# Mechanisms and Models of Attenuation of Shock Waves through Rock Formations
## 4 Discussion
### Model Establishment  
As the location of the impact shock source is usually the location of the fracture in the hard roof, Figure 12 shows the transmission of impact energy through the rock formation to the roadway when the impact dynamic load occurs. Assuming that the impact source is located in the rock formation above the roof of the roadway, the transmission of the shock wave from the impact source to the roof of the roadway requires passing through \(n\) layers of rock in sequence, where the thickness of the rock formation is \(h_{1}\), \(h_{2}\) to \(h_{nt}\), in that order. Figure 11: A model of shock wave transmission through rock formations. Compared to large structures such as working faces and chambers, the size and projected area of the roadway section is very small. Simplifying the roadway section to a circular hole, the radial and tangential stresses (\(\sigma_{r}\) and \(\sigma_{\theta}\)) at the center of the circle for a small hole of radius R in a uniform stress field, according to the theory of elastodynamics, are:  
\[\sigma_{r} =\gamma h\Big{(}1-\frac{r^{2}}{R^{2}}\Big{)} \tag{2}\] \[\sigma_{\theta} =\gamma h\Big{(}1+\frac{r^{2}}{R^{2}}\Big{)}\]  
where \(r\) is the distance from the center of the circle, and the total distance \(h=\sum\limits_{i=1}^{n}h_{i}\) from the shock source to the roadway where the mine tremor occurs is compared with the radius \(r\) of the roadway excavated to satisfy \(h\gg r\). Usually, these two physical quantities differ by one to two orders of magnitude. At the same time, the shock wave will fracture and break the rock layer in the process of transmission, and the mine seismic energy will be continuously attenuated; here, the attenuation coefficient of each layer of rock layer is set as \(\eta_{n}\). To simplify the study, here, we set the overall lithology of the rock formation as a homogeneous, isotropic rock, while ignoring creep and the viscous characteristics of the rock, and simplify it to a planar model for analytical solution. Figure 12: Mechanical modeling of shock wave attenuation through layers. Figure 13 shows the simplified form of the shock wave penetration decay model. The shock energy wave penetrates the top plate of n layers in turn to reach the location of the roadway, and acts on the roadway envelope to produce the damage effect. Here, we set the impact intensity at the source to be \(\sigma_{\rm d}\). Assuming there is no delamination between rock formations, there is no stress or energy loss when the dynamic wave passes between the formations. When the shock wave penetrates the first formation of rock, according to the propagation characteristics and energy attenuation characteristics in the elastomeric medium, it is known that the intensity of the shock wave arriving at the second formation is as follows:  
\[\sigma_{A}=\sigma_{\rm d}\cdot{h_{1}}^{-\eta_{1}} \tag{3}\]  
Therefore, it can be recursively obtained that when the shock wave reaches the roadway, the impact strength of the surrounding rock is as follows:  
\[\sigma_{h_{n}}=\sigma_{A}\cdot{\prod_{i=1}^{n}{h_{i}}^{-\eta_{i}}} \tag{4}\]  
When the shock wave is transmitted to the vicinity of the roadway support system, the shock wave, the surrounding rock endowment stress and the roadway support system interact with each other. Assuming that the shock source is located directly above the roadway roof, the resulting shock pressure will mainly act on the roadway roof in the form of radial stress. As the above research shows, the support strength of the anchor cable combined with the strength of the roof resists the damage of the shock wave. The shock transferred to the support system, the surrounding rock's stresses superimposed on each other and the roadway roof's own impact strength and support strength coupling balance; if the roadway can resist the impact load, then the following conditions will be met:  
\[\sigma_{h_{n}}+\sigma_{r}\leq\sigma_{s}+\sigma_{c} \tag{5}\]  
Figure 13: Simplification of the model.