# A Method for Predicting Radiated Acoustic Field in Shallow Sea Based on Wave Superposition and Ray
## 3 Accuracy Analysis of Combined Wave Superposition Method
### Accuracy Analysis of Green's Function in Channel  
The basis of the wave superposition method is the Green's function. The accuracy of the Green's function, especially in the near field, directly determines the accuracy of the transfer matrix. The Green's function is expressed as the sum of the normal modes. The high-order normal mode has little contribution to the far field; however, it has a great influence on the near field. Therefore, the normal order determines the near field accuracy of the Green's function. When the order is too small, the result of the Green's function is inaccurate in the near field, but gradually approaches the exact value as the distance increases.  
In order to illustrate the near field accuracy of the Green's function, a case is selected for analysis. In the shallow sea of 200 m depth, the sea surface is the Dirichlet boundary and the sea floor is the Neumann boundary. A point source with a frequency of 20 Hz is located at a depth of 25 m. According to the boundary conditions, the following constraints exist in Equation (5) as shown as:  
\[\Psi(0)=0 \tag{18}\]  
\[\left(\frac{\mathrm{d}\Psi}{\mathrm{d}z}\right)_{D}=0 \tag{19}\]  
The eigen function \(\Psi\) that satisfies the constraints is \(\Psi(x)=\sin(x)\). The specific form of the Green function can be obtained according to the constraint conditions, which is the analytical solution under boundary condition as shown as:  
\[G(r,z)=-\frac{i}{2D\rho(z_{s})}\sum_{n=1}^{N}\sin(k_{zn}z)\sin(k_{zn}z_{s})H_ {0}^{(2)}(k_{m}r) \tag{20}\]  
where, \(z_{s}\) shows the source depth, \(D\) indicates the depth of the sea, and \(k_{zn}\) and \(k_{m}\) represent the vertical and horizontal components of the wave number \(k\), respectively. The parameter \(r\) is the horizontal distance, and \(n\) is the normal order.  
The model is built using finite element software COMSOL and compared with the [PERSON]'s function method when the mesh density is sufficiently dense, and the results are shown in Figure 2.  
It can be seen that under far field conditions, the calculation result of the [PERSON]'s function is nearly consistent with the finite element method. In fact, the situation is different under near field conditions, which is what needs to be discussed next.  
We assume the error threshold 1 dB, that is, the accuracy of the near field calculation is characterized by the horizontal distance when the absolute error between the transmission loss of the [PERSON]'s function method and the finite element method is less than 1 dB. The further the distance is, the more accurate are the results of the near field of the Green's function we obtain. Figure 3 below shows the near field accuracy of the Green's function after changing the normal mode order of the Green's function at different frequencies or different depths, while the sea environment and point source parameters are consistent with the above case.  
The overall trend is that if the order is larger, the results of the near field of the Green's function are more accurate. In order to ensure that the wave superposition method can effectively reverse the acoustic field information, it needs to increase the order and to ensure that the horizontal distances from the equivalent sources to the positions of vibration velocity point are not too close. As the frequency increases, the energy of the acoustic field concentrates on the low-order normal modes, and loss due to the lack of energy of the high-order is also reduced. However, in order to get the accuracy of the near-field, it is necessary to keep the order large enough. In terms of greater depth, the requirement of the normal order is higher in this method. Therefore, theoretically the accuracy of the wave superposition method can be guaranteed with a small depth.