# A Method for Extracting Acoustic Water Surface Waves Based on Phase Compensation
## 2 Models
### Phase Discontinuity Model  
The frequency-modulated continuous-wave (FMCW) radar emits linear frequency-modulated (LFM) continuous waves and detects the displacement changes in the water surface from the phase difference in the echo, which contains the time-frequency information of underwater acoustic signals. In the experiments, it is assumed that the radar is located directly above the water surface with a height of \(h\) and shines vertically on the water surface. The start frequency of the LFM signal is \(f_{c}\). The signal bandwidth is \(B\). The frequency modulation rate is \(K_{r}\). The pulse duration is \(T_{p}\). The pulse repetition period is \(T\). The transmitted signal is represented as follows:  
\[s_{tx}\big{(}\hat{t},\eta\big{)}=rect\bigg{(}\frac{\hat{t}}{T_{p}}\bigg{)}exp \bigg{[}j\Big{(}2\pi f_{c}t+\pi K_{r}\hat{t}^{2}\Big{)}\bigg{]} \tag{4}\]  
Among them, \(\eta\) is the slow time, \(\hat{t}\) is the fast time, and \(t=\hat{t}+nT\) is the full time. Supposing the distance from the point target of water surface to the radar is \(R(t)\), the radar echo is represented as follows:  
\[s_{rx}\big{(}\hat{t},\eta\big{)}=Arect\bigg{(}\frac{\hat{t}-2R(t)/c}{T_{p}} \bigg{)}\cdot exp\bigg{[}j\Big{(}2\pi f_{c}(t-2R(t)/c)+\pi K_{r}\big{(}\hat{t}- 2R(t)/c\big{)}^{2}\Big{)}\bigg{]} \tag{5}\]  
Mixing the received signal with the transmitted signal, the expression for the intermediate frequency signal is as follows:  
\[ \begin{array}{ll}s_{IF}\big{(}\hat{t},\eta\big{)}&=s_{tx}\big{(}\hat{t},\eta \big{)}\cdot s_{rx}{}^{*}\big{(}\hat{t},\eta\big{)}\\ &=Arect\Big{(}\frac{\hat{t}-2R(t)/c}{T_{p}}\bigg{)}\cdot exp\bigg{[}j2\pi \bigg{(}-\frac{2K_{r}R(t)}{c}\hat{t}-\frac{2R(t)}{\lambda}+\frac{2K_{r}R(t)^{ 2}}{c^{2}}\bigg{)}\bigg{]}\end{array}  \tag{6}\]  
The exponential term \(\frac{2K_{r}R(t)^{2}}{c^{2}}\) is very small and can be ignored. Performing a Fourier transform on Equation (6) in the fast time domain gives the following:  
\[S_{IF}(f,\eta)=AT_{p}sinc\bigg{[}T_{p}\bigg{(}f+\frac{2K_{r}R(t)}{c}\bigg{)} \bigg{]}\cdot exp\bigg{[}j\frac{4\pi R(t)}{\lambda}\bigg{]} \tag{7}\]  
Among them, the distance from the point target of the water surface to the radar \(R(t)\) includes the distance \(H\) from the radar to the initial water surface and the amplitude of acoustic water surface waves \(y(t)\). The phase of the range bin where the target is located is extracted, that is, \(f=-\frac{2K_{r}R(t)}{c}\). The phase is given as follows:  
\[\phi(t)=\frac{4\pi R(t)}{\lambda}=\frac{4\pi y(t)}{\lambda}+\frac{4\pi H}{\lambda} \tag{8}\]  
The water surface vibration caused by underwater sound waves is of the order of micrometers, while the radar wavelength is of the order of millimeters. Therefore, the change in \(\phi(t)\) does not exceed \(\pi\) and the phase entanglement phenomenon will not occur. In actual communication scenarios, the sea surface fluctuates randomly, and the significant wave height is related to the sea surface wind speed. It is assumed that the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Quantity** & **Value** \\ \hline \(p_{i}\) & Sound pressure level & 170 dB \\ \(f\) & Frequency of the underwater source & 100 Hz \\ c & Sound speed in the water & 1450 m/s \\ \(\rho\) & Water density & 1000 Kg \(\cdot\) m\({}^{-3}\) \\ \(g\) & Gravitational acceleration & 9.8 N \(\cdot\) kg\({}^{-1}\) \\ \(\sigma^{\prime}\) & Surface tension coefficient & 0.07275 N \(\cdot\) m\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters of acoustic water surface waves. fluctuation of the sea surface, denoted as \(h(t)\), is linearly superimposed with acoustic water surface waves, denoted as \(y(t)\). In anticipated application scenarios, drones will operate close to the water surface, resulting in a limited illumination range. Consequently, the fluctuation of the sea surface \(h(t)\) focuses solely on temporal variations, disregarding the spatial variations in the model.