# Acoustic Shooting and Bounce Ray Method for Calculating Echoes of Complex Underwater Targets
###### Abstract  
The acoustic scattering characteristics of a target are of great significance to the design of underwater acoustic detection systems. With the improvement in underwater weapon detection ability, the precision and accuracy of underwater target echo characteristic simulations are required to be higher and higher. The traditional underwater target simulation based on bright spot can no longer meet the needs of an underwater acoustic detection system for targeting fine feature recognition. This paper proposes a calculation using the fine complex underwater target echo bounce beam line method, in addition to the establishment of a bandwidth signal time domain calculation model and the integral dimension reduction to accelerate the algorithm on the basis of thorough target subspace division, combined with the geometrical acoustics method (GA) and physical acoustics (PA) for the pipeline space beam propagation tracking and aperture sound field to solve and realize the acoustic scattering characteristic calculation of a complex underwater target. Finally, the proposed method was verified by the Benchmark standard submarine. The accuracy and computational efficiency of the PA acceleration method are given, and the integrity of the target information calculated by the method is verified by acoustic ISAR imaging analysis  
receiving, and \(r\) represents the coordinate origin \(S\) in the basic coordinate system to the sound source (receiving) point \(Q\) vector. Point \(O\) is the geometric center of any surface element on the target surface, \(n\) is the normal vector of the surface element, and \(r_{0}\) represents the vector from the coordinate origin to the center point of the surface element in the basic coordinate system;  
Point \(P\) is any point on the surface element, and \(x\) represents the vector from the geometric center of the element under the basic coordinate system from point \(O\) to point \(P\),\(r^{\prime}\) represents the vector from the coordinate origin \(S\) to point \(P\) in the basic coordinate system, and \(z^{\prime}\) represents the projection of \(r^{\prime}\) to \(r\) vector. The Kirchhoff integral formula in the frequency domain in the far-field is as follows:  
\[P_{s}(\textbf{r},\omega)=-\frac{ik}{2\pi r}e^{-ikr}\iint\limits_{S}\Gamma( \phi)e^{ikz^{\prime}}P_{i}(\textbf{r}^{\prime},\omega)cos\phi dS \tag{1}\]  
where \(P_{s}(\textbf{r},\omega)\) is the frequency response of the target scattered sound field at any field point determined by the space vector \(r\). \(P(\textbf{r}^{\prime},\omega)\) is the frequency response of the incident sound field at any position vector \(\textbf{r}^{\prime}\) on the target surface. \(G(\textbf{r},\textbf{r}^{\prime},\omega)\) is the Green's function, and \(\frac{\partial G(\textbf{r},\textbf{r}^{\prime},\omega)}{\partial n}\) is the partial differential of the Green's function normal to the target surface [5; 6]. Under far-field conditions, \(r\rightarrow\infty\), then the Green's function \(G(\textbf{r},\textbf{r}^{\prime},\omega)\) satisfies the following approximation:  
\[G(\textbf{r},\textbf{r}^{\prime},\omega)=\frac{e^{-ik_{i}(\textbf{r}-\textbf{ r}^{\prime})}}{4\pi|\textbf{r}-\textbf{r}^{\prime}|}\approx\frac{e^{-ik_{i}r}}{4 \pi r}e^{ik_{i}r^{\prime}} \tag{2}\]  
then:  
\[\frac{\partial G(\textbf{r},\textbf{r}^{\prime},\omega)}{\partial n}\approx-( ik_{i}+\frac{1}{r})cos\phi_{i}G(\textbf{r},\textbf{r}^{\prime},\omega)\approx-ik_{i}cos \phi_{i}G(\textbf{r},\textbf{r}^{\prime},\omega) \tag{3}\]  
where \(k_{i}=r/r\) is the incident wave-front vector, and \(\phi_{i}\) is the angle between the incident wave-front vector \(k_{i}\) and the surface element normal vector \(n\).