## 2 Boundary Value Problem (BVP)  
Let us consider an inclined porous plate fixed at the end wall of a two-dimensional wave flume in the water depth of \(h\). The length of a porous plate is \(l\) and the inclined angle \(\theta\). The coordinates \((x,z)\) with the \(z\) axis pointing vertically upward may be chosen. The incident waves of frequency \(\omega\) and amplitude \(A\) propagate from \(x\sim-\infty\). For an inviscid irrotational flow, the velocity \((u,w)\) can be expressed as the gradient of a scalar potential \(\
abla\Phi\). The conservation of mass requires that the potential satisfies Laplace's equation. \[\
abla^{2}\Phi=0 \tag{1}\]  
and under the assumption of the small amplitude wave, the boundary conditions are listed below. \[g\frac{\partial\Phi}{\partial z}+\Phi_{tt}=0,\ \ \ \text{on}\ z=0 \tag{2}\]  
\[\frac{\partial\Phi}{\partial z}=0,\ \ \text{on}\ z=-h \tag{3}\]  
\[\frac{\partial\Phi}{\partial x}=0,\ \ \text{on}\ x=0 \tag{4}\]  
where \(g\) is the gravitational acceleration. Sharp constriction on the porous plate makes flow separate downstream and vortices are formed from adjacent holes to create eddy zones. Therefore, boundary conditions at the porous plate are needed to realize energy dissipation by these vortices. The authors of [10] established the following boundary conditions. \[\frac{\partial\Phi^{+}}{\partial n}=-\frac{\partial\[PERSON]^{-}}{\partial n},\ \ \ \text{on}\ \text{porous plate} \tag{5}\]  
\[\frac{\partial\[PERSON]^{+}}{\partial t}-\frac{\partial\[PERSON]^{-}}{\partial t}= \frac{1}{2}C_{D}\frac{\partial\[PERSON]^{-}}{\partial n}\left|\frac{\partial\[PERSON] ^{-}}{\partial n}\right|+2C_{I}\frac{\partial^{2}\[PERSON]^{-}}{\partial t\, \partial n},\ \ \text{on}\ \text{porous plate} \tag{6}\]  
where \(\mid\mid\) denotes the modulus of the complex number. The above equations provide the matching conditions on the porous plate's upper (+) and lower (\(-\)) sides. The drag coefficient \(C_{D}\) representing energy dissipation and inertial coefficient \(C_{I}\) are determined empirically. The inertial term can be disregarded when the porous plate is thin and the hole size is small (\(C_{I}=0\)). The drag coefficient for a porous plate featuring uniformly distributed circular holes was empirically determined as a function of porosity using CFD simulations (Star-CCM+) for channel flow through the circular holes [15; 48]. \[C_{D}=0.6\bigg{(}\frac{1}{PC_{c}}-1\bigg{)}^{2} \tag{7}\]  
In steady flows with a high Reynolds number past sharp-edged orifices, the discharge coefficient \(C_{c}(=0.6+0.4P^{3})\) depends primarily on the porosity. For simple harmonic motion with frequency \(\omega\), the linearity of the BVP allows the separation of the time factor \(e^{-i\omega t}\) as follows:  
\[\Phi(x,z,t)=\text{Re}\Big{\{}\phi(x,z)e^{-i\omega t}\Big{\}} \tag{8}\]  
The linearized BVP including the radiation condition can be reduced to  
\[\left\{ \begin{array}{l}\
abla^{2}\phi=0,\ \text{in}\ \ \Omega\\ \frac{\partial\phi}{\partial z}-\frac{\omega^{2}}{g}\phi=0,\ \ \text{on}\ \Gamma^{F}\\ \frac{\partial\phi}{\partial z}=0,\ \ \text{on}\ \Gamma^{B}\\ \frac{\partial\phi}{\partial x}=0,\ \ \text{on}\ \Gamma^{W}\\ (\frac{\partial\phi}{\partial x}+ik)(\phi-\phi_{I})\to 0\ \text{as}\ x\rightarrow-\infty \end{array} \right. \tag{9}\]where the wave number \(k\) and frequency \(\omega\) are related by the dispersion relation \(\omega^{2}=gk\text{tanh}kh\). The incident wave potential \(\phi_{I}\) with the wave amplitude \(A\) propagating from \(x\sim-\infty\) is given by  
\[\phi_{I}(x,z)=-\frac{igA}{\omega}\frac{\cosh k(z+h)}{\cosh kh}e^{ik(x+L)} \tag{10}\]  
The quadratic drag term in Equation (6) makes the entire problem nonlinear, and the response to a simple harmonic \(\omega\) input should contain many harmonics (2\(\omega\),3\(\omega\),...). By employing equivalent linearization that the response is dominated by the first harmonic, the boundary conditions on the porous plate can be expressed as follows:  
\[\left\{