## 4 Feature Extraction of Dynamic Vortex Acoustic Signals
### Unified Motion Vortex  
The term \"unified motion vortex\" refers to the unified motion of the vortex center and the unified radial expansion of the vortex. The unified motion of the vortex center means that the position of the vortex center moves continuously in a certain direction at a constant speed. The unified radial expansion of the vortex means that the radius of the vortex expands continuously at a constant speed. Vortex center movement speed \(V_{w}\) and vortex radius diffusion speed \(V_{r}\) are introduced to better describe the dynamics and acoustic characteristics of a vortex field. In unified motion, the vortex center moves at speed \(V_{w}\) in the positive direction of the y-axis uniformly and the vortex radius diffuses at speed \(V_{r}\). To reasonably extract the phase characteristics of acoustic waves passing through a vortex field in uniform motion and diffusion, the extraction time interval \(t\) for the moving vortex field phase characteristics is specified. Based on \(t\) and the moving speed, the vortex center coordinates and the vortex radius at each time reversal can be determined. Assuming that the vortex center remains at the center of the transmit-receive array, the time interval \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Time Reversal Count** & **Numerical Simulation/rad** & **Experimental Fit/rad** & **Experimental Data/rad** & **Error between Experimental Fit and Numerical Simulation/\%** \\ \hline
1 & 0.1682 & 0.1814 & 0.1395 & 7.85\% \\ \hline
2 & 0.355 & 0.3485 & 0.3168 & \(-1.83\%\) \\ \hline
3 & 0.5032 & 0.5156 & 0.6370 & 2.46\% \\ \hline
4 & 0.710 & 0.6827 & 0.7027 & \(-3.84\%\) \\ \hline
5 & 0.8482 & 0.8499 & 0.7821 & 0.19\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Phase jump measurement error analysis. Figure 5: Dynamic vortex acoustic signal characteristic extraction. for each time reversal processing (dynamic vortex change interval) is approximately equal to the sound signal propagation time between arrays. At a given array spacing, the dynamic vortex change interval in this example is calculated to be \(\Delta t=8\)s.  
1. Vortex position remains unchanged, while vortex radius diffuses. If the position of the vortex remains unchanged, the vortex field is under diffusion effects, that is, the characteristic radius of the vortex increases with each time reversal. The impact of the vortex radius diffusion speed \(V_{r}\) on the phase extraction effect is investigated in this study. For \(V_{r}\), the vortex field radius for each time reversal \(r_{i}\) is given by \(r_{i}~{}=~{}\mathrm{V_{r}}\times(i-1)\Delta t+r_{0}\), where \(i=1\), 2... 6. After extracting the vortex acoustic signal for six times, the phase jump curve (blue) is obtained. Taking the phase jump of a steady-state vortex field (\(V_{w}=V_{r}=0\)) as a reference, the relative error between the diffusion vortex field phase jump under six time reversals and the steady-state vortex field phase jump is calculated to form an error curve (red). Different vortex radius diffusion speeds, i.e., \(V_{r}\) = 1.25 \(\times\) 10\({}^{-4}\) m/s, 6.25 \(\times\) 10\({}^{-4}\) m/s, 1.25 \(\times\) 10\({}^{-3}\) m/s, 1.88 \(\times\) 10\({}^{-3}\) m/s, 2.50 \(\times\) 10\({}^{-3}\) m/s, and 3.13 \(\times\) 10\({}^{-3}\) m/s, are tested, yielding six phase jump curves and error curves, as shown in Figure 6.  
As assumed in this paper, if the relative error of the phase jump is less than 20%, the phase amplification based on time reversal is considered to satisfy the requirements of linear phase amplification, and the difference from the steady-state vortex condition is not significant. As shown in Figure 6, when \(V_{r}\) is less than 2.50 \(\times\) 10\({}^{-3}\) m/s and the number of time reversals \(N\) is less than 6, TRM can effectively extract the vortex acoustic signal characteristics, and the dynamic characteristics of the diffusing vortex is ignorable. 2. Vortex position changes, while vortex radius remains unchanged.