## 4 Conclusions  
The method of reverberation-ray matrix (MRRM) is formulated based on analytical beam theory and matrix operation for analyzing the dispersion characteristics of bending waves (coupled flexural and thickness-shear waves) in bi-coupled periodic multi-component beams. It entails a numerically well-condition so as to be an accurate and uniform method, especially applicable for relatively high frequency analysis, where the wavelength is still larger than the size of beam cross-section and where other methods often cease to be effective. Through plentiful numerical examples on the dispersion curves and band properties of bending waves in exemplified periodic multi-component beams, the proposed MRRM formulation has been validated and its advantages as compared to the classical transfer matrix method have been illustrated. Special attention is paid to the comprehensive dispersion characteristics represented as the relationship between any two among three in frequency, wavenumber (or wavelength) and phase velocity. The dependence of band structures (properties) on the geometrical and material parameters of the constituent beams and the unit-cell configuration are studied as well. The investigation in this paper leads to the following conclusions:  
1. MRRM is an analytical method based on a distributed-parameter model and yields a unified formulation possessing a numerically well condition. It provides an alternative accurate method for analyzing bending waves in bi-coupled periodic multi-component beams. The method is effective as long as the adopted beam theory, such as the Timoshenko or Euler-Bernoulli beam theory, is valid for modeling the structural members. The advantages of MRRM as compared to TMM should be the consistently well-condition of MRRM, even in cases of relatively high frequency and long unit cell where TMM is ill-conditioned and may fail to give reasonable results. 2. Being different from the dispersion curves of longitudinal waves in periodic multi-component rods, essentially a kind of mono-coupled periodic structure, the dispersion curves of the bending \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Pass-Bands** & \multicolumn{2}{c}{**Periodic Binary Beams**} & \multicolumn{2}{c}{**Periodic Ternary Beams**} & \multicolumn{2}{c}{**Periodic Quaternary Beam**} \\ \hline
**Order** & **Bounding Frequencies (Hz)** & **Binary-I** & **Binary-II** & **Ternary-I** & **Ternary-II** & **Quaternary** \\ \hline \multirow{2}{*}{First} & Lower & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\  & Upper & 162.13 & 102.88 & 165.24 & 83.63 & 88.54 \\ \hline \multirow{2}{*}{Second} & Lower & 171.35 & 126.66 & 196.46 & 107.72 & 141.59 \\  & Upper & 520.61 & 353.47 & 589.39 & 296.93 & 340.82 \\ \hline \multirow{2}{*}{Third} & Lower & 731.37 & 484.77 & 690.65 & 472.36 & 497.86 \\  & Upper & 1052.51 & 686.77 & 1120.80 & 617.87 & 673.13 \\ \hline \multirow{2}{*}{Fourth} & Lower & 1338.31 & 893.94 & 1253.65 & 885.74 & 937.24 \\  & Upper & 1684.79 & 1040.70 & 1650.56 & 984.39 & 1096.64 \\ \hline \multirow{2}{*}{Fifth} & Lower & 1796.83 & 1348.76 & 1822.11 & 1335.82 & 1238.31 \\  & Upper & 2075.91 & 1500.30 & 2105.90 & 1443.82 & 1349.03 \\ \hline \multirow{2}{*}{Sixth} & Lower & 2479.84 & 1515.88 & 2437.78 & 1525.62 & 1909.45 \\  & Upper & 2506.23 & 1535.84 & 2475.43 & 1541.03 & 1910.74 \\ \hline \multirow{2}{*}{Seventh} & Lower & 2628.36 & 1713.27 & 2574.76 & 1648.44 & 2063.38 \\  & Upper & 2687.30 & 1863.85 & 2745.19 & 1735.87 & 2139.10 \\ \hline \multirow{2}{*}{Eighth} & Lower & 2699.99 & 2038.70 & 3018.43 & 2038.54 & 2422.14 \\  & Upper & 3005.43 & 2203.39 & 3105.16 & 2110.07 & 2498.37 \\ \hline \multirow{2}{*}{Ninth} & Lower & 3468.91 & 2203.39 & 3221.10 & 2234.20 & 2968.38 \\  & Upper & 3542.20 & 2268.12 & 3224.44 & 2262.24 & 3010.78 \\ \hline \multirow{2}{*}{Tenth} & Lower & 3622.93 & 2540.88 & 3473.08 & 2545.36 & 3068.38 \\  & Upper & 3830.39 & 2677.25 & 3584.17 & 2597.76 & 3077.64 \\ \hline \hline \end{tabular}
\end{table}