## 2 Underwater Sound Source Excitation and Detection Principle
### Detection Principle  
We choose an FMCW radar to measure the WSAW, as such radars have unique advantages that are not available in other radar types. Those are:  
High precision: Since the micro-amplitude changes on the water surface are in the order of micrometers, a terahertz-level bandwidth is required to directly measure the distance changes caused by the WSAW, which is unrealistic for a pulsed radar design requirement. Therefore, by measuring the phase change of the beat signal after mixing and filtering the received and the transmitted signals, we estimate the WSAW's wave height change. The phase change \(\phi(t)\) of the beat signal can be expressed as:  
\[\phi(t)=\frac{4\pi(h+y)}{\lambda_{\mathrm{r}}} \tag{3}\]  
where \(h\) is the distance between the radar and the calm water surface, \(y\) is the wave height of the WSAW, and \(\lambda_{\mathrm{r}}\) is the wavelength of the radar signal.  
From Equation (3), the shorter the wavelength, e.g., millimeter wave or terahertz frequency band, the greater the phase change caused, which is conducive to achieving high-precision phase estimation. However, a too-short wavelength will impose rapid phase wrapping.  
Being robust against interference: The WSAW information is mainly reflected in the phase change of the FMCW radar's beat signal, which is different from the amplitude-modulated signal and is less affected by noise.  
Multi-objective separation: An FMCW radar can identify reflective targets at different distances. Indeed, the radar transmits signals to measure the WSAW and can simply separate the WSAW signals from the other interference signals in the environment through different distance bins, extract specific distance bins separately, and analyze the WSAW parameters.  
The FMCW expression in the effective interval \(T_{\mathrm{e}}\in\left[-\frac{T}{2},\frac{T}{2}\right]\) is:  
\[S_{\mathrm{T}}(t)=A_{0}\mathrm{e}^{2\pi(ft+\frac{\mu^{2}}{2})+\varphi_{0}},t \in T_{\mathrm{e}} \tag{4}\]  
where \(A_{0}\) denote the amplitude of the transmitted signal, \(f_{0}\) is the center frequency, \(t\) is the fast-time, \(\mu=B_{\mathrm{r}}/T_{\mathrm{r}}\) is the chirp rate, \(B_{\mathrm{r}}\) is the bandwidth, \(T_{\mathrm{r}}\) is the pulse width, and \(\varphi_{0}\) is the initial phase.  
The radar illuminates the water surface, generating WSAW due to the excitation of the underwater sound source. Since the echo delay of the received signal is \(\tau(t)=(h+y)/c_{\mathrm{r}}\), (\(c_{\mathrm{r}}\) is the light speed), then the received signal can be expressed as:  
\[S_{\mathrm{R}}(t)=K_{\mathrm{r}}S_{\mathrm{T}}(t-\tau(t)) \tag{5}\]  
where \(K_{\mathrm{r}}\) is related to the target's ability to reflect electromagnetic waves and the signal loss in the propagation process.  
The received and the transmitted signals are mixed to obtain the beat signal. Considering the actual situation, the vibration speed of the WSAW is much lower than the speed of light. Therefore, we can obtain mixed signals,  
\[S_{\text{b,up}}(t)=\frac{1}{2}K_{\text{r}}A_{0}{}^{2}\text{e}^{\left[2\pi\left[ \left(\frac{2\pi(k+y)}{\alpha_{\text{r}}}\right)t+\frac{4\pi\theta_{\text{b}}}{ \lambda_{\text{r}}}+\frac{4\pi y}{\lambda_{\text{r}}}\right],t\in\left[-\frac{T }{2},\frac{T}{2}\right] \tag{6}\]  
The wave height change \(y\) of the WSAW is according to the phase change \(\Delta\varphi=4\pi y/\lambda_{\text{r}}\), and is also directly related to the underwater speaker frequency. Finally, the frequency is obtained through spectral estimation.