# Pseudo-Spin Polarized One-Way Elastic Wave Eigenstates in One-Dimensional Phononic Superlattices
## 2 Model and Method
### Model System and Dynamic Equations of Motion
The equations of the motion of masses at interfaces I and II are as follows:  
\[\frac{d^{2}u_{A,0}^{N}}{dt^{2}}=K_{1}\Big{(}u_{A,1}^{N}-u_{A,0}^{N}\Big{)}-K_{ 2}\Big{(}u_{A,0}^{N}-u_{B,p-1}^{N-1}\Big{)} \tag{4a}\] \[\frac{d^{2}u_{B,0}^{N}}{dt^{2}}=K_{2}\Big{(}u_{B,1}^{N}-u_{B,0}^{N}\Big{)}-K_{ 1}\Big{(}u_{B,0}^{N}-u_{A,n-1}^{N}\Big{)} \tag{4b}\]  
Inserting the null terms \(-K_{1}\Big{(}u_{A,0}^{N}-u_{A,p-1}^{N-1}\Big{)}\mp K_{1}\Big{(}u_{A,0}^{N}-u_{ A,p-1}^{N-1}\Big{)}\) into Equation (4a) and \(-K_{2}\Big{(}u_{B,0}^{N}-u_{B,-1}^{N}\Big{)}\mp K_{2}\Big{(}u_{B,0}^{N}-u_{B,-1 }^{N}\Big{)}\) into Equation (4b), we obtain, by virtue of the bulk equations of motion, the conditions of the continuity of forces at interfaces I and II:  
\[K_{1}\Big{(}u_{A,0}^{N}-u_{A,p-1}^{N-1}\Big{)}=K_{2}\Big{(}u_{A,0}^{N}-u_{B,p-1 }^{N-1}\Big{)} \tag{5a}\] \[K_{2}\Big{(}u_{B,0}^{N}-u_{B,-1}^{N}\Big{)}=K_{1}\Big{(}u_{B,0}^{N}-u_{A,n-1}^{N} \Big{)} \tag{5b}\]  
We also introduce the following conditions of the continuity of displacement at I and II:  
\[\frac{d^{2}u_{A,m}^{N}}{dt^{2}}=K_{1}\Big{(}u_{A,m+1}^{N}-u_{A,m}^{N}\Big{)}-K_{1 }\Big{(}u_{A,m}^{N}-u_{A,m-1}^{N}\Big{)} \tag{1a}\]0 \[\frac{d^{2}u_{A,m}^{N}}{dt^{2}}=K_{1}\Big{(}u_{A,m+1}^{N}-u_{A,m}^{N}\Big{)}-K_{1 }\Big{(}u_{A,m}^{N}-u_{A,m-1}^{N}\Big{)} \tag{1a}\]1  
With these conditions, Equation (5a,b) becomes:  
\[\frac{d^{2}u_{A,m}^{N}}{dt^{2}}=K_{1}\Big{(}u_{A,m+1}^{N}-u_{A,m}^{N}\Big{)}-K_{1 }\Big{(}u_{A,m}^{N}-u_{A,m-1}^{N}\Big{)} \tag{1a}\]2 \[\frac{d^{2}u_{A,m}^{N}}{dt^{2}}=K_{1}\Big{(}u_{A,m+1}^{N}-u_{A,m}^{N}\Big{)}-K_{1 }\Big{(}u_{A,m}^{N}-u_{A,m-1}^{N}\Big{)} \tag{1a}\]3In Equation (7a), we use \(u_{A,p-1}^{N-1}=u_{A,-1}^{N}\). Inserting the general solutions given by Equation (2a,b) and the Bloch wave form into Equations (6a,b) and (7a,b) yields the following set of dynamic equations:  
\[ \begin{pmatrix}\alpha_{1}&\beta_{1}&-1&-1\\ f\alpha_{1}\delta_{-}^{(1)}&f\beta_{1}\delta_{+}^{(1)}&-\delta_{-}^{(2)}&- \delta_{+}^{(2)}\\ 1&1&-e^{-i\overline{q}L}\alpha_{2}&-e^{-i\overline{q}L}\beta_{2}\\ f\delta_{-}^{(1)}&f\delta_{+}^{(1)}&-e^{-i\theta L}\alpha_{2}\delta_{-}^{(2)}&- e^{-i\theta L}\beta_{2}\delta_{+}^{(2)}\end{pmatrix}\begin{pmatrix}A_{+}\\ A_{-}\\ B_{+}\\ B_{-}\end{pmatrix} =0 \tag{8}\]  
where \(f=\frac{K_{1}}{K_{2}}\), \(\alpha_{1}=\frac{1}{P_{1}}=e^{ik_{1}ua}\), \(\alpha_{2}=\frac{1}{P_{2}}=e^{ik_{2}pa}\), and \(\delta_{\pm}^{(j)}=1-e^{\pm ikj,a}\), with \(j=1\), \(2\). We also enforce the periodicity of the system, requiring that the solutions in Equation (2a,b) take the form of Bloch waves; that is, \(B_{\pm}^{N-1}=B_{\pm}^{N}e^{-i\theta L}\). Equation (8) possesses non-trivial solutions when the determinant of the dynamic matrix is equal to zero. This condition leads to the dispersion relation:  
\[ \begin{split} f\Big{(}\delta_{+}^{(2)}-\delta_{-}^{(2)}\Big{)} \Big{(}\delta_{+}^{(1)}-\delta_{-}^{(1)}\Big{)}2\text{cos}\,qL-(\alpha_{1} \beta_{2}+\beta_{1}\alpha_{2})\Big{(}\delta_{+}^{(2)}-f\delta_{+}^{(1)}\Big{)} \Big{(}\delta_{-}^{(2)}-f\delta_{-}^{(1)}\Big{)}+(\alpha_{1}\alpha_{2}+\beta_ {1}\beta_{2})\\ \Big{(}\delta_{+}^{(2)}-f\delta_{-}^{(1)}\Big{)}\Big{(}\delta_{-}^{(2)}-f \delta_{+}^{(1)}\Big{)}=0.\end{split}  \tag{9}\]  
This dispersion relation can be reformulated as:  
\[\cos qL=\cos k_{1}d_{1}\cos k_{2}d_{2}+\left[-\frac{1}{2}\left(\frac{1}{f} \frac{4\Big{(}\sin k_{2}\frac{a}{2}\Big{)}^{2}}{\sin k_{1}a\sin k_{2}a}+f\frac {4\big{(}\sin k_{1}\frac{a}{2}\big{)}^{2}}{\sin k_{1}a\sin k_{2}a}\right)+ \frac{4\Big{(}\sin k_{1}\frac{a}{2}\Big{)}^{2}\Big{(}\sin k_{2}\frac{a}{2} \Big{)}^{2}}{\sin k_{1}a\sin k_{2}a}\right]\sin k_{1}d_{1}\sin k_{2}d_{2} \tag{10}\]