# Acoustic Shooting and Bounce Ray Method for Calculating Echoes of Complex Underwater Targets
###### Abstract
Let \(P_{i}(\textbf{r}^{\prime},\omega)\) denote the incident sound pressure at any point \(P\) on the target surface represented by the vector \(\textbf{r}^{\prime}\), \(P_{i}(\textbf{r}_{0},\omega)\) denote the incident sound pressure at the geometric center \(O\) of the target surface represented by the vector \(r_{0}\), and \(P_{s}(\textbf{r}^{\prime},\omega)\) denote the scattered sound pressure on the target surface at any point \(P\) represented by the vector \(\textbf{r}^{\prime}\). E is the  
Figure 1: Time domain discrete integral acceleration algorithm object. angle between the backscattered wavefront vector \(\mathbf{k}\mathbf{s}=(\mathbf{r}-\mathbf{r^{\prime}})/(\mathbf{r}-\mathbf{r^{\prime}})\) and the normal vector \(\mathbf{n}\). As the far-field condition \(r\rightarrow\infty\) is satisfied under the co-location of the transceiver, there are \(\phi_{i}\approx\phi_{s}=\phi\), \(\mathbf{k}_{i}\approx-k_{s}\), \(\mathbf{k}_{i}\cdot\mathbf{r^{\prime}}=z^{\prime}\), \(\mathbf{k}_{i}\cdot(\mathbf{r}-\mathbf{r^{\prime}})=\mathbf{r}-z^{\prime}\) (that is, the projection of the vector \(\mathbf{r}-\mathbf{r^{\prime}}\) on the vector-\(\mathbf{k}_{i}\)). Considering the scale of the panel itself \(l<<r\rightarrow\infty\), it is reasonable to use \(P_{i}(\mathbf{r}_{0},\omega)\) to represent the sound pressure value of the entire panel. What needs to be changed is the phase of the panel, and the change in the phase is reflected by exp (\(i\mathbf{k}_{i}\cdot(\mathbf{r}-\mathbf{r^{\prime}})\)). According to the above conditions, the following formulas can be obtained:  
\[P_{i}(\mathbf{r^{\prime}},\omega)\approx P_{i}(\mathbf{r}_{0},\omega)e^{i\mathbf{k}_{i} \cdot\mathbf{r^{\prime}}} \tag{4}\]  
\[\frac{\partial P_{i}(\mathbf{r^{\prime}},\omega)}{\partial n}\approx-ik_{i}cos \phi_{i}P_{i}(\mathbf{r^{\prime}},\omega)\approx-ik_{i}cos\phi_{i}P_{i}(\mathbf{r}_{0},\omega)e^{i\mathbf{k}_{i}\cdot\mathbf{r^{\prime}}} \tag{5}\]  
\[P_{s}(\mathbf{r^{\prime}},\omega)\approx\Gamma(\phi)P_{i}(\mathbf{r^{\prime}},\omega) \approx\Gamma(\phi)P_{i}(\mathbf{r}_{0},\omega)e^{i\mathbf{k}_{i}\cdot\mathbf{r^{\prime}}} \tag{6}\]  
\[\frac{\partial P_{s}(\mathbf{r^{\prime}},\omega)}{\partial n}\approx-\Gamma(\phi )\frac{\partial P_{i}(\mathbf{r^{\prime}},\omega)}{\partial n}\approx ik_{i}cos \phi_{i}\Gamma(\phi)P_{i}(\mathbf{r}_{0},\omega)e^{i\mathbf{k}_{i}\cdot\mathbf{r^{\prime}}} \tag{7}\]  
where \(\Gamma(\phi)\) is the reflection coefficient, which can be obtained according to the layered medium acoustic definition regardless of whether the target surface is rigid or not at a high frequency approximation. Substitute the non-approximate scheme of Equation (2) into Equation (1), and simplify Equation (1) by using \(-\mathbf{k}_{i}\cdot(\mathbf{r}-\mathbf{r^{\prime}})=r-z^{\prime}\),and \(\mathbf{k}_{i}\cdot\mathbf{r^{\prime}}=z^{\prime}\), then the scattered sound pressure at \(\mathbf{r}\), \(P_{s}(\mathbf{r},\omega)\) can be written as:  
\[P_{s}(\mathbf{r},\omega)=-\frac{ik}{2\pi r}e^{-ikr}\iint\limits_{S}\Gamma(\phi)e ^{ikz^{\prime}}P_{i}(\mathbf{r^{\prime}},\omega)cos\phi dS \tag{8}\]  
The scattered field is divided into surface elements of the target surface for discrete integration:  
\[P_{ss}(\mathbf{r},\omega)=-\frac{ik}{2\pi r}e^{-ikr}\sum\limits_{m=1}^{M}\Gamma_{ m}(\phi_{m})\iint\limits_{S_{m}}e^{ikz^{\prime}}P_{i}(\mathbf{r}_{i},\omega) cos\phi_{m}dS \tag{9}\]  
or  
\[G(\textbf{r},\textbf{r}^{\prime},\omega)=\frac{e^{-ik_{i}(\textbf{r}-\textbf{ r}^{\prime})}}{4\pi|\textbf{r}-\textbf{r}^{\prime}|}\approx\frac{e^{-ik_{i}r}}{4 \pi r}e^{ik_{i}r^{\prime}} \tag{2}\]0  
\(P_{ss}(\mathbf{r},\omega)\) represents the total scattered sound field in the target frequency domain, \(M\) represents the total number of surface elements on the target surface, \(\Gamma_{m}(\phi_{m})\) represents the reflection coefficient of surface elements, \(S_{m}\) represents the region of each surface element, and \(\phi_{m}\) represents the angle between the normal vector \(\mathbf{n}\) on each surface element.