# Investigations on the Complex Band Diagram of Flexural Wave through the Fluid-Loaded Phononic Plate
## 2 Descriptions and Model Formulations
### The \(\mathbf{k}(\omega)\) Numerical Approach
\begin{array}{l}\mathbf{K}_{s0}=\iint\limits_{S_{\Gamma}}\mathbf{B}_{\Theta }\mathbf{D}\mathbf{B}_{\Theta}dxdy\\ \mathbf{K}_{s1}=\iint\limits_{S_{\Gamma}}\mathbf{B}(\mathbf{B}_{\Theta}^{ \mathsf{T}}\mathbf{D}\mathbf{B}_{\Gamma}-\mathbf{B}_{\Gamma}^{\mathsf{T}} \mathbf{D}\mathbf{B}_{\Theta})dxdy\\ \mathbf{K}_{s2}=\iint\limits_{S_{\Gamma}}\mathbf{B}(\mathbf{B}_{\Theta}^{ \mathsf{T}}\mathbf{D}\mathbf{H}\mathbf{N}_{s}+\mathbf{N}_{s}^{\mathsf{T}} \mathbf{H}^{\mathsf{T}}\mathbf{D}\mathbf{B}_{\Theta}-4\mathbf{B}_{\Gamma} \mathbf{D}\mathbf{B}_{\Gamma})dxdy\\ \mathbf{K}_{s3}=\iint\limits_{S_{\Gamma}}\mathbf{2}\big{(}-\mathbf{B}_{\Gamma} ^{\mathsf{T}}\mathbf{D}\mathbf{H}\mathbf{N}_{s}+\mathbf{N}_{s}^{\mathsf{T}} \mathbf{H}^{\mathsf{T}}\mathbf{D}\mathbf{B}_{\Gamma}\big{)}dxdy\\ \mathbf{K}_{s4}=\iint\limits_{S_{\Gamma}}\mathbf{N}_{s}^{\mathsf{T}}\mathbf{H} ^{\mathsf{T}}\mathbf{H}\mathbf{N}_{s}dxdy\\ \mathbf{M}_{s}=\rho_{s}\mathbf{f}\iint\limits_{S_{\Gamma}}\mathbf{N}_{s}^{ \mathsf{T}}\mathbf{N}_{s}dxdy\end{array}  \tag{20}\]  
where \(\mathbf{N}_{s}\) is the matrix of shape functions for \(w_{k}\), \(\mathbf{B}_{\Theta}=\mathbf{\Theta}\mathbf{N}_{s}\), and \(\mathbf{B}_{\Gamma}=\mathbf{\Gamma}\mathbf{N}_{s}\). \(\rho_{s}\) represents the density of the plate. The discrete matrixes to describe wave motion in a fluid are given as:  
\[ \begin{array}{l}\mathbf{K}_{f0}=\iint\limits_{\Omega_{f}}\mathbf{B}_{\Gamma }^{\mathsf{T}}\mathbf{B}_{\Gamma}dxdydz\\ \mathbf{K}_{f1}=\iint\limits_{\Omega_{f}}\left(\mathbf{B}_{\Gamma}^{\mathsf{T }}\mathbf{h}\mathbf{N}_{p}-\mathbf{N}_{p}^{\mathsf{T}}\mathbf{B}_{\Gamma} \right)dxdydz\\ \mathbf{K}_{f2}=-\iint\limits_{\Omega_{f}}\mathbf{N}_{p}^{\mathsf{T}}\mathbf{ h}\mathbf{N}_{p}dxdydz\\ \mathbf{M}_{f}=\frac{1}{c_{f}^{2}}\iint\limits_{\Omega_{f}}\mathbf{N}_{p}^{ \mathsf{T}}\mathbf{N}_{p}dxdydz\\ \mathbf{C}_{f}=\frac{1}{c_{f}}\iint\limits_{S_{\Gamma}}\mathbf{N}_{p}^{ \mathsf{T}}\mathbf{N}_{p}dxdy\end{array}  \tag{21}\]  
where \(\mathbf{N}_{p}\) is the matrix with shape functions for \(p_{k}\) and \(\mathbf{B}_{\Gamma}=\mathbf{L}\mathbf{N}_{p}\). In addition, the coupling matrix \(\mathbf{R}\) is formulated as: \(\mathbf{R}=\iint\limits_{S_{\Gamma}}\mathbf{N}_{p}^{\mathsf{T}}\mathbf{N}_{s} dxdy\). As the unknown nodal value is transformed from \(w(p)\) to \(w_{k}(p_{k})\) after using the wave field transformation, the periodic boundary conditions also need to be modified. The mechanical field of the displacement and pressure should satisfy the following relation at the unit cell boundaries [39]:  
\[\left\{\begin{array}{l}\mathbf{W}_{k,k_{2}}=\mathbf{W}_{k,k_{1}},\mathbf{W} _{k,k_{4}}=\mathbf{W}_{k,k_{3}}\\ \mathbf{W}_{k,c_{4}}=\mathbf{W}_{k,c_{3}}=\mathbf{W}_{k,c_{2}}=\mathbf{W}_{k,c _{1}}\\ \left\{ \begin{array}{l}\mathbf{P}_{k,S_{2}}=\mathbf{P}_{k,S_{1}},\,\,\mathbf{P }_{k,S_{4}}=\mathbf{P}_{k,S_{3}}\\ \mathbf{P}_{k,k_{4}}=\mathbf{P}_{k,h_{3}}=\mathbf{P}_{k,l_{2}}=\mathbf{P}_{k,l_ {1}}\\ \end{array} \right. \tag{22}\]  
Thus, \(\mathbf{H}_{\mathbf{u}}\) and \(\mathbf{H}_{p}\) are reduced to matrixes without any phase constants, i.e., Bloch wave vectors. The applied force induced by the adjacent cell should also satisfy the new relation:  
\[ \begin{array}{l}\mathbf{F}_{L_{1}}+\mathbf{F}_{L_{2}}=\mathbf{0}\\ \mathbf{F}_{L_{3}}+\mathbf{F}_{L_{4}}=\mathbf{0}\\ \mathbf{F}_{C_{1}}+\mathbf{F}_{C_{2}}+\mathbf{F}_{C_{3}}+\mathbf{F}_{C_{4}}= \mathbf{0}\\ \end{array}  \tag{23}\]After substituting the modified periodic boundary conditions into Equation (19), a new polynomial eigenvalue problem with respect to \(\lambda\) can be formed:  
\[\Big{(}\mathbf{D}_{0}+\lambda\mathbf{D}_{1}+\lambda^{2}\mathbf{D}_{2}+\lambda^{3 }\mathbf{D}_{3}+\lambda^{4}\mathbf{D}_{4}\Big{)}\mathbf{U}_{k}=\mathbf{0} \tag{24}\]  
where \(\mathbf{U}_{k}\) represents the general displacement vector that consists of node values of \(w_{k}\) and \(p_{k}\). From Equation (24), we can see that the wave vector is independent of the assembled matrixes of the system.