## 4 Simulation Verification
\tag{13}\]  
where the incident wave potential function \(\varphi_{i}\) = (A/\(r_{1}\))exp(\(\mathrm{i}kr_{1}\)) (omitting the time factor \(\mathrm{e}^{-\mathrm{i}\omega t}\)), and A is the amplitude constant. In the case of transceiver co-location, the surface boundary conditions lead to:  
\[\varphi_{s}=\frac{-\mathrm{i}k_{0}\mathrm{A}}{2\pi}\int_{S_{0}}\frac{\mathrm{ e}^{\mathrm{i}2k_{0}r}}{r^{2}}\cos\theta ds \tag{14}\]  
where \(r\) is the scattering point vector diameter and \(\theta\) is the angle of incidence. We extend the above results to non-rigid surfaces. When the surface radius of curvature \(R\) is large (the product of the radius of curvature and wavelength is much larger than 1), the local plane wave approximation can be applied by setting the surface reflection coefficient to \(V(\theta)\), and the surface acoustic impedance to \(Z_{n}\):  
\[\left\{ \begin{array}{l}|\varphi_{s}|=|V(\theta)|\cdot|\varphi_{i}|\\ \frac{\mathrm{i}\omega(\varphi_{s}+\varphi_{i})}{\alpha(\varphi_{s}+\varphi_{ i})ds}=-Z_{n}\end{array} \right. \tag{15}\]  
where \(\omega\) is the incident wave angular frequency and \(\rho\) is the target surface density. The following equation relates the plane wave reflection coefficient to the total surface acoustic impedance of any complex interface at an infinitely large plane interface:  
\[\frac{\mathrm{e}^{\mathrm{i}\varphi/\cos\theta}}{Z_{n}}=\frac{1-V(\theta)}{1 +V(\theta)} \tag{16}\]  
where the underwater sound speed is c, usually 1500 m/s. Equation (15) can be substituted for Equation (16). In the case of transceiver co-location, the surface boundary conditions lead to:  
\[\varphi_{s}=\frac{-\mathrm{i}k_{0}\mathrm{A}}{2\pi}\int_{S_{0}}\frac{\mathrm{ e}^{\mathrm{i}2k_{0}r}}{r^{2}}V(\theta)\cos\theta ds \tag{17}\]  
We can obtain the target intensity in the far-field condition according to Equation (17):  
\[TS=10\log\left|-\frac{\mathrm{i}k_{0}}{2\pi}I\right| \tag{18}\]Among them:  
\[I=\int\limits_{s_{0}}\mathrm{e}^{2\mathrm{i}k_{0}\overrightarrow{p}\cdot \overrightarrow{r_{0}}}(\overrightarrow{n_{0}}\cdot\overrightarrow{r_{0}})V( \theta)\mathrm{d}s \tag{19}\]  
Among them, \(\overrightarrow{\rho}=x\overrightarrow{i}+y\overrightarrow{j}\) is the vector from the point where the face element is located to the reference point, \(\overrightarrow{n_{0}}=\overrightarrow{k}\) is the unit normal vector of the face element, \(\overrightarrow{r_{0}}=u\overrightarrow{i}+v\overrightarrow{j}+w \overrightarrow{k}\) is the unit vector from the receiving point to the reference point, and \(\overrightarrow{n_{0}}\cdot\overrightarrow{r_{0}}=\cos\theta=w\). Suppose that the reflection coefficient \(V(\theta)\) is constant within a plate element; then, the integral of a plate element is:  
\[I_{\Delta s}=\int\limits_{\Delta s}\mathrm{e}^{2\mathrm{i}k_{0}(ux+vy)}V( \theta)\mathrm{d}x\mathrm{d}y=V(\theta)w\sum\limits_{n=1}^{K}\frac{\mathrm{e }^{2\mathrm{i}k_{0}(x_{n}u+y_{n}v)}(p_{n-1}-p_{n})}{(2k_{0}u+2k_{0}p_{n-1}v)(2 k_{0}u+2k_{0}p_{n}v)} \tag{20}\]  
where \(K\) is the number of polygon vertices and \((x_{n},y_{n})\) is the coordinate of the polygon vertex and set \(p_{0}=\frac{y_{1}-y_{K}}{x_{1}-x_{K}}\). The target intensity of a complex target can be calculated by the following slab metric method. The target surface is first divided into a grid of many small slabs.