# A Method for Predicting Radiated Acoustic Field in Shallow Sea Based on Wave Superposition and Ray
## 2 Theoretical Model
### Wave Superposition Theory  
The acoustic field radiated by the elastic structure in the fluid environment meet the Sommerfeld radiation condition in the far field, and the fluid-solid coupling boundary satisfies the condition of the normal vibration velocity. Additional boundary conditions for the Helmholtz equation can be expressed as:  
\[\
abla^{2}p+k^{2}p=0 \tag{1}\]  
\[v_{n}={v_{n}}^{+} \tag{2}\]  
\[\lim_{r\rightarrow\infty}\left[r\left(\frac{\partial p}{\partial r}+jkp\right) \right]=0 \tag{3}\]  
where, \(v_{n}\) is the normal velocity of the fluid medium, and \({v_{n}}^{+}\) is the normal vibration velocity of the structural surface in the fluid medium. Equation (3) shows the Sommerfeld far field radiation condition. The equation can be solved by the Helmholtz integral formula. For any shape of radiator, the radiated pressure at the field is the integral of the internal volume source of the structure as shown as:  
\[p(P)=j\rho_{0}\omega\int\limits_{V}q(Q)G(P,Q)dV \tag{4}\]  
where, \(P\) and \(Q\) are coordinates of the field point and virtual source in the global coordinate system, respectively. The \(\rho_{0}\) shows medium density in field, \(\omega\) is the angular frequency, \(q(Q)\) is strength of the virtual source, and \(G(P,Q)\) is the Green's function under the corresponding boundary conditions. A schematic diagram is shown in Figure 1.  
For the typical channel environment (i.e., the sea surface is the Dirichlet boundary and the sea floor is the Neumann boundary), the normal mode Green's function in cylindrical coordinates can be shown as [14]:  
\[G(r,z)=-\frac{i}{2D\rho(z_{\rm s})}\sum_{n=1}^{N}\Psi(k_{2n}z)\Psi(k_{2n}z_{\rm s })H_{0}^{(2)}(k_{m}r) \tag{5}\]  
where, \(z_{\rm s}\) shows the source depth, \(D\) is depth of the sea, \(k_{2n}\) and \(k_{m}\) represent the vertical and horizontal components of the wave number \(k\), respectively, and \(\Psi(k_{2n}z)\) is Eigen function with different forms under different boundary conditions. The parameter \(r\) is the horizontal distance, and \(N\) is the normal order. Introducing the linear Euler formula to transform Equation (4) into an integral representation of normal vibration velocity on the surface can be expressed as:  
\[u_{n}(P)=-\int\limits_{V}q(Q)\frac{\partial G(P,Q)}{\partial n}dV \tag{6}\]  
where, Equation (6) relates the surface normal velocity to the Green's function. In theory, there is no limit for the placement of virtual sources and they can be located anywhere within the structure. Related studies have shown that the virtual source surface and the surface of the structure are in the same shape, which can improve the inversion accuracy of the configured equivalent source. It can be assumed that the virtual source is distributed over a virtual spherical shell surface \(\sigma\) of thickness \(\delta_{\tau}\) inside the structure; hence, Equation (6) can be written as:  
\[u_{n}(P)=-\delta_{\tau}\oint\limits_{\sigma}q(Q)\frac{\partial G(P,Q)}{ \partial n}d\sigma \tag{7}\]  
where \(P\) is the node coordinate on the surface of the structure. Dividing the virtual spherical shell into \(I\) parts, each surface of the structure is shown by \(\sigma_{i}\). If each spherical shell crown \(\sigma_{i}\) is small enough, the summation coefficient \(\delta_{\tau}\) can be regarded as constant, and the normal vibration velocity at different points \(P_{s}\) on the structural surface can be approximately written as:  
\[u_{n}(P_{s})=-\sum_{i=1}^{I}q(Q_{i})\frac{\partial G(P_{s},Q_{i})}{\partial n} \tag{8}\]  
where, \(Q_{i}\) represents the coordinate of the \(i\)th virtual source, and \(q(Q_{i})\) represents the strength of the \(i\)th source on the virtual surface. Hence, Equation (8) can be written in the matrix as:  
\[Q=[D]^{-1}U \tag{9}\]  
The structural surface normal velocity matrix \(U\) in Equation (9) can be expressed by the source strength matrix \(Q\) and transfer matrix \(D\), where \(D\) can be expressed as:  
\[v_{n}={v_{n}}^{+} \tag{2}\]0  
\[v_{n}={v_{n}}^{+} \tag{2}\]1 Schematic diagram of wave superposition theory.