## 3 Solution Procedure
### Spectral Element Method
\begin{array}{l}\mathbf{u}_{\text{L}}=\left[w_{1,\text{A}}\;\theta_{1, \text{A}}\right]^{\mathsf{T}},\;\mathbf{u}_{\text{R}}=\left[w_{2,\text{B}}\; \theta_{2,\text{B}}\right]^{\mathsf{T}},\;\mathbf{u}_{\text{M}}=\left[w_{ \text{M}}\;\theta_{\text{M}}\right]^{\mathsf{T}},\\ \mathbf{f}_{\text{L}}=\left[Q_{1,\text{A}}\;M_{1,\text{A}}\right]^{\mathsf{T}}, \;\mathbf{f}_{\text{R}}=\left[Q_{2,\text{B}}\;M_{2,\text{B}}\right]^{\mathsf{ T}},\\ \mathbf{S}_{\text{LL}}=\mathbf{S}_{\text{A1}},\;\mathbf{S}_{\text{LM}}=\mathbf{ S}_{\text{A2}},\;\mathbf{S}_{\text{ML}}=\mathbf{S}_{\text{A3}},\;\mathbf{S}_{\text{MR}}= \mathbf{S}_{\text{B2}},\\ \mathbf{S}_{\text{RM}}=\mathbf{S}_{\text{B3}},\;\mathbf{S}_{\text{RR}}=\mathbf{ S}_{\text{B4}},\;\mathbf{S}_{\text{MM}}=\mathbf{S}_{\text{A4}}+\mathbf{S}_{\text{B1}}, \\ \mathbf{S}_{\text{A}}=\left\{\begin{array}{ccc}\mathbf{S}_{\text{A}}(j,1)& \mathbf{S}_{\text{A}}(j,2)\\ \mathbf{S}_{\text{A}}(j+1,1)&\mathbf{S}_{\text{A}}(j+1,2)\\ \mathbf{S}_{\text{A}}(j-1,3)&\mathbf{S}_{\text{A}}(j-1,4)\\ \mathbf{S}_{\text{A}}(j,3)&\mathbf{S}_{\text{A}}(j,4)\end{array}\right\}, \;j=1,\;3\\ \mathbf{S}_{\text{B}}=\left\{\begin{array}{ccc}\left[\begin{array}{ccc} \mathbf{S}_{\text{B}}(j,1)&\mathbf{S}_{\text{B}}(j,2)\\ \mathbf{S}_{\text{B}}(j+1,1)&\mathbf{S}_{\text{B}}(j+1,2)\\ \mathbf{S}_{\text{B}}(j-1,3)&\mathbf{S}_{\text{B}}(j-1,4)\\ \mathbf{S}_{\text{B}}(j,3)&\mathbf{S}_{\text{B}}(j,4)\end{array}\right],\;j=2, \;4\\ \end{array}\right.\\ \end{array}  \tag{18}\]  
\([\;]\)\((j,k)\) represents the element at the \(j\)th row and \(k\)th column of the matrix \([\;]\). Further, since the displacement at the joint of the two segments is not necessary to be determined, one can eliminate \(\mathbf{u}_{\text{M}}\) from Equation (17), and Equation (17) is then simplified to  
\[\left[ \begin{array}{cc}\mathbf{S}1&\mathbf{S}2\\ \mathbf{S}3&\mathbf{S}4\end{array} \right]\left[ \begin{array}{c}\mathbf{u}_{ \text{L}}\\ \mathbf{u}_{\text{R}}\end{array} \right]=\left[ \begin{array}{c}\mathbf{f}_{ \text{L}}\\ \mathbf{f}_{\text{R}}\end{array} \right], \tag{19}\]  
where  
\[ \begin{array}{l}\mathbf{S}1=\mathbf{S}_{\text{LL}}-\mathbf{S}_{\text{LM}} \mathbf{S}_{\text{MM}}^{-1}\mathbf{S}_{\text{ML}},\;\mathbf{S}2=-\mathbf{S}_{ \text{LM}}\mathbf{S}_{\text{MM}}^{-1}\mathbf{S}_{\text{MR}},\\ \mathbf{S}3=-\mathbf{S}_{\text{RM}}\mathbf{S}_{\text{MM}}^{-1}\mathbf{S}_{ \text{ML}},\;\mathbf{S}4=\mathbf{S}_{\text{RR}}-\mathbf{S}_{\text{MM}} \mathbf{S}_{\text{MR}}^{-1}\mathbf{S}_{\text{MR}}.\end{array}  \tag{20}\]  
Thus, the local stiffness matrix for a single cell is yielded as  
\[\mathbf{S}_{1}=\left[ \begin{array}{cc}\mathbf{S}1&\mathbf{S}2\\ \mathbf{S}3&\mathbf{S}4\end{array} \right], \tag{21}\]  
which is a 4 \(\times\) 4 matrix. Based on \(\mathbf{S}_{1}\) and [PERSON]'s theorem of plane wave propagation in phononic crystals, we can obtain  
\[\mathbf{u}_{\text{R}}=\text{e}^{\text{i}q/\text{\tiny PC}}\mathbf{u}_{\text{L}},\;\mathbf{f}_{\text{R}}=-\text{e}^{\text{i}q/\text{\tiny PC}}\mathbf{f}_{ \text{L}}, \tag{22}\]  
where \(q\) is the Bloch wave vector. In general, the real and imaginary parts of \(q\) represent, respectively, the phase difference and amplitude decay rate of wave motion between two adjacent cells in a PC structure. Substituting Equation (22) into Equation (19), the following linear equations are yielded  
\[\left(\mathbf{S}3+\mathbf{S}41\;\text{e}^{\text{i}q/\text{\tiny PC}}+\mathbf{S }2\;\text{e}^{2\text{i}q/\text{\tiny PC}}\right)\mathbf{u}_{\text{L}}=0. \tag{23}\]  
If Equation (23) has a nontrivial solution, the determinant of the coefficient matrix must be vanished, that is,  
\[\det\left(\mathbf{S}3+\mathbf{S}41\;\text{e}^{\text{i}q/\text{\tiny PC}}+ \mathbf{S}2\;\text{e}^{2\text{i}q/\text{\tiny PC}}\right)=0. \tag{24}\]Hence, the relationship between the incident frequency and wave vector can be determined by Equation (24) once the other system parameters are provided.