## 3 Numerical Examples
### The Advantages of the Proposed MRRM as Compared to the Transfer Matrix Method (TMM)
Within each pass-band, the phase  
Figure 5: Comparison of the band structures obtained by MRRM and TMM for bi-coupled periodic beam consisting of three Binary-IIs as the super cell. (MRRM: Method of reverberation-ray matrix; TMM: Transfer matrix method.) constant spectrum is periodic about wavenumber with the minimum positive period being \(ql/\pi=2\). Within each stop-band, the attenuation constant spectrum is a closed loop and non-periodic about the wavenumber; (3) in the frequency-wavelength spectra, the wavelength associated with a given frequency in pass-band is multi-valued. As we take \(\lambda/l=1\) as the dividing line, the spectra below it become more and more intensive with the decrease of dimensionless wavelength, while those above it are sparse. The frequency-wavelength spectra within the second and higher pass-bands take the bounding frequencies with phase 0 as their asymptotes as the wavelength tends to infinity. This means that near \(ql=0\) all the unit cells are displaced in the same direction without phase delay, thus the periodic structure displaces as a whole; (4) in the frequency-phase velocity spectra, the phase velocity associated with a given frequency in pass-band are also multi-valued. There is not any cut-off value for phase velocity in the second and higher order pass-bands. The spectra become more and more intensive as the phase velocity decreases. By contrast with the frequency-wavelength spectra, the frequency-velocity spectra do not have clear division between the intensive and sparse regions. The frequency-velocity spectra also take the bounding frequencies with phase 0 as their asymptotes as the phase velocity tends to infinity; (5) the phase velocity-related dispersion curves show the dispersion properties of the frequency-related dispersion curves in different viewpoints. As observed from the phase-velocity related dispersion curves, the spectrum in the first pass-band has cut-off phase velocity, but does not have cut-off wavelength, as noted in Figure 6c. The special features of bending-wave dispersion curves in bi-coupled periodic multi-component beams, as compared to the longitudinal-wave dispersion-curves in a mono-coupled periodic multi-component rod, include: (A) the frequency-related dispersion curves in some pass-bands may not be entirely constrained between the bounding frequencies corresponding to \(ql=0\) and \(ql=\pi\), which means that the frequencies corresponding to the Brillouin zone boundary may not be the demarcation between pass-band and stop-band. This should be the most innovative dispersion feature for bending waves in bi-coupled periodic multi-component beams, which never exist in mono-coupled periodic structures; (B) the pass-bands and stop-bands no longer appear alternately due to the existence of two kinds of characteristic bending waves. There may be frequency regions that are pass-bands/stop-bands for both characteristic modes, because in these regions the phase/attenuation constant spectra associated with different characteristic modes may overlap. There are regions that have neither phase constant spectra nor attenuation constant spectra, in which the wavenumbers are neither purely real nor purely imaginary, but are complex; (C) since the attenuation spectra in some stop-bands are concave, more than one forms of attenuations at a given frequency occur in this case; (D) the cut-off phase velocity should be obtained by setting \(ql=\pi\), which is different from the determination of the cut-off phase velocity in the mono-coupled periodic multi-component rods by setting \(ql=0\) (\(\omega=0\)). Figure 6: Frequency-related dispersion curves of bending waves in the bi-coupled periodic beam with Ternary-I unit cell: (**a**) the attenuation constant spectra; (**b**) the phase constant spectra; (**c**) the wavelength spectra; (**d**) the phase velocity spectra.