# Investigations on the Complex Band Diagram of Flexural Wave through the Fluid-Loaded Phononic Plate
## 2 Descriptions and Model Formulations
### The \(\omega(\mathbf{k})\) Numerical Approach
\begin{array}{cc}\mathbf{H}_{u}^{\mathrm{H}}\mathbf{M}_{s} \mathbf{H}_{u}&\mathbf{0}\\ \rho_{f}\mathbf{H}_{p}^{\mathrm{H}}\mathbf{R}^{\mathrm{T}}\mathbf{H}_{u}&\mathbf{ H}_{p}^{\mathrm{H}}\mathbf{M}_{f}\mathbf{H}_{p}\end{array} \right]+j\omega\left[ \begin{array}{cc}\mathbf{H}_{u}^{\mathrm{H}}\mathbf{C}_{s}\mathbf{H}_{u}& \mathbf{0}\\ \mathbf{0}&\mathbf{H}_{p}^{\mathrm{H}}\mathbf{C}_{f}\mathbf{H}_{p}\end{array}  \right]+\left[ \begin{array}{cc}\mathbf{H}_{u}^{\mathrm{H}}\mathbf{K}_{s} \mathbf{H}_{u}&-\mathbf{H}_{u}^{\mathrm{H}}\mathbf{R}\mathbf{H}_{p}\\ \mathbf{0}&\mathbf{H}_{p}^{\mathrm{H}}\mathbf{K}_{f}\mathbf{H}_{p}\end{array}  \right]\right)\left[ \begin{array}{c}\mathbf{u}_{s}\\ \mathbf{p}_{s}\end{array} \right]=\left[ \begin{array}{c}\mathbf{0}\\ \mathbf{0}\end{array} \right] \tag{6}\]  
To compare with the results in previous work and validate the proposed method, the rigid wall boundary conditions are also adopted and the formulations to describe the motion of the fluid-loaded phononic plate with the rigid wall boundary are also derived as below. If a rigid wall boundary condition is imposed on the outer surface of the fluid domain, the damping matrix \(\mathbf{C}_{f}\) from the radiation boundary is negligible. Hence, Equation (6) can be rewritten as the following when the structural model is assumed to be undamped:  
\[\left(-\omega^{2}\left[ \begin{array}{cc}\mathbf{H}_{u}^{\mathrm{H}}\mathbf{M }_{s}\mathbf{H}_{u}&0\\ \rho_{f}\mathbf{H}_{p}^{\mathrm{H}}\mathbf{R}^{\mathrm{T}}\mathbf{H}_{u}& \mathbf{H}_{p}^{\mathrm{H}}\mathbf{M}_{f}\mathbf{H}_{p}\end{array} \right]+ \left[ \begin{array}{cc}\mathbf{H}_{u}^{\mathrm{H}}\mathbf{K}_{s}\mathbf{H}_{ u}&-\mathbf{H}_{u}^{\mathrm{H}}\mathbf{R}\mathbf{H}_{p}\\ \mathbf{0}&\mathbf{H}_{p}^{\mathrm{H}}\mathbf{K}_{f}\mathbf{H}_{p}\end{array}  \right]\right)\left[ \begin{array}{c}\mathbf{u}_{s}\\ \mathbf{p}_{s}\end{array} \right]=\left[ \begin{array}{c}\mathbf{0}\\ \mathbf{0}\end{array} \right] \tag{7}\]  
It can be seen that the dispersion relation is able to be achieved by solving a linear eigenvalue problem. To further reduce the computation costs of the underlying equations, the fluid medium can be assumed as incompressible for which the sound speed is infinitely large. In that case, there is no need to take into account the matrix \(\mathbf{M}_{f}\), and Equation (7) becomes:  
\[\left(-\omega^{2}\left[ \begin{array}{cc}\mathbf{H}_{u}^{\mathrm{H}}\mathbf{M }_{s}\mathbf{H}_{u}&\mathbf{0}\\ \rho_{f}\mathbf{H}_{p}^{\mathrm{H}}\mathbf{R}^{\mathrm{T}}\mathbf{H}_{u}& \mathbf{0}\end{array} \right]+\left[ \begin{array}{cc}\mathbf{H}_{u}^{\mathrm{ H}}\mathbf{K}_{s}\mathbf{H}_{u}&-\mathbf{H}_{u}^{\mathrm{H}}\mathbf{R}\mathbf{H}_{p}\\ \mathbf{0}&\mathbf{H}_{p}^{\mathrm{H}}\mathbf{K}_{f}\mathbf{H}_{p}\end{array}  \right]\right)\left[ \begin{array}{c}\mathbf{u}_{s}\\ \mathbf{p}_{s}\end{array} \right]=\left[ \begin{array}{c}\mathbf{0}\\ \mathbf{0}\end{array} \right] \tag{8}\]  
Then, the pressure vector can be easily solved:  
\[\mathbf{p}_{s}=\omega^{2}\rho_{f}\left(\mathbf{H}_{p}^{\mathrm{H}}\mathbf{K}_{ f}\mathbf{H}_{p}\right)^{-1}(\mathbf{H}_{p}^{\mathrm{H}}\mathbf{R}^{\mathrm{T}} \mathbf{H}_{u})\mathbf{u}_{s} \tag{9}\]  
Substituting this relation to the structural dynamic equation, the numerical formulations for obtaining the band structure of periodic plates in the presence of fluid loading can be achieved:  
\[(\mathbf{H}_{u}^{\mathrm{H}}\mathbf{K}_{s}\mathbf{H}_{u})\mathbf{u}_{s}+(- \omega^{2}(\mathbf{H}_{u}^{\mathrm{H}}\mathbf{M}_{s}\mathbf{H}_{u}+\rho_{f} \mathbf{H}_{u}^{\mathrm{H}}\mathbf{R}\mathbf{H}_{p}(\mathbf{H}_{p}^{\mathrm{H}} \mathbf{K}_{f}\mathbf{H}_{p})^{-1}(\mathbf{H}_{p}^{\mathrm{H}}\mathbf{R}^{ \mathrm{T}}\mathbf{H}_{u})))\mathbf{u}_{s}=\mathbf{0} \tag{10}\]  
This approach allows one to calculate the spatial dispersion for an arbitrary phononic plate with heavy fluid loading. The lower and upper edge frequencies of the band gap can be captured by specifying the Bloch wave vector at the boundary lines of the first irreducible Brillouin zone, i.e., IBZ.