# A Method for Predicting Radiated Acoustic Field in Shallow Sea Based on Wave Superposition and Ray
## 3 Accuracy Analysis of Combined Wave Superposition Method
### Calculation of Acoustic Radiation Field by Combined Method  
The key to the combined method is the solution of the acoustic inverse problem. The results of the virtual source inversion have a significant impact on the calculation of the acoustic field [19]. Research shows that the accuracy of wave superposition inversion has a great relationship with virtual source configuration. These researches are based on free space or half space Green's function, but the Green's function in sea is a cylinder functions, which has its own particularity. In order to illustrate the calculation accuracy of the method proposed in this paper and analyze the influence of different virtual source configuration methods, an excited spherical shell is selected to calculate its far field pressure using the proposed method and compared with the finite element model in COMSOL software. The reason why this axisymmetric structure was selected is that the calculation distance of the three-dimensional field by COMSOL is limited by the performance of the computer. The selected spherical shell has a radius of 2 m and a thickness of 0.1 m. The Material elastic Young's modulus is \(E=2\times 10^{11}\) N/m\({}^{2}\) and [PERSON]'s ratio is \(\sigma=0.3\); density is \(\rho_{s}=7850\) kg/m\({}^{3}\). The outside medium is water, in which the density is \(\rho_{1}=1024\) kg/m\({}^{3}\), and the speed of sound is \(c=1500\) m/s. The inside of the spherical shell is a vacuum. The center of the spherical shell is located at a water depth of 25 m. A normal force of 1 N with a frequency of 100 Hz is applied at the bottom of the spherical shell. The depth of the sea is 200 m, and the sea surface and seabed boundary conditions satisfy typical channel conditions. The schematic diagram of the COMSOL model is shown in Figure 5. The sound pressure level (SPL) of the radiated field of this model is calculated using COMSOL. As the spherical shell is circumferentially symmetrical, the vibration velocity of the two-dimensional structure surface calculated by COMSOL can be scanned every 36 degrees in the circumferential direction to obtain the three-dimensional structure surface vibration velocity distribution. We also calculate SPL of the radiated field using combined WSM based on 3D vibration velocity distribution data. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Frequency (Hz)** & **Depth of Sea (m)** & **Far Field MSE (\textgreater{}100 m)** \\ \hline  & 50 & 0.7274 \\
20 & 100 & 0.2573 \\  & 200 & 0.4015 \\ \hline  & 50 & 1.2416 \\
50 & 100 & 0.6347 \\  & 200 & 0.8860 \\ \hline  & 50 & 0.8024 \\
100 & 100 & 0.5367 \\  & 200 & 0.3707 \\ \hline  & 50 & 0.4219 \\
1000 & 100 & 0.2159 \\  & 200 & 0.0756 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: MSE of the solution of the ray solution compared to the Green’s function for different frequencies and different sea depths. The virtual sources are distributed on a surface that has the same shape as the surface of the structure, which is an appropriate virtual source configuration. The virtual source surface is generally obtained by shrinking the surface of the structure inwardly. Here the radius ratio is chosen to be 0.5. Except for the upper and lower vertices, the virtual source surface is divided into 10 and 6 intervals in the circumferential and vertical directions respectively, that is, a total of 72 virtual source points (\(10\times 7\) virtual sources) are created, as shown in Figure 6.  
The sound pressure level is calculated at different depths as shown in Figure 7.  
Figure 5: The schematic diagram of the COMSOL model. (**a**) The schematic diagram of the spherical shell in COMSOL; (**b**) The schematic diagram of the sea channel in COMSOL. Figure 6: Schematic diagram of virtual source placement (virtual sources); (**a**) Top view; (**b**) Front view. Figure 7: _Cont._  
The SPL is calculated at the received depth of 25 m, 50 m, 100 m, and 150 m. The results of combined WSM are compared with the finite element calculation results.