## 4 Feature Extraction of Dynamic Vortex Acoustic Signals
### Oscillating Motion Vortex  
In the previous sections, the impact of uniform motion of the vortex center and the diffusion movement of the vortex radius on phase extraction effectiveness was discussed. However, the uniform motion is a relatively simple form of movement; therefore, to explore more complex forms of motion in the vortex field, this section will use a trigonometric function model to simulate the random oscillation phenomenon of the vortex field under the assumption that the oscillation trajectory of the vortex center is a sine curve. The oscillation amplitude \(L_{D}\) is introduced to represent the degree of oscillation of the vortex center. Taking \(L_{D}\) = 0.16 m as an example, the oscillation trajectory of the vortex center is shown in Figure 9, where the y-axis represents the y-coordinate position of the vortex center, and the x-axis refers to the duration of the vortex center's oscillation. The sampling is carried out at a time interval of \(\Delta t=8\)s with the symbol * marking the oscillation positions of the vortex field's core center. To describe the effectiveness of vortex acoustic signal characteristic extraction under different vortex field oscillation amplitudes, \(L_{D}\) is set to 0.04 m, 0.08 m, 0.12 m, 0.16 m, and 0.20 m, respectively. Considering the diffusion of the vortex radius and the oscillatory motion of the vortex center, the phase characteristics of the vortex acoustic signals are extracted after six time  
Figure 9: Oscillation trajectory of the vortex center. reversals with \(L_{D}\) values of 0.04 m, 0.08 m, 0.12 m, 0.16 m, and 0.20 m, and \(V_{r}\) values of 0, \(1.25\times 10^{-3}\) m/s, and \(2.5\times 10^{-3}\) m/s, respectively. Since the initial state of the oscillating vortex field is steady, the relative error of the phase jump measurement of the initial time-reversal vortex acoustic signal is not considered. And its comparison with those from a steady-state vortex field for time reversal counts \(N=2\) to 6 is shown in Figure 10 plotted using the built-in 'pcolor' function in Matlab 2019b, with the y-axis representing the vortex field oscillation amplitude and the x-axis standing for the vortex radius diffusion speed. This comprehensive analysis indicates that when the vortex field is subjected to both oscillatory motion and radius diffusion, the behavior of the oscillation amplitude \(L_{D}\) and the radius diffusion speed \(V_{r}\) changes in a specific manner as the number of time reversals increases. Initially, as the number of time reversals increases, the oscillation amplitude \(L_{D}\) that meets the error criteria for accurately extracting the phase characteristics of vortex acoustic signals tends to decrease. This means that for higher time reversal counts, the allowable oscillation amplitude within which TRM can still effectively extract phase characteristics becomes smaller. Conversely, the vortex radius diffusion speed \(V_{r}\) exhibits a different trend. Initially, \(V_{r}\) decreases, indicating that for fewer time reversals, a lower radius diffusion speed is required to meet the error criteria. However, as the number of time reversals continues to increase, \(V_{r}\) begins to increase. This suggests a more complex interaction between the time reversal process and the dynamic behavior of the vortex field. One possible explanation for these observations is the position of the vortex center due to its oscillatory motion. When the vortex center moves to the edge of the acoustic observation range, it leads to a cancelation effect in the phase jump measurements at the radius of the vortex. This positional shift impacts the ability of TRM to accurately capture the phase characteristics, necessitating adjustments in both \(L_{D}\) and \(V_{r}\) to maintain accuracy. Figure 10: Error heatmap of phase jump measurements for oscillatory motion in the vortex field. (a) \(N=2\); (b) \(N=3\); (c) \(N=4\); (d) \(N=5\); (e) \(N=6\).