## 3 Dual Boundary Element Method (DBEM)
### Inner Solution  
The DBEM-based inner solutions belonging to \(-L\leq x\leq 0\) are determined numerically. For the inclined porous plate shown in Figure 1, the inner solutions must satisfy the BVP composed by Equations (9), (11), and (16). [PERSON]'s second identity, when applied to the combination of the fundamental solution \(G\) and the complex potential \(\phi\), results in the following boundary integral equation (BIE) that satisfies the full boundary conditions in the domain \(\Omega\). \[\varepsilon(P)\phi(P)=\int\limits_{\Gamma}\left[\phi(Q)\frac{\partial G(P,Q)} {\partial n_{Q}}-G(P,Q)\frac{\partial\phi(Q)}{\partial n_{Q}}\right]d\Gamma(Q) \tag{17}\]  
with \(\varepsilon(P)\) being defined by \(\varepsilon(P)=\left\{ \begin{array}{ll}2\pi,&\text{if }P\in Q\\ \pi,&\text{if }P\in\Gamma\\ 0,&\text{if }P\
otin Q+\Gamma\end{array} \right.\) where \(P(x,z)\) is the field point and \(Q(\xi,\eta)\) the source point, and \(\Gamma\) corresponds to the entire boundaries enclosing the computational domain. The fundamental solution \(G\) can be described as \(\
abla^{2}G=\delta(\overset{\rightarrow}{r})\), where \(\delta\) is the Dirac delta function and \(r=\sqrt{(x-\xi)^{2}+(z-\eta)^{2}}\) is the distance between the source point and field point. The two-dimensional fundamental solution is given by \(G=\ln r\). The BIE of Equation (17) can be used to compute the unknown potentials on the boundary. The numerical solutions can be obtained by subdividing the boundary into small elements and then integrating numerically over these elements. However, for a numerical model including a plate with zero thickness inside the domain, Equation (17) leads to a rank-deficiency problem, disturbing the determination of a unique solution. To resolve this difficulty, for each pair of coincident source points on the degenerate plate boundary, two distinct boundary integral equations are necessary. This involves applying the singular boundary integral equation (SBIE) on one side of the degenerate boundary and the hyper-singular boundary integral equation (HSBIE) on the other with a higher order singularity on the opposite side \(\Gamma^{-}\) by taking the derivatives of the kernels in the SBIE. Figure 1: The definition sketch of an inclined porous plate as a wave absorber. By shifting the field point \(P(x,z)\) to the boundary, the dual boundary integral equations on the smooth boundaries yield the following integral equations:  
\[ \begin{array}{l}\pi\phi(P)=\int\limits_{\Gamma}\left[\phi(Q)\frac{\partial G(P,Q )}{\partial n_{Q}}-G(P,Q)\frac{\partial\phi(Q)}{\partial n_{Q}}\right]d\Gamma(Q ),\ P\in\Gamma^{R}\cup\Gamma^{+},\\ \pi\frac{\partial\phi(P)}{\partial n_{P}}=\int\limits_{\Gamma}\left[\phi(Q) \frac{\partial^{2}G(P,Q)}{\partial n_{Q}\partial n_{P}}-\frac{\partial G(P,Q )}{\partial n_{P}}\frac{\partial\phi(Q)}{\partial n_{Q}}\right]d\Gamma(Q),\ P \in\Gamma^{-},\end{array}  \tag{18}\]  
where \(\Gamma^{+}\) and \(\Gamma^{-}\) represent the two sides of the degenerate boundary and \(\Gamma^{R}(=\Gamma^{F}+\Gamma^{M}+\Gamma^{B}+\Gamma^{W})\) the remaining non-degenerate boundary. \(\Gamma\) denotes the whole boundary, i.e., \(\Gamma=\Gamma^{R}+\Gamma^{+}+\Gamma^{-}\). To transform the above dual BIEs into algebraic equations, the entire boundary \(\Gamma\) must be discretized into \(N\) segments with constant value of \(\phi\) and \(\frac{\partial\phi}{\partial n}\) on each segment. The distribution direction of elements traverses counterclockwise for the boundary \(\Gamma^{R}\) and follows a clockwise rotation for the degenerate boundary \(\Gamma^{\pm}\). Applying the discretized form of Equation (17) at the centroid of each element, gives rise to an assembled matrix given by  
\[