## 3 Vibration Wave Splitting and Loading
### Longitudinal and Transverse Wave Propagation and Loading Modes  
The vibration waves caused by pressure bumping are mainly P-waves (longitudinal waves) and S-waves (transverse waves). Because of the different propagation modes of the two waves, they need to be applied according to their respective vibration propagation modes [25]. The direction of particle vibration in P-waves aligns with their propagation direction, while the vibration direction of S-waves is perpendicular to their propagation direction. In underground mining operations, a roof collapse can precipitate the abrupt release of elastic energy stored within the rock mass, thereby resulting in considerable dynamic hazards. From a mechanical perspective, pressure bumping functions as a comprehensive representation of external loads, internal structures, structural characteristics, and the physical and mechanical properties of materials. The formation of pressure bumping is a complex process that involves multiple interdisciplinary fields, such as geology, mining, geophysics, rock mechanics, and nonlinear dynamics. Furthermore, it demonstrates the unique characteristics of temporal and spatial evolution. The geographical positioning of the mine's microseismic monitoring system indicates that sources located near the roadway are typically modeled as point sources. However, the inherent characteristics of split vibration waves, which are defined by time-velocity curves, render direct application to grid nodes impractical. To address this challenge, a source sphere is proposed as a simplified representation of the point source. In this model, the split P-wave is oriented in the normal direction relative to the surface of the sphere, whereas the split S-wave is oriented in the tangential direction [25; 26]. To investigate the influence of sphere size on the propagation of vibration waves, a linear elastic model is utilized within the FLAC\({}^{\text{3D}}\) software (7.00.126). A spherical boundary with radii of 0.2 m and 2 m is established as the source, while a viscous boundary is employed to absorb the reflected vibration wave, as depicted in Figure 8. Figure 9 presents the observed particle vibration velocities associated with different source sizes. The mechanical parameters utilized in this simulation consist of a density of 2580 kg/m\({}^{3}\), a bulk modulus of 9.73 \(\times\) 10\({}^{9}\) Pa, and a shear modulus of 5.84 \(\times\) 10\({}^{9}\) Pa. When a vibrational wave is applied to a smaller sphere, the energy released from the inner surface is more concentrated, resulting in a relatively rapid dissipation of energy during the propagation process. The larger sphere can disperse energy more gently, and the monitoring points receive a higher amplitude at the same distance. As depicted in Figure 9, the waveforms generated by different sources demonstrate a general trend of similarity; however, the vibration amplitude produced by the 2 m sphere is significantly greater. This enhanced amplitude is beneficial for investigating the response characteristics of the surrounding rock, which is anchored to the roadway, in reaction to vibrational waves. Therefore, the 2 m sphere has been chosen as the source point for the subsequent numerical analysis.  
Figure 8: Numerical model.