# Active Tunable Elastic Metasurface for Abnormal Flexural Wave Transmission
## 2 Mechanisms of Actively Tunable Elastic Metasurface
### Theoretical Formulations of the Transmission Matrix Method
\begin{bmatrix}e^{-ik_{3}l_{1}}&e^{ik_{3}l_{1}}&e^{- k_{3}l_{1}}&e^{k_{3}l_{1}}\\ -ik_{3}e^{-ik_{3}l_{1}}&ik_{3}e^{ik_{3}l_{1}}&-k_{3}e^{-k_{3}l_{1}}&k_{3}e^{k_{3 }l_{1}}\\ -D_{3}k_{2}^{3}e^{-ik_{3}l_{1}}&-D_{3}k_{2}^{3}e^{ik_{3}l_{1}}&D_{3}k_{2}^{3}e^ {-k_{3}l_{1}}&D_{3}k_{2}^{3}e^{k_{3}l_{1}}\\ iD_{3}k_{3}^{3}e^{-ik_{3}l_{1}}&-iD_{3}k_{3}^{3}e^{ik_{3}l_{1}}&-D_{3}k_{3}^{3} e^{-k_{3}l_{1}}&D_{3}k_{3}^{3}e^{k_{3}l_{1}}\end{bmatrix}\begin{bmatrix}a_{+3}\\ a_{-3}\\ b_{+3}\\ b_{-3}\end{bmatrix}  \tag{21}\]  
where \(k_{3}\) is the wavenumber in region III and \(D_{3}\) is the flexural rigidity of region III. According to the above equation, \(\mathbf{V}_{\mathrm{L3}}\) can be expressed as \(\mathbf{V}_{\mathrm{L2}}=\mathbf{N}_{\mathrm{3}}\mathbf{A}\), which can be obtained as \(\mathbf{A}=\mathbf{N}_{\mathrm{3}}^{-1}\mathbf{V}_{\mathrm{L3}}\). The state vector \(\mathbf{V}_{\mathrm{R3}}\) can be expressed by the following equation:  
Figure 2: The basic unit for deriving the transmission coefficient and the phase shift. \[\mathbf{V}_{\text{R3}}= \begin{bmatrix}e^{-ik_{3}(l_{1}+l_{2})}&e^{ik_{3}(l_{1}+l_{2} )}&e^{-k_{3}(l_{1}+l_{2})}&e^{k_{3}(l_{1}+l_{2})}\\ -ik_{3}e^{-ik_{3}(l_{1}+l_{2})}&ik_{3}e^{ik_{3}(l_{1}+l_{2})}&-k_{3}e^{-k_{3}(l_ {1}+l_{2})}&k_{3}e^{k_{3}(l_{1}+l_{2})}\\ -D_{3}k_{3}^{2}e^{-ik_{3}(l_{1}+l_{2})}&-D_{3}k_{3}^{2}e^{ik_{3}(l_{1}+l_{2})}&D _{3}k_{3}^{2}e^{-k_{3}(l_{1}+l_{2})}&D_{3}k_{3}^{2}e^{k_{3}(l_{1}+l_{2})}\\ iD_{3}k_{3}^{3}e^{-ik_{3}(l_{1}+l_{2})}&-iD_{3}k_{3}^{3}e^{ik_{3}(l_{1}+l_{2})}& -D_{3}k_{3}^{3}e^{-k_{3}(l_{1}+l_{2})}&D_{3}k_{3}^{3}e^{k_{3}(l_{1}+l_{2})}\\ \end{bmatrix}\begin{bmatrix}a_{+3}\\ a_{-3}\\ b_{+3}\\ b_{-3}\end{bmatrix}  \tag{22}\]  
According to the above equation, \(\mathbf{V}_{\text{R3}}\) can be expressed as \(\mathbf{V}_{\text{R3}}=\mathbf{M}_{3}\mathbf{A}\), and from Equations (21) and (22), it can be expressed as \(\mathbf{V}_{\text{R3}}=\mathbf{M}_{3}\mathbf{N}_{3}^{-1}\mathbf{V}_{\text{L2} }=\mathbf{U}_{3}\mathbf{V}_{\text{L2}}\). The above equation gives the following relationship  
\[\mathbf{V}_{\text{L4}}=\mathbf{V}_{\text{R4}}=\mathbf{U}_{3}\mathbf{V}_{\text {L3}}=\mathbf{U}_{3}\mathbf{V}_{\text{R2}}=\mathbf{U}_{3}\mathbf{U}_{2} \mathbf{V}_{\text{L2}}=\mathbf{U}_{3}\mathbf{U}_{2}\mathbf{V}_{\text{R1}} \tag{23}\]  
It is possible to solve for the transmission coefficients and phase shifts of the flexural waves from Equation (23). For this purpose, the wavefields in regions I, II, III and IV are prepared as follows:  
\[ \begin{array}{l}w_{1}(x)=e^{-ik_{1}x}+re^{ik_{1}x}+r^{*}e^{ik_{1}x}\\ w_{2}(x)=a_{+2}e^{-ik_{2}x}+a_{-2}e^{ik_{2}x}+b_{+2}e^{-kx}+b_{-2}e^{k_{2}x}\\ w_{3}(x)=a_{+3}e^{-ik_{3}x}+a_{-3}e^{ik_{3}x}+b_{+3}e^{-k_{3}x}+b_{-3}e^{k_{3}x} \\ w_{4}(x)=te^{-ik_{4}x}+t^{*}e^{-k_{4}x}\end{array}  \tag{24}\]  
where \(r\), \(r^{*}\), \(t\), and \(t^{*}\) are the amplitude ratios of the reflected propagating wave, the reflected attenuating wave, the transmitted propagating wave, and the transmitted attenuating wave to the incident wave, respectively. \(k_{\mathbf{4}}\) is the wavenumber in region IV, and \(k_{1}\) is the wavenumber in region I. The equation of the \(\mathbf{V}_{\text{R1}}\) relationship is as follows:  
\[\mathbf{V}_{\text{R1}}= \begin{bmatrix}1&1\\ ik_{1}&k_{1}\\ -D_{1}k_{1}^{2}&D_{1}k_{1}^{2}\\ -iD_{1}k_{1}^{3}&D_{1}k_{1}^{3}\end{bmatrix}\begin{bmatrix}r\\ r^{*}\end{bmatrix} + \begin{bmatrix}1\\ -ik_{1}\\ -D_{1}k_{1}^{2}\\ iD_{1}k_{1}^{3}\end{bmatrix}  \tag{25}\]  
where \(D_{1}\) is the flexural rigidity of region I. According to the above equation, the mathematical relationship of \(\mathbf{V}_{\text{R1}}\) can be written as \(\mathbf{V}_{\text{R1}}=\mathbf{U}_{1}\mathbf{r}+\mathbf{H}_{1}\). \(\mathbf{V}_{\text{L4}}\) can be expressed as  
\[\mathbf{V}_{\text{L4}}=