## 3 Numerical Simulation Results and Analysis  
### Failure Characteristics of the Double U-Shaped Caverns under Horizontally Incident Wave  
As a plane P-wave is incident from the left side of the model horizontally, there are certain differences of the wave propagation process around caverns 1 and 2, leading to distinct variations in their stress evolution processes. When the cavern clearance \(d\) is 2.5 m and the stress wave amplitude \(P_{d}\) is 15 MPa, the evolution processes of the tangential stress at different positions of the two caverns are illustrated in Figure 3. When the stress wave propagates to the caverns, dynamic compressive stress concentrations occur at both  
Figure 1: The model of double U-shaped caverns under horizontally incident wave. Figure 2: The layout of measurement circles around the two caverns. the crown and floor areas of both caverns. The peak tangential stresses at the crowns of caverns 1 and 2 are 60.0 and 57.4 MPa, respectively, while the peaks at the floor are 39.8 and 32.0 MPa, respectively. By comparing, the dynamic stress concentration at the roof and floor of the incident side cavern (cavern 1) is more significant than that at the non-incident side cavern (cavern 2). Furthermore, the incoming stress wave results in dynamic tensile stress concentrations at the sidewalls. However, due to the existence of static compressive stress induced by in situ stress, the sidewall areas remain under compression. As shown in Figure 3b, the stress wave diffraction leads to a period of dynamic compressive stress concentration at the B2 position before the occurrence of dynamic tensile stress concentration. Figures 4 and 5 depict the evolution processes of strain energy and kinetic energy at different positions of the two caverns under dynamic disturbances, respectively. Comparing Figure 3 with Figure 4, it can be found that the evolution pattern of strain energy is similar to the evolution pattern of tangential stress at the same position. The strain energy accumulates at the cavern crown and floor, and the peak strain energies at the crown and floor areas of cavern 1 are higher than those at cavern 2. Comparing the peak value of kinetic energy at different positions of the two caverns, it is found that the maximum peak kinetic energy occurs at position B1, and the corresponding value is 2.8 kJ, significantly higher than the peak kinetic energy at position B2. This indicates that the presence of the incident side cavern weakens the reflection of stress waves on the facing-wave side of the non-incident side cavern. For the back-wave side of the caverns, the peak kinetic energy at position C1 is also significantly higher than that at position C2, suggesting that the presence of cavern 2 intensifies the particle vibration of the surrounding rock on the incident side cavern's back-wave side. Figure 3: The evolution process of tangential stress at different positions of the caverns under horizontally incident stress wave: (**a**) midpoints of crown; (**b**) midpoints of facing-wave sidewall; (**c**) midpoints of back-wave sidewall; (**d**) midpoints of floor. Figure 4: The evolution process of strain energy at different positions of the caverns under horizontally incident stress wave: (**a**) midpoints of crown; (**b**) midpoints of facing-wave sidewall; (**c**) midpoints of back-wave sidewall; (**d**) midpoints of floor. Figure 5: The evolution process of kinetic energy at different positions of the caverns under horizontally incident stress wave: (**a**) midpoints of crown; (**b**) midpoints of facing-wave sidewall; (**c**) midpoints of back-wave sidewall; (**d**) midpoints of floor. Previous studies have indicated that cavern clearance is a critical factor for the stress distribution in surrounding rock. Three cavern clearances of \(d=\) 1.25, 2.50, and 3.75 m were specified to explore the influence of the clearance between caverns on the stress field of surrounding rock, and the corresponding ratios of the cavern clearance to cavern width (\(d/w_{1}\)) equal to 0.5, 1.0, and 1.5, respectively.