## 4 Discussion  
It was found that not only the displacement but also the stress at the front and rear interfaces of the filled joint were discontinuous when loaded by the stress wave [20; 45]. This attributes to the inertial effect of the mass of the filled medium. Both the stress and the displacement at the interface between the rock and filled material are assumed to be continuous in the current study. It can be shown that the sandwiched element surrounded by two interfaces as shown in Figure 1b can characterize the stress and displacement discontinuities at the filled joint according to Equation (34). Therefore, our model can simulate the discontinuous response of stress and the displacement at the filled joint, which has similar physics to DSDM by [20]. Additionally, our model can automatically consider the wave reflection inside the filled layer while DSDM loses the capability. However, the influence  
Figure 11: Variation of the transmission coefficient (**a**) \(T_{\text{P}\rightarrow\text{P}}\) and (**b**) \(T_{\text{P}\rightarrow\text{sv}}\) versus dimensionless joint spacing \(\xi\) after the P wave propagates through four parallel joints considering different water contents. The incident P wave has a central frequency of 42.5 Hz and the incident angle of 20\({}^{\circ}\). of the wave reflection is prominent and cannot be neglected when the ratio of the filled thickness to the wavelength is large enough. As described in Section 3.1, our analytical model was suitable to characterize the seismic response of the filled joint considering different water contents. Properties of the filled joints can be simply described by several common parameters such as the filled thickness, the density, the wave velocity, and the quality factors of the filled medium. These parameters have definite physical meanings and can be obtained relatively more readily compared with those of DSDM [20], i.e., the viscous stiffness. The water content has a significant effect on the dynamic mechanical characteristics of the filled joint. As shown in Table 1, the increase in the water content can increase the density but reduce the wave velocity and quality factor of the filled medium. As a result, the transmission of the stress wave was significantly influenced. Especially, the variation of the quality factor stands for the variation of the viscosity. As derived in Equations (7) and (8), the viscoelastic wave equations have complex forms composed of a real part and imaginary part. The increase in the imaginary part has double effects, not only can it increase the equivalent dynamic modulus to let more wave energy pass through the filled joint (positive effect), but it can also cause energy loss (negative effect) [21]. For \(T_{\rm P\to p}\), the negative effect first plays the domain role and the positive effect plays the domain role in turn when the quality factor increases. For \(R_{\rm P\to p}\), \(R_{\rm P\to sv}\) and \(T_{\rm P\to sv}\), the positive effect always plays the domain role. When the quality factor is large, the viscous effect of the filled medium is not prominent. In this case, the reflection-transmission coefficients are independent of the quality factor. We can now understand the variation of the reflection-transmission coefficients versus the quality factor as shown in Figure 8 as well as the energy loss with quality factor in Figure 9. The transmitted waves after propagating through multiple parallel joints are the superposed waves consisting of the first-arriving wave and the later-arriving waves caused by multiple reflections. When the water content increases, the wave reflection increases while the wave transmission decreases (see Figures 5-7). According to this, the multiple reflections at the joint should increase with increasing water content. However, the multiple reflection decreases with increasing water content instead (see Figures 10 and 11).