# A Method for Predicting Radiated Acoustic Field in Shallow Sea Based on Wave Superposition and Ray
## 3 Accuracy Analysis of Combined Wave Superposition Method
### Applicability of Rays Superimposed  
For the [PERSON]'s function requiring high-order superposition, the calculation time of each virtual source propagating the acoustic field increases rapidly with the increase of the calculated range and  
Figure 3: The near field accuracy of the [PERSON]’s function at different frequencies or different depths. (**a**) Accuracy of different frequencies at a depth of 200 m; (**b**) Accuracy of different depths at a frequency of 20 Hz.  
Figure 2: Comparison of far field transmission loss between [PERSON]’s function method and finite element method.  
order, making the calculation efficiency low. The large number of virtual sources inversion by the wave superposition method makes the superposition with the [PERSON]'s function not the most ideal choice. The ray theory performs ray tracing in the entire acoustic field and extracts the eigenray, which is less computationally expensive. Since the ray theory is derived under the high frequency approximation, it is necessary to verify that the ray solution can meet the accuracy requirements of the acoustic field superposition at different frequencies and depths.  
The channel is a typical shallow sea waveguide (i.e., the sea surface boundary is the Dirichlet boundary; the sea bottom boundary is the Neumann boundary), and the six point sources are located at depths of 25 to 30 m at intervals of 1 m. The depth of water and frequency are changed, and the acoustic field is superimposed by the ray theory and the [PERSON]'s function method, respectively. The field comparison chart after superposition is shown in Figure 4 as:  
The ray solution will have a slight oscillation near the true value at low frequency, and the superposition of multiple point sources will amplify the amplitude of the oscillation. A large number of virtual sources can cause an increase in the amplitude of the oscillation. The amplitude of the oscillations is different at different frequencies and depths. The error caused by oscillation is characterized by mean square error (MSE).  
\[MSE=\frac{1}{L}\sum_{i=1}^{M}{(TL_{R}-TL_{G})^{2}} \tag{21}\]  
where \(M\) represents the number of measurement points, and \(TL_{R}\) and \(TL_{G}\) are the transmission losses obtained by ray theory and the [PERSON]'s function methods respectively. The table below gives the MSE for certain frequencies and depths.  
Figure 4: The field comparison chart after superposition at different frequencies and depths. (**a**) Comparison at 20 Hz and 50 m; (**b**) Comparison at 50 Hz and 50 m; (**c**) Comparison at 50 Hz and 100 m; (**d**) Comparison at 1000 Hz and 50 m.  
In Table 1, in the case where the frequency is too low or the depth is too small, there is a large error in ray theory. At high frequency, the solution of the ray theory is very accurate. The field of virtual source can be directly superimposed, and the result is reliable. As a whole, the oscillation of the ray theory has a limited influence on the calculation results. The performance at high frequency is superior, and the application range of the wave superposition method can be expanded.