## 3 Numerical Simulation of Shockwave Soil Loosening
### Results and Observations  
Utilizing the configuration shown in Figure 4, a numerical analysis of the shock phenomenon at the output tube was conducted using computational fluid dynamics (CFD) simulations. The study examined the shockwave behavior at injection holes A, B, and C, as well as the shockwave dynamics within the pipe, from upstream to downstream.  
Figure 5 illustrates the propagation of shockwaves at 0.18 ms, 0.4 ms, 0.6 ms, 1 ms, and 2 ms after generation. At 0.18 ms, a shockwave was emitted from hole A, exiting the nozzle and propagating into the soil. In its initial phase, the shockwave formed a near-spherical shape with a uniform pressure distribution. By 0.4 ms, a second shockwave was emitted from hole B, with a time interval of 0.18 ms between emissions from holes A and B. At 0.6 ms, hole C emitted another shockwave, with a 0.2 ms interval between holes B and C--slightly longer than the interval between holes A and B (by 0.02 ms). This delay was attributed to energy attenuation as the shockwave propagated through the output tube. At 0.8 ms, a reflected shockwave traveled back toward hole C and interacted with the original shockwave emitted from that location, inducing a resonance phenomenon. Similar resonance effects were observed at 1.0 ms and 2.0 ms at holes B and A, respectively. Since holes A, B, and C were evenly spaced, it was noted that as the shockwave traveled further downstream, its propagation speed gradually decreased. Additionally, when the reflected shockwave moved upstream within the output tube, resonance effects were observed.  
Figure 5: Propagation of shockwaves at different times.  
At 0.18 ms, when a low-intensity shockwave was emitted from hole A, a low-pressure bubble was observed at the exit of hole A. The presence of a notch on the surface of the shockwave output tube led to the formation of an oblique shockwave within hole A. As the shockwave continued propagating, similar phenomena were detected at holes B and C, as depicted in Figures 6 and 7. By 0.8 ms, the reflected shockwave traveling upstream interacted with the shockwave within the output tube at hole C, triggering resonance effects. This interaction resulted in shockwave superposition, leading to periodic fluctuations in high- and low-pressure jet flows, as shown in Figure 6.  
The further analysis of Mach number variations (Figure 7) revealed a characteristic nozzle-interacting shockwave jet. Although the Mach numbers of the shockwave jets from holes A, B, and C exhibited slight differences, the overall pressure distribution remained relatively consistent, as demonstrated in Figure 8. Figure 8 shows the pressure variations over time at the shockwave output holes. Initially, the shockwaves emitted from holes A, B, and C exhibited a pressure of approximately 1.5 atm. However, due to interactions between the reflected and upstream shockwaves, the pressure of the emitted shockwaves eventually increased to 3.4 atm. Furthermore, the interaction region exhibited periodic pressure oscillations, inducing the formation of standing waves or resonance effects.  
Figure 6: Pressure variations at each nozzle exit at different times.  
Figure 7: Mach number variations at each nozzle exit at different times.