# 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}\frac{-\omega^{2}}{c^{2}}p_{k}-\mathbf{L}^{T}(\mathbf{L}p_{ k})-\lambda\left(\mathbf{h}^{\text{T}}\mathbf{L}p_{k}+\mathbf{L}^{\text{T}} \mathbf{h}p_{k}\right)-\lambda^{2}\mathbf{h}^{\text{T}}\mathbf{h}p_{k}=0\ ;\ p_{k}\in V_{in}\\ \mathbf{n}_{\text{I}}^{\text{T}}\mathbf{L}p_{k}+\lambda\mathbf{n}_{\text{I}}^ {\text{T}}\mathbf{h}p_{k}-\rho_{f}^{2}\omega^{2}w_{k}=0\ ;\ p_{k}\in S_{\text{I}}\\ \mathbf{n}_{\text{II}}^{\text{T}}\mathbf{L}p_{k}+\lambda\mathbf{n}_{\text{II}}^ {\text{T}}\mathbf{h}p_{k}-j\omega_{\text{c}}^{1}p_{k}=0\ ;\ p_{k}\in S_{\text{II}}\end{array}  \tag{16}\]  
where the vector \(\mathbf{h}\) is: \(\mathbf{h}=\left[\cos(\theta)\sin(\theta)\,0\right]^{\text{T}}\). The angle \(\theta\) denotes the propagation direction of the flexural wave. Then, we consider the effects of fluid loading on the periodic plate. The transverse motion of the plate in the presence of external fluid can be characterized by:  
\[\rho_{s}t\ddot{w}+\Theta^{\text{T}}\mathbf{D}(\Theta w)=p \tag{17}\]  
where \(\Theta=\left[\partial^{2}/\partial x^{2}\ \ \partial^{2}/\partial y^{2}\ \ 2 \partial^{2}/\partial x\partial y\right]^{\text{T}}\), and \(\mathbf{D}\) is the flexural modulus matrix. In addition, substituting the displacement and pressure with the expressions in Equation (12) to the governing equation of the host phononic plate, the governing equation is rewritten as:  
\[-\omega^{2}\rho_{s}tw_{k}+\left\{ \begin{array}{l}\mathbf{\Theta}^{\text{T}} \mathbf{D}\mathbf{\Theta}w_{k}+\lambda\left(2\mathbf{\Theta}^{\text{T}} \mathbf{D}\mathbf{\Gamma}+\mathbf{\Gamma}^{\text{T}}\mathbf{D}\mathbf{\Theta} \right)w_{k}+\lambda^{2}\left(\mathbf{\Theta}^{\text{T}}\mathbf{D}\mathbf{H} +\mathbf{\Gamma}^{\text{T}}\mathbf{D}\mathbf{\Theta}\right)w_{k}\\ +\lambda^{3}\left(\mathbf{2}\mathbf{\Gamma}^{\text{T}}\mathbf{D}\mathbf{H}+2 \mathbf{H}^{\text{T}}\mathbf{D}\mathbf{\Gamma}\right)w_{k}+\lambda^{4} \mathbf{H}^{\text{T}}\mathbf{D}\mathbf{H}w_{k}\end{array} \right\}-p_{k}=0 \tag{18}\]where the operator \(\Gamma\) is: \(\Gamma= \begin{bmatrix}\cos(\theta)&0\\ 0&\sin(\theta)\\ \sin(\theta)&\cos(\theta)\\ \end{bmatrix} \left[ \begin{array}{c}\frac{\partial}{\partial x}\\ \frac{\partial}{\partial y}\\ \end{array} \right]\). Applying the standard Galerkin approach on Equations (16) and (18), the discrete forms of the coupled system are eventually derived:  
\[\left\{ \begin{array}{l}\left[\begin{array}{cc}-\omega^{2}\mathbf{M}_{s}+ \mathbf{K}_{s0}&-\mathbf{R}^{\mathsf{T}}\\ -\omega^{2}\rho_{f}\mathbf{R}&-\omega^{2}\mathbf{M}_{f}-j\omega\mathbf{C}_{f}+ \mathbf{K}_{f0}\\ \end{array}\right]+\lambda\left[\begin{array}{cc}\mathbf{K}_{s1}&\mathbf{0} \\ \mathbf{0}&\mathbf{K}_{f1}\\ \end{array}\right]+\\ \lambda^{2}\bigg{[}\begin{array}{cc}\mathbf{K}_{s2}&\mathbf{0}\\ \mathbf{0}&\mathbf{K}_{f2}\\ \end{array}\bigg{]}+\lambda^{3}\bigg{[}\begin{array}{cc}\mathbf{K}_{s3}& \mathbf{0}\\ \mathbf{0}&\mathbf{0}\\ \end{array}\bigg{]}+\lambda^{4}\bigg{[}\begin{array}{cc}\mathbf{K}_{s4}& \mathbf{0}\\ \mathbf{0}&\mathbf{0}\\ \end{array}\bigg{]}\\ \end{array} \right\}\left[ \begin{array}{c}\mathbf{W}_{k}\\ \mathbf{P}_{k}\\ \end{array} \right]=\left[ \begin{array}{c}\mathbf{0}\\ \mathbf{0}\\ \end{array} \right] \tag{19}\]  
The assembled matrixes for the periodic plate are formulated as:  
\[