## 3 Numerical Simulations
### The Measurement Point Arrangement Determined by SVM
It could be seen that, with the existence of reflection boundary, the modal natural frequency of every mode could change. When the structure is quite close to the boundary that is considered to be'soft', the modal natural frequency for the certain mode increases because the added mass of water decreases. Conversely, the modal natural frequency is close to the free-field-case when the structure is not close to the boundary. For example, when \(h_{U}=0.2\) m and the excitation frequency is \(31.4\) Hz, mode \((1,5)\) is not included. However, in the free field, mode \((1,5)\) should be included in the total modes. In other words, when in the half-space and the submerged depth is quite small, the variations of SVM details make it necessary to adjust the measurement point arrangement to meet the practical requirements. The sound prediction is accomplished with ESM. The quantities of equivalent sources and the measurement points are determined with the rule presented in the study [13]. The equivalent source surface and the structure are conformal and the Recess ratio of the equivalent source is chosen to be 0.6\(\sim\)0.8 to avoid the ill-condition problem of the dipole matrix [9]. The coordinate system and the measurement points on the structure are shown in Figure 4.  
Define \(f_{0}\) as the excitation frequency. For example, when \(f_{0}=71\) Hz, if the structure is in free field, the highest order for the axial mode is 6, while it is 7 for the circumferential mode. However, when \(h_{U}=0.2\) m, the highest order for both the axial mode and the circumferential mode is 6. In other words, with the effect of reflection boundary, the highest circumferential mode changes. According to the theory of SVM, the vibration measurement along the circumferential direction should change with the modal variation. To verify the theory of SVM, a prediction plane composed of observation points is set. The coordinates of the observation points are defined as follows: \(x=-5\) m \(\sim 5\) m, \(z=-15\) m \(\sim-5\) m, and \(y=50\) m (the interval between two adjacent points is 0.2 m). In the predictions of FEM-BEM, the measurement points are sufficiently dense (axial: 80; circumferential: 80) to ensure the accuracy of sound prediction. In order to quantitively  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Mode**} & \multicolumn{3}{c}{**(\(n_{a}\),\(n_{c}\))**} & \multicolumn{3}{c}{**(1,5)**} & \multicolumn{3}{c}{**(3,6)**} & \multicolumn{3}{c}{**(3,7)**} \\ \hline \multirow{3}{*}{Shape} & \multirow{3}{*}{\(\begin{array}{c}\\ \\ \  
evaluate the performance of the prediction methods, the prediction error of sound pressure amplitude is defined as \(p_{E}=20\lg(p_{i}-p_{i}^{\prime})\), the prediction error of sound pressure phase is defined as \(\varphi_{E}=\varphi_{i}-\varphi_{i}^{\prime}\), \(p_{i}\) and \(\varphi_{i}\) are prediction results of sound pressure amplitude and phase, and \(p_{i}^{\prime}\) and \(\varphi_{i}^{\prime}\) are the references of sound pressure amplitude and phase computed by FEM-BEM. The prediction errors by ESM with sparse measurement points are shown in Figure 5 when the structure is in the free field. The number of measurement points is 270 (axial: 18; circumferential: 15), which has been proven to not be able to be reduced in either direction. The shell is excited with a point force of 50 N, which is located at (0,1,0) in Cartesian coordinates. Random noise with a signal-noise ratio (SNR) of 30 dB is added to the measured normal velocity to simulate the actual measurement. The medium is homogenous, with the density of 1000 kg/m\({}^{3}\) and sound speed of 1500 m/s. It could be seen in Figure 5 that, with the measurement points determined by SVM, using ESM, the prediction results could still be precise as the error of sound pressure level is always lower than 2 dB, while the error of sound pressure phase is always lower than 0.3 rad/s. Prediction errors with the levels are quite low such that they could meet the engineering requests.