## 2 Validation of FLAC Modeling on Wave Transmission across Rock Joints
### Normally Incident P Wave Transmission through a Single Joint  
The FLAC model for P wave transmission through a single joint is illustrated in Figure 1. The origin of the X-Z coordinates is located at the left bottom of the model. The model size is 500 m \(\times\) 500 m, and the joint is horizontally located at Z = 250 m. Brick elements is adopted as the mesh type, and all the mesh size is 1 m. Non-reflection viscous boundaries are placed at the bottom and upper boundaries to avoid wave reflections from the artificial boundaries. The left and right boundaries are fixed in X-direction. And a half-cycle sinusoidal incident stress pulse with amplitude 1.662276 \(\times\) 10\({}^{7}\) Pa (to make sure that the velocity amplitude is 1.0 m/s) and frequency 50 Hz is normally applied at the bottom boundary.  
The rock mass is assumed to be purely elastic, and material damping is not included in this paper for two reasons [15]. First, when a wave propagates across jointed rock mass, the wave attenuation is mainly due to joints. Secondly, this study is focused on effects of rock joints on wave attenuation and thus the attenuation from material is ignored. The joint model adopts the Coulomb slip model. In order to ensure that the joint is linear and elastic, high values are assigned to the cohesion and tension of the joint. In the numerical simulation, cohesion is 10 GPa, the friction angle is 30\({}^{\circ}\), and the tension strength is 10\({}^{5}\) GPa. The physico-mechanical properties of a rock matrix are listed in Table 1[20].  
\begin{table}
\begin{tabular}{c c c} \hline
**Density (kg/m\({}^{3}\))** & **Bulk Modulus (GPa)** & **Shear Modulus (GPa)** \\ \hline
2570 & 13.44 & 10.08 \\ \hline \end{tabular}
\end{table}
Table 1: Physico-mechanical properties of rock mass.  
Figure 1: Fast Lagrangian Analysis of Continua (FLAC) model for P wave transmission through a single joint.  
In this section, only the transmission coefficients are studied. The FLAC results and analytical solutions are compared in Figure 2. In this model, the incident and transmitted measure point are located at (X, Z) = (250, 50) and (250, 450) of Figure 1, respectively. To focus on the study of the relation between the transmission coefficient and joints stiffness, the normal joint stiffness \(k_{n}\) is equal to the shear joint stiffness \(k_{s}\); for convenience, both of them are expressed as \(k\). The X-coordinate is normalized joint stiffness \(K\), which is defined as:  
\[K=k/\omega z \tag{1}\]  
where \(z\) is given by:  
\[z=\rho C \tag{2}\]  
where \(k\) is the normal or shear joint stiffness, \(\omega\) is the angular frequency of the incident wave that is the product of wave frequency f and \(2\pi,z\) is the wave impedance, \(\rho\) is the rock density, and \(C\) refers to \(C_{P}\) or \(C_{S}\), which is the velocity of the P or Sv wave. The vertical coordinate is the magnitude of transmission coefficient across a single joint for P wave incidence \(T_{1P}\), which is defined as [18]:  
\[T_{1P}=\max\mid v_{TP}\mid/\max\mid v_{IP}\mid \tag{3}\]  
where \(v_{IP}\) is the particle velocity of the incident P wave and \(v_{TP}\) is the particle velocity of the transmitted P wave. From Figure 2, it can be found that \(T_{1P}\) from the FLAC results agrees well with the analytical solutions. Therefore, FLAC is applicable to study P wave transmission.