## 3 Numerical Simulation Analysis
### Applicable Conditions for Directional Point Sources Equivalence  
This paper proposes the prediction method for deep-sea target sound scattering based on the equivalence of the directional point sound source. The complex elastic scattering field is equivalent to the radiation field of a directional point sound source, and the equivalent is only held in the far field. In the near-field area, it can be regarded as the interference superposition of incoming waves at different positions of the structure, according to the Helmholtz integral formula. The sound field does not only have radial radiation components but also lateral non-radiation components. Only at a distance far enough, the lateral non-radiation components account for enough small, it can be considered that the far-field condition is satisfied, and the far-field scattered field of the elastic structure is approximately equivalent to the radiation field of a directional point sound source. The following is a simulation analysis of the applicability of this equivalence. The FEM is used to analyse the sound scattering field of an elastic spherical shell in the free field. The spherical shell has a radius of 10 m, a thickness of 0.1 m, and a plane wave with a frequency of 50 Hz incident on the elastic spherical shell horizontally. The density of the spherical shell is \(\rho_{\mathrm{s}}\) = 7850 kg/m\({}^{3}\), the Young's modulus is \(E_{\mathrm{s}}\) = 2 \(\times\) 10\({}^{11}\) Pa, and the Poisson's ratio is \(\mu_{\mathrm{s}}\) = 0.33. The far-field sounds of the sound scattering field pressure directivity diagram (Figure 2a) and the near-field acoustic energy flow vector diagram (Figure 2b) are drawn, which indicate the significant spatial directional difference of the scattering field. The scattering main lobe is formed in the 0\({}^{\circ}\) direction, whilst multiple scattering side lobes exist in other directions. The near-field acoustic energy flow vector diagram of the scattering field presented in Figure 2b shows that, in the near field, theacoustic energy flowing in different directions do not flow out from the same sound centre. Transverse acoustic energy flow components exist. The proportion of transverse acoustic energy flow components decreases with the increase of the propagation distance, and the acoustic energy flow in different directions tends to flow out from the same sound centre. Additionally, the colour indicates the level of sound pressure, the red area has a higher sound pressure, and the blue area has a lower sound pressure. Figure 2c shows the transverse acoustic energy flow and the radial sound in the 0\({}^{\circ}\) direction from near and far. The ratio of energy flow further illustrates this point. The sound pressures of the discrete points on the spherical surface are extracted at the outer radii of 20 and 70 m, the spherical wave is orthogonally decomposed, and the sound pressure of the scattering field from near and far is calculated, according to the finite element simulation. Figure 3 shows the comparison of the scattering sound pressure calculated based on the spherical wave decomposition of the sound pressure at different positions in the 0\({}^{\circ}\) direction with the finite element direct calculation result. The scattering field calculation result is consistent with the finite element direct calculation result at a position that is greater than the distance from the decomposition surface. The results indicate that accurate and reliable far field scattering fields can be obtained based on the spherical sound pressure at different positions, providing convenience for the finite element simulation calculation, which is, the spherical sound pressure interface can be selected close to the spherical shell to reduce the amount of finite element sound field grids and improve the calculation efficiency. Subsequently, the spherical surface at 20 m is selected, and the discrete sound pressure on the spherical surface is calculated.