## 2 Theoretical Basis
### Sound Field Separation Technology Based on Vibration Measurement  
When a structure is located in the non-free-field, the reflection sounds could change the vibration of the structure. In NAH based on ESM, the interference of the reflection sounds must be excluded to obtain the 'useful' strength of equivalent sources in the free field. Most relevant studies chose double sound pressure holographic layers, double particle velocity holographic layers, or single sound pressure-particle velocity holographic layer to complete the process of SFS [36; 37; 38]. However, with the above methods, the scale of the measurement system would be quite enormous. In this section, the separation of normal velocity on the structural surface based on SFS will be introduced [1]. Suppose that the structure is submerged in the water with a finite depth. In that case, only the reflection sounds from the water-air boundary need to be considered. The schematic diagram of the ESM in half-space could be shown in Figure 2.  
Figure 2: The schematic diagram of the equivalent source method in half-space. In the figure, \(S_{0}\) is the surface of the structure, \(S_{R}\) is the 'image surface' of the virtual structure, \(\Omega\) is the surface of equivalent sources, while \(\Omega^{\prime}\) is the surface of image equivalent sources. The measured normal velocity on the structural surface \(v\) consists of three parts: the normal velocity in the free field \(v_{0}\), the normal velocity caused by reflection sounds \(v_{R}\), and the normal velocity related to the scattering sounds \(v_{s}\):  
\[v=v_{0}+v_{R}+v_{s} \tag{9}\]  
Obviously, the radiated sounds and scattering sounds are the components that radiate outwards, while the reflection sounds radiate inwards. The strength of equivalent sources \(Q\) could be also divided into three parts, which are related to the above three vibration components: \(q_{0}\), \(q_{R}\), and \(q_{S}\). As ESM is based on the Huygens' principle, which has been proven to be equivalent to the Helmholtz integral Equation [7], it could be easily concluded that \(v_{R}\) does not make contribution to the radiated sound field:  
\[\oint\!\!\!\!\!\!\!\int_{S_{R}}\!\!\!\left[G(\mathbf{r},\mathbf{r}_{R})v_{R}(\mathbf{r}_{ R})-\frac{\partial G(\mathbf{r},\mathbf{r}_{R})}{\partial\mathbf{n}_{s}}p_{R}(\mathbf{r}_{R}) \right]\!dS_{R}=0 \tag{10}\]  
where \(\mathbf{r}_{R}\) is the location of the node on \(S_{R}\) and \(p_{R}\) is the sound pressure variation caused by reflection sounds on \(S_{0}\). Thus, to precisely predict the radiated sound of a structure in half-space, \(\mathbf{v}_{R}\) should be separated from the measured normal velocity \(v\). For simplicity, the normal derivative is rewritten as \(\
abla_{\mathbf{n}}(.)\). With Equations (5) and (9), the vibration components could be written as follows:  
\[\left\{ \begin{array}{l}\mathbf{v}_{0}+\mathbf{v}_{s}=(\
abla_{\mathbf{n}}\mathbf{G}(\mathbf{r}_{ 0},\mathbf{r}_{s}))(\mathbf{q}_{0}+\mathbf{q}_{s})\\ \mathbf{v}_{R}=(\
abla_{\mathbf{n}}\mathbf{G}(\mathbf{r}_{1},\mathbf{r}_{s}))\mathbf{q}_{R}\end{array}  \right.,\left(\mathbf{r}_{0}\in\Omega,\mathbf{r}_{1}\in\Omega^{\prime},\mathbf{r}_{s}\in S _{0}\right) \tag{11}\]  
The relationship between \(\mathbf{v}_{R}\) and \(\mathbf{v}_{0}+\mathbf{v}_{s}\) is as follows:  
\[\mathbf{v}_{R}=\eta\Big{(}\
abla_{\mathbf{n}}\mathbf{G}(\mathbf{r}_{1},\mathbf{r}_{s})(\
abla_{ \mathbf{n}}\mathbf{G}(\mathbf{r}_{0},\mathbf{r}_{s}))^{+}\Big{)}(\mathbf{v}_{0}+\mathbf{v}_{s}) \tag{12}\]  
Then, the 'useful' normal velocity \(\mathbf{v}_{0}+\mathbf{v}_{s}\) could be separated from \(\mathbf{v}\) to calculate the strength of the equivalent sources:  
\[(\mathbf{v}_{0}+\mathbf{v}_{s})=\Big{[}\mathbf{E}+\eta.\
abla_{\mathbf{n}}\mathbf{G}(\mathbf{r}_{1}, \mathbf{r}_{s})(\
abla_{\mathbf{n}}\mathbf{G}(\mathbf{r}_{0},\mathbf{r}_{s}))^{+}\Big{]}^{-1}\mathbf{v} \tag{13}\]  
where \(\mathbf{E}\) is an identity matrix with \(N\) dimensions.