# Numerical Simulation of Seismic-Wave Propagation in Specific Layered Geological Structures
## 2 Seismic Event Conditions
### Wave Propagation  
The attenuation characteristics of seismic energy are influenced by the differing rates at which various materials absorb and dissipate seismic energy. At interfaces between different geological layers, wave reflection, and refraction occur, affecting the amplitude of the waves and the propagation paths.  
When seismic waves propagate below the Earth's surface, they approximate spherically spreading waves. The wavefronts of seismic waves can be considered concentric spherical surfaces. Due to the spherical symmetry of medium motion in spherical coordinates (\(r\), \(\theta\), \(\varphi\)), only the radial displacement component \(u(r,t)\) is considered.  
The radial displacement component \(u(r,t)\) is non-zero, and all state parameters are functions of the radial distance \(r\) and time \(t\), independent of \(\theta\) and \(\varphi\). It can be demonstrated that [43; 44]  
\[\varepsilon_{r}(r,t)=\frac{\partial u(r,t)}{\partial r},v(r,t)=\frac{\partial u (r,t)}{\partial t}, \tag{2}\]  
\[\varepsilon_{\theta}(r,t)=\varepsilon_{\varphi}(r,t)=\frac{u(r,t)}{r}, \tag{3}\]  
\[\sigma_{r}=\sigma_{r}(r,t),\sigma_{\theta}(r,t)=\sigma_{\varphi}(r,t). \tag{4}\]  
Here, \(\varepsilon_{r},\varepsilon_{\theta}\), and \(\varepsilon_{\varphi}\) denote strains along the \(r\), \(\theta\), and \(\varphi\) directions, respectively, and \(\sigma_{r}\), \(\sigma_{\theta}\), and \(\sigma_{\varphi}\) represent the corresponding stresses. The function \(u(r,t)\) denotes radial displacement related to propagation distance and time, while \(v(r,t)\) denotes the radial velocity function. Assuming \(\sigma_{\theta}=\sigma_{\varphi}\), we derive the following:  
\[\frac{\partial\sigma_{r}}{\partial r}+\frac{2(\sigma_{r}-\sigma_{\theta})}{r} =\rho_{0}\frac{\partial v}{\partial t}. \tag{5}\]  
For elastic waves, as they propagate to an interface, the incident stress \(\sigma_{I}\) partially transmits to the next interface as \(\sigma_{T}\) and partially reflects bs \(\sigma_{R}\). In accordance with the force balance equation [45],  
\[\sigma_{I}+\sigma_{R}=\sigma_{T} \tag{6}\]  
Moreover, the continuity of motion at the aforementioned interface indicates that  
\[v_{I}+v_{R}=v_{T} \tag{7}\]  
Considering the relationship between particle velocity and stress, for example, when a seismic wave propagates from basalt to granite [46],  
\[\sigma_{I}=v_{I}\rho_{\text{basalt}}C_{\text{basalt}}, \tag{8}\]  
\[\sigma_{R}=v_{R}\rho_{\text{basalt}}C_{\text{basalt}}, \tag{9}\]  
\[\sigma_{T}=v_{I}\rho_{\text{granite}}C_{\text{granite}}, \tag{10}\]  
where \(\rho_{\text{basalt}}\) and \(\rho_{\text{granite}}\) are the densities of basalt and granite, respectively, and \(C_{\text{basalt}}\) and \(C_{\text{granite}}\) are the P-wave velocities in these media. From these equations, it can be determined that  
\[\frac{\sigma_{T}}{\sigma_{I}}=\frac{2\rho_{\text{granite}}C_{\text{granite}}}{ \rho_{\text{granite}}C_{\text{granite}}+\rho_{\text{basalt}}C_{\text{basalt}}}, \tag{11}\]  
\[\frac{\sigma_{R}}{\sigma_{I}}=\frac{\rho_{\text{granite}}C_{\text{granite}}- \rho_{\text{basalt}}C_{\text{basalt}}}{\rho_{\text{granite}}C_{\text{granite}}+ \rho_{\text{basalt}}C_{\text{basalt}}}. \tag{12}\]  
The combination of Equations (5), (11) and (12) clarifies the propagation pattern of seismic waves. In practice, the intricate structural characteristics of geological conditions necessitate the use of numerical simulation to ascertain the propagation behavior of stress waves within such media.