## 4 Discussion
### The Influence of Submerged Depth  
In the above content, the submerged depth of the source is quite small, such that the influence of reflection sounds on the structural vibration could not be neglected. Thus, when decomposing the power resistance matrix \(\mathbf{R}_{\rm{p}}\), the half-space Green function has to be used in the construction of dipole matrix \(\mathbf{D}_{0}+\eta\mathbf{D}_{1}\) to eliminate the disturbance of reflection sounds, which leads to extra computation. However, with the increase of submerged depth, the influence of \(\mathbf{D}_{1}\) is weakened. When \(\mathbf{D}_{1}\) could be neglected, the dipole matrix is approximated to \(\mathbf{D}_{0}\) and almost half of the computation is saved. To verify the conclusion, a simulation is proposed when \(f_{0}=300\) Hz. Define the prediction where \(\mathbf{D}_{0}+\eta\mathbf{D}_{1}\) is the dipole matrix as the SFS case and the prediction where \(\mathbf{D}_{0}\) is the dipole matrix as the Quick case. The prediction errors with the variations of \(h_{U}/R_{c}\) are shown in Figure 19, where \(R_{c}\) is the radius of the structure.  
It could be seen that, in the SFS case, the prediction result could be accurate at an arbitrary submerged depth. However, in the Quick case, the prediction errors are quite large at a small submerged depth because the ARMs could not be calculated accurately when the influence of \(\mathbf{D}_{1}\) could not be neglected. When the submerged depth increases, the  
Figure 19: Prediction errors at different submerged depths in the SFS case and Quick case when 90 measurement points are used (\(h_{U}=0.2\) m, \(f_{0}=300\) Hz ). (**a**) \(\varepsilon_{p}\) and (**b**) \(\varepsilon_{\Phi}\).  
Figure 18: Prediction errors with the variation of the signal–noise ratio (SNR) (\(f_{0}=300\) Hz, \(h_{U}=0.2\) m, 90 measurement points optimized with EFI). (**a**) \(\varepsilon_{p}\) and (**b**) \(\varepsilon_{\Phi}\).  
prediction results become more accurate. In the above simulation, when \(h_{U}/R_{c}\) exceeds 5, the prediction results in the Quick case could also be acceptable. In fact, for a real elastic structure, the influence of reflection sounds on the structural vibration depends on many factors such as shape, size, thickness, material, and frequency. It is a complex problem that involves fluid-solid coupling, scattering, and sound propagation. Because of the limited space, only the influence of submerged depth is discussed in this paper. Because all the ARMs derive from the equivalent sources inside the structure, suppose that the maximum dimension of the structure is \(D_{m}\), the mean distance between \(S_{0}\) and \(\Omega\) is usually less than \(D_{m}/2\). In contrast, the mean distance between \(S_{0}\) and \(\Omega^{\prime}\) is more than \(2h_{U}\). According to the attenuation rule of monopole and Equation (11), it is easy to conclude that \(\mathbf{v}_{R}\) is an order of magnitude less than \((\mathbf{v}_{0}+\mathbf{v}_{s})\) when \(2h_{U}>10(D_{m}/2)\) is satisfied. In other words, when the submerged depth is large enough, \(\mathbf{D}_{0}\) could be used as the dipole matrix instead of \(\mathbf{D}_{0}+\eta\mathbf{D}_{1}\) and the Quick case goes into effect. There are two advantages of using the Quick case when the application condition is satisfied. Firstly, it is not necessary to compute \(\mathbf{D}_{1}\) and almost half of the computation is reduced, which is beneficial for fast prediction. Secondly, the stability of the prediction is reinforced without the participant of \(\mathbf{D}_{1}\), as the condition number of \(\mathbf{D}_{1}\) is usually large such that the regularization process is necessary.