## 4 Conclusions

Table 3: The bounding frequencies of the first 10 pass-bands of bending waves in the five exemplified bi-coupled periodic multi-component beams. waves in bi-coupled periodic multi-component beams no longer show alternately distributed pass-bands and stop-bands due to the existence of two characteristic wave modes. Pass-bands and stop-bands for both characteristic wave modes may arise due to the overlaps of the dispersion curves of different wave modes in the same regions. The wavenumber of the characteristic waves may be complex in some frequency ranges other than purely real or purely imaginary in other ranges. In some stop-bands, the attenuation constants may be multi-valued. The most innovative finding is that the bounding frequencies corresponding to the Brillouin zone boundary are not always the demarcations between the pass-band and stop-band. The cut-off phase velocity should be obtained by setting \(ql=\pi\) rather than \(ql=0\) (or \(\omega=0\)). 3. The cross-sectional area and the cross-sectional inertia moment as involved in the geometrical parameters, the modulus, the Poisson's ratio and the density as involved on the material parameters, as well as the unit-cell configuration, all affect both the central frequencies and the widths of the frequency bands. In particular, the increasing (decreasing) of the cross-sectional area and the density have similar effects on the variation of band structures with the decreasing (increasing) of the cross-sectional inertia moment, the modulus and the Poisson's ratio, which all push the band structures toward the lower frequency side and vice versa. The unit cell configuration does have an effect on the band structures and properties, but there seems not to be any general influence rule. All these findings can guide the adjustment of bending wave propagation/attenuation characteristics. In summary, we believe the presented MRRM and the deep understanding of bending-wave dispersion curves in this paper will push forward the quantitative design and optimization of bi-coupled periodic multi-component beams for wave filtering and vibration isolation applications. The authors appreciate the financial support for this work by the National Natural Science Foundation of China (No. 11372119). The authors also wish to thank the anonymous reviewers for their constructive comments. [PERSON] conceived the main ideas, derived the theoretical formulation and wrote the computer code. [PERSON] and [PERSON] conducted the numerical examples. All authors contributed in writing the manuscript. The authors declare no conflict of interest. The funding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; and in the decision to publish the results. The following abbreviations are used in this manuscript:  
MRRM Method of reverberation-ray matrix  
MTM Modified transfer matrix method  
TMM Transfer matrix method  
FEM Finite element method  
WFEM Wave finite element method  
RM Receptance method  
SEM Spectral element method  
WSEM Wave spectral element method  
LMM Lumped mass method  
PWE Plane wave expansion  
DQM Differential quadrature method  
EDQEM Extended differential quadrature element method  
TBT Timoshenko beam theory  
EBT Euler-Bernoulli beam theory  
DOF Degree of freedom  
FRF Frequency response function  
EMAT Electromagnetic acoustic transducer