## 1 Introduction
### Spectral System Equation and Dispersion Relation  
The global scattering relation (12) gives altogether 4 \(m\) equations for the 4 \(m\) unknown arriving wave amplitudes and the 4 \(m\) unknown departing wave amplitudes in vectors \(\mathbf{a}\) and \(\mathbf{d}\), respectively. Obviously, another 4 \(m\) equations should be supplemented for solving the wave problem. Fortunately, the global phase relation \(\mathbf{a}=\mathbf{P}(\omega)\widetilde{\mathbf{d}}=\mathbf{P}(\omega)\mathbf{ U}\mathbf{d}\) exactly provides another 4 \(m\) equations for the global arriving and departing wave vectors \(\mathbf{a}\) and \(\mathbf{d}\), whose derivation is detailed in Appendix A. Substitution of this global phase relation into the global scattering relation (12) gives the spectral system equation:  
\[[\mathbf{A}(\omega,q)\mathbf{P}(\omega)\mathbf{U}+\mathbf{D}(\omega,q)] \mathbf{d}=\mathbf{R}_{d}(\omega,q)\mathbf{d}=\mathbf{0}, \tag{15}\]  
where \(\mathbf{R}_{d}=\mathbf{A}\mathbf{P}\mathbf{U}+\mathbf{D}\) is the 4 \(m\ \times\ 4\ m\) system matrix. Equation (15) in fact poses a non-linear and implicit eigenproblem with the eigenvector \(\mathbf{d}\) and the eigenvalue \(\mathbf{e}^{\mathrm{i}q\mathbf{I}}\) as the frequency \(\omega\) is specified or the eigenvalue \(\omega\) in case of given wavenumber \(q\). For an existing non-trivial solution to the system Equation (15), the determinant of the system matrix should vanish to give  
\[\det[\mathbf{R}_{d}(\omega,q)]=\det[\mathbf{A}(\omega,q)\mathbf{P}(\omega) \mathbf{U}+\mathbf{D}(\omega,q)]=\mathbf{0}, \tag{16}\]  
which is a transcendental dispersion equation determining the dispersion characteristics of bending waves in the infinite bi-coupled periodic beams. In practice, except for the closed-form transcendental dispersion equation provided in Reference [74] for simply configured bi-coupled periodic beams with a binary unit cell having only two constituent beams, the concrete dispersion equation is difficult to write out in closed-form for the general bi-coupled periodic multi-component beams. This is because complicated operations of matrix multiplication and matrix determinant are involved in deriving the closed-form of Equation (16). Consequently, the dispersion equation is usually solved numerically. If any one among three in frequency \(\omega\), wavenumber \(q\) (or wavelength \(\lambda=2\pi/q\)) and phase velocity \(c=\omega/q\) is specified within a prescribed range, the other two can be determined by properly searching the root of Equation (16). The solving process and the adopted techniques are summarized as follows:  
1. Initially preset a beginning value and the increment to the known quantity. In later circulation computations, add an increment to the current value of the known quantity to give it an updated value. 2. Specify two tentative values to the requested quantity. The first tentative value should be specified at the initial computation and obtained from step (6) during the later calculations. The second tentative value is obtained from the first tentative value by adding a specified increment. Based on the golden section method, the two golden points between the first and the second tentative values are calculated. Then, the wavenumbers \(q\) and frequencies \(\omega\) can be computed corresponding to the value of known quantity and the four values of requested quantity including two tentative values and two golden section values. 3. Substitute the resulting four (\(q\),\(\omega\)) groups into the pertinent formulas and matrices to form the system matrices associated with the four values of requested quantities. Then calculate the determinants of the system matrices in these four cases. 4. According to the golden section method and referring to the determinant amplitudes of the four system matrices, decide whether there is a root to the requested quantity within the two tentative values.