## 1 Introduction
### Wave-Based Modeling of Structural Members  
For any structural member described in either of its local right-handed coordinates, the governing equations based on the Timoshenko beam theory [76] are:  
\[\kappa GA\left(\frac{\partial^{2}v}{\partial x^{2}}-\frac{\partial\varphi}{ \partial x}\right)=\rho A\frac{\partial^{2}v}{\partial t^{2}},EI\frac{\partial ^{2}\varphi}{\partial x^{2}}+\kappa GA\left(\frac{\partial v}{\partial x}- \varphi\right)=\rho I\frac{\partial^{2}\varphi}{\partial t^{2}}. \tag{1}\]  
Assuming time harmonic variation \(\mathrm{e}^{\mathrm{i}\omega t}\) to the generalized displacements with \(\omega\) the circular frequency, Equation (1) can be reduced to the wave equations as:  
\[\frac{\mathrm{d}^{2}\phi}{\mathrm{d}x^{2}}-\frac{\mathrm{d}\phi}{\mathrm{d}x} +\frac{\omega^{2}}{c_{l}^{2}}\vartheta=0,\,\frac{\mathrm{d}^{2}\phi}{\mathrm{ d}x^{2}}+\frac{Ac_{t}^{2}}{Ic_{l}^{2}}\frac{\mathrm{d}\phi}{\mathrm{d}x}+ \left[\frac{\omega^{2}}{c_{l}^{2}}-\frac{Ac_{t}^{2}}{Ic_{l}^{2}}\right]\phi=0, \tag{2}\]  
where \(c_{l}=\sqrt{E/\rho}\) and \(c_{t}=\sqrt{\kappa G/\rho}\) are the velocities of quasi-longitudinal and quasi-transverse waves in the structural beam, respectively; and an over caret signifies the quantities in the frequency domain. Equation (2) can be solved by virtue of the theory of ordinary differential equations to give the solutions to the generalized displacements as:  
\[ \begin{split}\vartheta(x)&=(a_{2}\mathrm{e}^{ \mathrm{i}k_{2}x}+a_{3}\mathrm{e}^{\mathrm{i}k_{3}x})+(d_{2}\mathrm{e}^{- \mathrm{i}k_{2}x}+d_{3}\mathrm{e}^{-\mathrm{i}k_{3}x}),\\ \phi(x)&=(\beta_{2}a_{2}\mathrm{e}^{\mathrm{i}k_{2}x }+\beta_{3}a_{3}\mathrm{e}^{\mathrm{i}k_{3}x})-(\beta_{2}d_{2}\mathrm{e}^{- \mathrm{i}k_{2}x}+\beta_{3}d_{3}\mathrm{e}^{-\mathrm{i}k_{3}x}),\end{split}  \tag{3}\]  
where \(\mathrm{i}=\sqrt{-1}\) is the imaginary unit. The terms in the former and latter parenthesis of Equation (3), as combined with \(\mathrm{e}^{\mathrm{i}\omega t}\), signify backward (arriving) and forward (departing) traveling waves. The wavenumbers \(k_{r}\) of flexural (\(r=2\)) and thickness-shear (\(r=3\)) modes are related to \(\omega\) by:  
\[k_{2,3}=\frac{\omega}{\sqrt{2}c_{l}}\sqrt{\left[\left(\frac{c_{l}^{2}}{c_{t}^{ 2}}+1\right)\pm\sqrt{\left(\frac{c_{l}^{2}}{c_{t}^{2}}-1\right)^{2}+4\frac{Ac_ {l}^{2}}{I\omega^{2}}}\right]}, \tag{4}\]  
where \(k_{2}\) is always purely-real indicating the propagation property of the flexural mode all along the frequency domain, and \(k_{3}\) is first purely-imaginary below \(\omega_{c}\) and then purely-real above \(\omega_{c}\) manifesting the first attenuation and then propagation property of the thickness-shear mode. \(\omega_{c}=\sqrt{\kappa GA/(\rho I)}\) is  
Figure 2: Description of the unit cell: the coordinates and the physical variables. (**a**) The global coordinates for joints and the local dual coordinates for structural members; (**b**) sign convention of the generalized displacements for joints and members; (**c**) sign convention of the generalized forces for joints and members; (**d**) convention of the wave amplitudes and wave vectors. the cut-off frequency. Note that in order to guarantee the numerically well condition of our formulation, \(k_{3}=-\mathrm{i}\omega\sqrt{-\left[\left(c_{l}^{2}/c_{t}^{2}+1\right)^{2}-\sqrt{ \left(c_{l}^{2}/c_{t}^{2}-1\right)^{2}+4 Ac_{l}^{2}/\left(I\omega^{2}\right)} \right]}/(\sqrt{2}c_{l})\) should be adopted as \(\omega<\omega_{c}\). It is indicated from Equation (3) that the flexural and the thickness-shear waves are coupled to each other, since the solutions to the generalized displacements contain contributions from both waves. \(\beta_{r}=[\omega^{2}/c_{t}^{2}-k_{r}^{2}]/(\mathrm{i}k_{r})\) representing the contribution of the flexural (\(r=2\)) or thickness-shear (\(r=3\)) wave on rotation displacement, is called the influence coefficient of rotation displacement, as the influence coefficients of transverse displacement are both unit.