# 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  
In the \(\mathbf{k}(\omega)\) approach, complex wave vectors are required to be obtained for a given real frequency. The real and imaginary components of wavenumbers can be used to predict the propagation and attenuation of flexural waves, respectively. Therefore, either the stop band or the degree of wave decaying can be examined or assessed by this approach. However, when using the general FEM formulations, it is impossible to evaluate the propagation and attenuation constants from a polynomial eigenvalue problem in the arbitrary direction of \(\mathbf{k}\) as the nature depends on whether the ratio between \(k_{y}\) and \(k_{x}\) is rational [28]. If \(k_{y}/k_{x}\) is rational, the polynomial eigenvalue equation can be formed; otherwise, the transcendental eigenvalue equation is preferred. For instance, when focusing on the \(\Gamma\)X direction and \(k_{x}\) is assumed as the unknown variable in Equation (7), the eigenvalue problem needs to be rewritten in the form:  
\[ \begin{array}{l}[\lambda_{x}^{-r-1}\mathbf{A}_{1}+\lambda_{x}^{-r}\mathbf{A} _{2}+\lambda_{x}^{-1}\mathbf{A}_{3}+\mathbf{A}_{4}+\lambda_{x}^{-r+1}\mathbf{ A}_{5}\\ +\lambda_{x}^{r-1}\mathbf{A}_{6}+\lambda_{x}\mathbf{A}_{7}+\lambda_{x}^{r} \mathbf{A}_{8}+\lambda_{x}^{r+1}\mathbf{A}_{9}]\mathbf{Q}=\mathbf{0}\end{array}  \tag{11}\]where \(\lambda_{x}=e^{ik_{x}a}=e^{i\mu_{x}}\) and parameter \(r\) denotes the ratio of \(k_{y}/k_{x}\). From Equation (11), we can see that the eigenvalue equation remains in polynomial form only if \(r\) is rational. In order to reduce the computation difficulties, a wave field transformation capable of eliminating the phase-constant-dependent matrixes is applied [39]. The framework for dealing with the fluid-structure interaction problem is reformulated and presented as follows. The transverse displacement and the pressure under the wave field transformation operator are:  
\[ \begin{array}{l}p=p_{k}(x,y,z,t)e^{(\lambda\cos\theta x+\lambda sin\theta y)} \\ w=w_{k}(x,y,t)e^{(\lambda\cos\theta x+\lambda\sin\theta y)}\end{array}  \tag{12}\]  
where \(\boldsymbol{\lambda}=j\mathbf{k}\) and \(\lambda=|\boldsymbol{\lambda}|\). In this case, the fluid domain is divided into three sections: the surfaces \(S_{\text{L}}\), \(S_{\text{II}}\) and the inner domain \(V_{in}\). \(S_{\text{I}}\) and \(S_{\text{II}}\) denote the fluid-structure interface and the exterior surface, respectively. The wave motion equation in the inviscid and nonflowing fluid medium is:  
\[\ddot{p}/c^{2}=\mathbf{L}^{T}(\mathbf{L}p),\ \ p\in V_{in} \tag{13}\]  
where the differential operator \(\mathbf{L}\) is expressed as: \(\mathbf{L}=\left[\partial/\partial x\ \ \partial/\partial y\ \ \partial/\partial z\right]^{\text{T}}\). The continuity between normal velocity and stress at the fluid-structure interface is ensured by:  
\[\partial p/\partial n_{\text{I}}=-\rho_{f}\ddot{w},\ p\in S_{\text{I}} \tag{14}\]  
where \(n_{\text{I}}\) denotes the normal direction of the interface \(S_{\text{I}}\), and in this study, it implies the positive \(z\) direction. The wave-absorbing boundary imposed on the exterior surface is assumed as:  
\[\partial p/\partial n_{\text{II}}=-\ddot{p}/c,\ p\in S_{\text{II}} \tag{15}\]  
where \(n_{\text{II}}\) denotes the normal direction of the exterior surface \(S_{\text{II}}\) of the fluid domain. Substituting the expressions of the displacement and the pressure of Equation (12) to Equation (13), Equation (14) and Equation (15) yield a new formulation to describe the time-harmonic motion of acoustic waves in the fluid medium:  
\[