# Mechanisms and Models of Attenuation of Shock Waves through Rock Formations
## 3 Numerical Simulation Study
### Results  
The location of the impact shock source is above the recovery roadway. During this period, the shock wave is transmitted from the impact shock source through multiple rock formations to the roadway. The dynamic shock stress and the energy are significantly decaying. The destructiveness of the shock waves on the roadway is mainly caused by the impact dynamic stresses accumulated on the roadway static stresses, which overpower the strength of the surrounding rock. The transmission process and attenuation characteristics of the shock waves are shown in Figure 6.  
As shown in Figure 6a-d, the peak stress decreased from 125 MPa to 40 MPa after the shock wave passed through the first mudstone layer. It decreased to 15 MPa after passing through the sandstone layer. After passing through the mudstone and roof coal, the vertical stresses were almost constant. For this reason, the stress characteristics of the different rock formations were recorded separately at different rock positions.  
Figure 7 shows the curve of Y displacement at the monitoring points A to G. The Y displacement at the monitoring point from the source of the shock to the roadway envelope increases. The rate of decrease in the center of the oscillation increases for each curve. The velocity of the oscillation is the fastest at the source, but the change in displacement is the smallest.  
Figure 5: Roadway support parameters.  
Figure 6: \(Cont.\)Figure 6: Block YY effective stress during dynamic simulation. (**a**–**f**) Represents a model of 2900, 3000, 3100, 3200, 3300, 3400 time steps, respectively.  
Figure 7: Y\(-\)displacement curve of monitoring point.  
Figure 9 shows the stress decay characteristics of the impact stress in the process of conduction. As seen in the process of stress transfer, the decay rate constantly slows down and tends to disappear. The decay rate during conduction is simply calculated using the following equation:  
\[\eta=-log_{h_{i}}(\frac{\sigma_{h_{i}}}{\sigma_{h_{i-1}}}) \tag{1}\]  
where \(\eta\) is the attenuation rate of the stress wave and \(\sigma_{h_{i}}\) is the vertical stress of the \(i\)th rock formation. The following conclusions can be drawn after carrying out calculations:  
1. There is a decreasing trend in the rate of attenuation during stress wave conduction.
2. There is a large difference in the attenuation coefficients during the passage through rock formations of the same properties.
3. The attenuation coefficients of the sandstone closer to the shock source are different from the second mudstone layer, which shows that different rock properties attenuate the shock waves to different degrees.  
By the time the stress wave is transmitted to the second sandstone layer, it has attenuated to less than 10 MPa, at which point the rock layer and the roadway support system can withstand the damage caused by the stress shock wave.  
Figure 10: _Cont._  
Figure 10 shows the stress\(-\)YY characteristics of the shock wave passing through roadway. The figure shows the stress distribution around the roadway with the axial force characteristics of the anchor cable when the shock wave reaches the roof of the roadway, the two ribs of the roadway and the floor of the roadway, respectively. For the no-support model, during this process, the shock wave always accumulates with the static load on both ribs of the roadway, accelerating roadway damage. Under the influence of the support, the integrity of the roadway envelope is better, and no large deformations occur when the shock wave arrives, slowing the damage to the roadway. In brief:  
1. The axial force of the anchor cable remains almost unchanged during this process. It does not proactively change its own axial force to resist damage from the shock wave.
2. The loose areas of the roadway in the no-support conditions are much greater than in supported structures.
3. The support structures will reinforce the roadway, making it resist shock waves as intact rock would. They change the stress distribution around the roadway.