## 1 Introduction
The works on bi-coupled periodic multi-component beams so far mainly have concerned bi-coupled periodic beams with the unit cell containing only two beam segments, particularly in the numerical examples, although most literature has presented the analysis on a unit cell model consisting of an arbitrary number of components. To the best of the authors' knowledge, only [PERSON] et al. [70] studied the complex band structures, the localization factors, and the characteristic modes of bending waves in periodic fourfold (quaternary) beams by TMM based on EBT. Nevertheless, as revealed by [PERSON] and [PERSON] [71], the types/properties of periodicity have clear influence on bending wave characteristics. Hence, deducing the properties of bending waves in bi-coupled periodic multi-component beams from those of the adequate studies on similar structures may not be feasible, unless the similarities and differences between them have been obtained. 2. Although many methods have been proposed or used in analyzing the bending waves in bi-coupled periodic multi-component beams, such as PWE, TMM, SEM, DQM among the analytical ones and FEM among the numerical ones, effectively analyzing the bending waves at a relatively high frequency range is still challenging due to the essential deficiency in some of the existing methods. For example, the PWE method and the FEM are too computationally inefficient (expensive) for providing satisfactory results at a high frequency range. The TMM suffers from numerical weakness in cases of high frequency and a long transfer path [72]. 3. The dispersion of bending waves in bi-coupled infinite periodic beams has been monotonously represented by the band structures, i.e., the frequency-wavenumber dispersion curves, in all previous research. There is very little knowledge on the properties of the wavelength and phase velocity of the characteristic bending waves in bi-coupled periodic beams. In particular, the general dispersion characteristics of bending waves in bi-coupled periodic multi-component beams have not been revealed so far. Considering the pending problems in the aforementioned three aspects, this paper takes bi-coupled periodic multi-component beams with unit cells composed of arbitrary beam segments of different materials and cross-sections. Various periodic binary, ternary and quaternary beams are analyzed in the numerical examples, with special emphasis on the effect of unit-cell configuration on the dispersion curves and properties. The method of reverberation-ray matrix (MRRM) [73; 74; 75] is introduced as an alternative method for analyzing bending wave propagation in bi-coupled periodic multi-component beams. Note that MRRM compromises somewhat the computational efficiency as compared to the very efficient transfer matrix method (TMM). This is because the global scattering matrix in MRRM is formed by grouping all the element-related scattering matrices as compared to the global transfer matrix in TMM which is formed by the multiplication of all element transfer matrices, and thus the dimension of the system equation in MRRM is four times the element number as compared to the dimension of the system equation in TMM always being four regardless of the element number. However, by avoiding the matrix multiplications in the MRRM formulation, a numerical well-condition is achieved, as will be validated in Section 3.2. The comprehensive dispersion curves have been presented to illustrate the general dispersion characteristics of bending waves in bi-coupled periodic multi-component beams; in particular, the wavelength-related and phase velocity related dispersion curves are newly introduced to represent the dispersion of waves from all viewpoints. The influences of various construction factors in the unit cell, especially the beam component number, on the band structures are studied.