# 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
Further, the relative error is defined as:  
\[\eta=\sqrt{\sum_{i=1}^{K}{\left(\left|p_{r}-p_{f}\right|\right)^{2}}}/\sqrt{\sum _{i=1}^{K}\left|p_{f}\right|^{2}}\times 100\% \tag{22}\]  
where, \(K\) is the number of measurement points. The \(p_{r}\) and \(p_{f}\) are the sound pressures generated by combined WSM and FEM, at the \(i\)th measurement point, respectively. The \(14\times 11\) and \(10\times 7\) virtual sources (excluding the upper and lower vertices) are selected respectively, to form a virtual source surface, as shown in Figure 8, and analyzed the effect of the radius ratio \(\lambda\) on the calculation result at the receive depth of 25 m, as shown in Figure 9.  
Theoretically, when the number of virtual sources is large, the dispersion degree of the volume source will be higher, and the inversion of the sound field will be more accurate. In fact, for the  
Figure 8: Different virtual source surface. (**a**) \(14\times 11\) Virtual sources; (**b**) \(10\times 7\) Virtual sources. Figure 7: The SPL at the depth of 25 m and 50 m (100 Hz). (**a**) The depth of 25 m; (**b**) The depth of 50 m; (**c**) The depth of 100 m; (**d**) The depth of 150 m.  
Figure 6.0 The effect of radius ratio and the number of virtual sources on the calculation result. situation analyzed in this section, after the number of virtual sources meets the basic requirements of the calculation, it does have some impact on the accuracy of the calculation when the radius ratio is in the range of 0.7 to 1. The impact of this factor is far greater than the impact of the number of virtual sources. Figure 9 shows that in the case where the number of virtual sources is constant, the radius ratio is too small or too large, which leads to an increase in error. When the radius ratio approaches 1, because the Green's function is a cylindrical function, the horizontal distance between the virtual source and the surface node of the structure becomes too small, and the calculated \(D\) matrix singularity increases. When the radius ratio is too small, the difference in surface vibration velocity is not sufficiently reflected on the virtual source. In a word, the appropriate radius ratio plays a critical role in the accuracy of the calculation. For the spherical shell, the optimum ratio of the radius ratio is in the range of 0.5 to 0.9. When the shape or other parameters of the structure change, the configuration of the virtual sources should be adjusted. The \(10\times 7\) virtual sources (excluding the upper and lower vertices) and a radius ratio of 0.5 are selected, which is same with the model in Figure 7 to get a two-dimensional color figure of SPL in 100 Hz using combined WSM and FEM respectively as shown in Figure 6.1  
The field interference fringes in Figure 10a are clear, and the results are similar to the finite element results in Figure 10b. Same as in Figure 10b, Figure 10a can also describe the phenomenon caused by the interaction of radiated sound with the sea surface and the sea bottom during the propagation of the sound, that is, the sound rays reflect back and forth between the sea surface and the sea bottom. However, the difference is that compared with Figure 10b, the peak and trough amplitudes of Figure 10a are not clear enough, which is also reflected in Figure 6.2  
It should be noted that in order for combined WSM to accurately predict the radiated field of the structure, the environment of the sea needs to meet the applicable range of ray theory. The following relationships can be used as a guide for high-frequency approximation of ray theory [1] as shown as:  
Figure 6.3  
where \(f\) is the frequency, \(c\) is sound speed in water, and \(D\) is the depth of the water. This relationship can also be used as guidance for the applicable conditions of combined WSM. Although the guidance given by this relationship is not necessarily accurate, it can indeed reflect the potential of combined WSM for the calculation of high-frequency problems.