequency ranges. This serves as the main advantages of the MRRM as compared to TMM. The General Dispersion Characteristics of Bending Waves in Bi-Coupled Periodic Multi-Component Beams  
Among all kinds of dispersion curves, the corresponding ones of bending waves in the five exemplified bi-coupled periodic multi-component beams illustrate qualitatively identical features although they show quantitatively different shapes and positions. As a representative, the first several frequency-related dispersion curves and the first several phase velocity-related dispersion curves of the bending waves in a bi-coupled periodic beam with Ternary-I unit cell are exhibited respectively in Figures 6 and 7. Figure 8 depicts those frequency-related dispersion curves within the frequency range 500-505 kHz of bending waves in the same bi-coupled periodic ternary beam, which again indicates that the present MRRM formulation is well-conditioned within the considered high frequency range. The comparison of the corresponding subfigures in Figures 6 and 8 verifies that the frequency related dispersion curves illustrate similar features at low and relatively high frequency ranges. In addition, when the bending-wave dispersion curves in Figures 6-8 are compared to the counterpart longitudinal-wave dispersion curves in a periodic multi-component rod [75], a typical mono-coupled periodic structure, features that are alike and unlike may be sought out. The features that are alike include: (1) among the diversified dispersion curves representing the propagation characteristics, only those frequency-related dispersion curves reflect the band property; (2) in the frequency-wavenumber spectra, the phase and the attenuation constant spectra are both symmetrical about \(ql=0\), since \(+ql\) and \(-ql\) denoting a pair of characteristic waves in opposite directions should have the same propagation property.