## 4 Feature Extraction of Dynamic Vortex Acoustic Signals
### Unified Motion Vortex
The vortex field does not undergo diffusion effects, and its position moves uniformly with increased time reversal counts. In this study, the impact of the vortex center movement speed \(V_{w}\) on the phase extraction effect is investigated. For \(V_{w}\), the vortex center position for each time reversal \(w_{i}\) is given by \(w_{i}=V_{w}\times(i-1)\Delta t\) 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 as a reference, the relative error between the phase jump with reversals for six times at the movement speed \(V_{w}\) and the steady-state vortex field phase jump is calculated to obtain an error curve (red). Different vortex center movement speeds, i.e., \(V_{w}\) = 1.38 \(\times\) 10\({}^{-3}\) m/s, 1.63 \(\times\) 10\({}^{-3}\) m/s, 1.88 \(\times\) 10\({}^{-3}\) m/s, 2.13 \(\times\) 10\({}^{-3}\) m/s, and 2.38 \(\times\) 10\({}^{-3}\) m/s, are tested, yielding five phase jump curves and error curves, as shown in Figure 7.  
As shown in Figure 7, when \(V_{w}\) is less than 2.38 \(\times\) 10\({}^{-3}\) m/s and the number of time reversal \(N\) is less than 6, TRM is effective in the extraction of the characteristics of the vortex acoustic signals from the moving vortex center, which allows the dynamic characteristics of the vortex center to be disregarded. Figure 6: Phase characteristics of vortex acoustic signals at different vortex radius diffusion speeds. Both vortex position and radius change. Considering the diffusion effects in the vortex field and the uniform motion of the vortex center, the phase characteristics of the vortex acoustic signals are extracted after six time reversals. The vortex radius diffusion speeds \(V_{r}\) are set to 0, \(1.25\times 10^{-4}\) 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, and the vortex center moving speeds \(V_{w}\) are set to 0, \(1.38\times 10^{-3}\) m/s, \(1.63\times 10^{-3}\) m/s, \(1.88\times 10^{-3}\) m/s, \(2.13\times 10^{-3}\) m/s, and \(2.38\times 10^{-3}\) m/s, respectively. The initial time-reversal vortex acoustic signal is received with minimal error when the acoustic wave passes through a steady-state vortex field. The comparison between relative errors of the phase jump measurements of vortex acoustic signals from time reversals \(N=2\) to 6 and those from a steady-state vortex field are shown in Figure 8. This figure is plotted by using the built-in 'pcolor' function in Matlab 2019b, where the horizontal axis represents the vortex radius diffusion speed and the vertical axis stands for the vortex center movement speed. As shown in Figure 8, the effectiveness of extracting phase characteristics from vortex acoustic signals decreases with the increase in the number of time reversals. Initially, TRM  
Figure 8: Uniform motion vortex field phase jump error heatmap. (**a**) \(N=2\); (**b**) \(N=3\); (**c**) \(N=4\); (**d**) \(N=5\); (**e**) \(N=6\). Figure 7: Phase characteristics of vortex acoustic signals at different vortex center movement speeds. is highly effective. When the number of time reversals, denoted as \(N\), is less than 5, TRM can successfully extract the phase characteristics of vortex acoustic signals. This remains true even under challenging conditions where the vortex field is moving at its maximum center speed and expanding at its maximum radius diffusion speed. However, the situation changes significantly from the fifth time reversal onward. When the vortex moving speed \(V_{w}\) exceeds \(2.2\times\)\(10^{-3}\) m/s, and the radius diffusion speed \(V_{r}\) exceeds \(2.5\times 10^{-3}\) m/s, the effectiveness of TRM in extracting the phase characteristics of vortex acoustic signals diminishes. This reduction in effectiveness indicates that the dynamic changes in the uniformly diffusing motion of the vortex field introduce complexities that TRM cannot adequately address beyond a certain number of reversals.