# Active Tunable Elastic Metasurface for Abnormal Flexural Wave Transmission
## 2 Mechanisms of Actively Tunable Elastic Metasurface
### Theoretical Formulations of the Transmission Matrix Method  
In this subsection, we will use the transmission matrix method (TMM) to resolve the transmission coefficients and phase shifts of the unit cell. It has been shown that the transmission coefficient and phase shift of a three-dimensional model is equivalent to those of a two-dimensional model [44]. Therefore, the wave propagation can be considered as a plane strain problem in the \(xz\) plane, which is independent of the \(y\)-axis direction, as shown by the basic unit in Figure 2. The general solution to Equation (10) is  
\[w(x)=a_{+}e^{-ikx}+a_{-}e^{ikx}+b_{+}e^{-kx}+b_{-}e^{kx} \tag{18}\]  
where \(a_{+}e^{-ikx}\) and \(b_{+}e^{-kx}\) are positive propagating and attenuating waves, respectively, and \(a_{-}e^{ikx}\) and \(b_{-}e^{kx}\) are negative propagating and attenuating waves, respectively. \(a_{+},a_{-},b_{+}\), and \(b_{-}\) represent the complex coefficients. The displacement \(w(x)\), slope \(\theta(x)\), bending moment \(\bar{M}(x)\), and shear force \(V(x)\) compose the state vector \(\mathbf{V}=\{w,\theta,M,V\}^{\mathrm{T}}\). The basic unit in Figure 2 can be divided into four regions, where region I is the incident region; regions II and III are sub-cell 1 and sub-cell 2, respectively; and region IV is the transmission region. Each region shares the same central axis. The materials of region I and IV are the same as the base beam material with thickness \(h_{0}=h_{1}\). As shown in Figure 2, the state vectors on the left and right sides of the connection between the regions working together can be defined as \(\mathbf{V}_{\mathrm{R1}}\), \(\mathbf{V}_{\mathrm{L2}}\), \(\mathbf{V}_{\mathrm{R2}}\), \(\mathbf{V}_{\mathrm{L3}}\), and \(\mathbf{V}_{\mathrm{R3}}\) and \(\mathbf{V}_{\mathrm{L4}}\). The connectivity at the boundaries can be determined from the fact that \(\mathbf{V}_{\mathrm{R1}}=\mathbf{V}_{\mathrm{L2}}\), \(\mathbf{V}_{\mathrm{R2}}=\mathbf{V}_{\mathrm{L3}}\) and \(\mathbf{V}_{\mathrm{R3}}=\mathbf{V}_{\mathrm{L4}}\). In this way, \(\mathbf{V}_{\mathrm{L2}}\) can be expressed as  
\[\mathbf{V}_{\mathrm{L2}}= \begin{bmatrix}1&1&1&1\\ -ik_{2}&ik_{2}&-ik_{2}&k_{2}\\ -D_{2}k_{2}^{2}&-D_{2}k_{2}^{2}&D_{2}k_{2}^{2}&D_{2}k_{2}^{2}\\ D_{2}k_{2}^{3}&-iD_{2}k_{2}^{3}&-D_{2}k_{2}^{3}&D_{2}k_{2}^{3}\end{bmatrix} \begin{bmatrix}a_{+2}\\ a_{-2}\\ b_{+2}\\ b_{-2}\end{bmatrix}  \tag{19}\]  
where \(k_{2}\) is the wavenumber in region II and \(D_{2}\) is the flexural rigidity of region II. According to the above equation, \(\mathbf{V}_{\mathrm{L2}}\) can be expressed as \(\mathbf{V}_{\mathrm{L2}}=\mathbf{N}_{\mathrm{2}}\mathbf{A}\), which can be obtained as \(\mathbf{A}=\mathbf{N}_{\mathrm{2}}^{-1}\mathbf{V}_{\mathrm{L2}}\). Similarly, \(\mathbf{V}_{\mathrm{R2}}\) can be written as  
\[\mathbf{V}_{\mathrm{R2}}= \begin{bmatrix}e^{-ik_{2}l_{1}}&e^{ik_{2}l_{1}}&e^{- k_{2}l_{1}}&e^{k_{2}l_{1}}\\ -ik_{2}e^{-ik_{2}l_{1}}&ik_{2}e^{ik_{2}l_{1}}&-k_{2}e^{-k_{2}l_{1}}&k_{2}e^{2k_ {2}l_{1}}\\ -D_{2}k_{2}^{2}e^{-ik_{2}l_{1}}&-D_{2}k_{2}^{2}e^{ik_{2}l_{1}}&D_{2}k_{2}^{2}e^ {-k_{2}l_{1}}&D_{2}k_{2}^{2}e^{k_{2}l_{1}}\\ iD_{2}k_{2}^{3}e^{-ik_{2}l_{1}}&-iD_{2}k_{2}^{3}e^{ik_{2}l_{1}}&-D_{2}k_{2}^{3} e^{-k_{2}l_{1}}&D_{2}k_{2}^{3}e^{k_{2}l_{1}}\end{bmatrix}\begin{bmatrix}a_{+2}\\ a_{-2}\\ b_{+2}\\ b_{-2}\end{bmatrix}  \tag{20}\]  
For brevity, \(\mathbf{V}_{\mathrm{R2}}\) can be expressed as \(\mathbf{V}_{\mathrm{R2}}=\mathbf{M}_{\mathrm{2}}\mathbf{A}\). Combining Equations (19) and (20) leads to \(\mathbf{V}_{\mathrm{R2}}=\mathbf{M}_{\mathrm{2}}\ \mathbf{N}_{\mathrm{2}}^{-1}\mathbf{V}_{\mathrm{L2}}= \mathbf{U}_{\mathrm{L2}}\). With respect to \(\mathbf{V}_{\mathrm{R3}}\), there is the following mathematical relationship. \[\mathbf{V}_{\mathrm{L3}}=