## 3 Results
### Experimental Calibration
#### 3.2.2 Case of Multiple Parallel Filled Joints  
When the stress wave propagates though a rock mass containing a set of joints, multiple reflections occur at each joint [33; 34; 43]. In this section, involved filled joints are assumed to have the same properties such as filled thickness, density, wave velocity, quality factor of the filled medium, etc. The rock mass is assumed to contain a set of joints with the identical joint spacing. A cycle of sinusoidal P wave with unit amplitude and frequency of 50 Hz was used to be the incident wave. The start time of the wave signal to oscillate has a delay of around 0.1 s. According to [21], the actual central frequency of the incident wave is 42.5 Hz for one cycle of sinusoidal wave with an input frequency of 50 Hz. It is reasonable to use the central frequency of 42.5 Hz to represent the frequency of the incident wave. The ratio between the joint spacing and wavelength have an important effect on the transmission coefficient because of the multiple reflections at filled joints [21; 33; 34; 35]. Therefore, it is convenient to study the dimensionless joint spacing which is defined as \(\xi=h_{r}/\chi=(h_{r}f)/V_{\mathrm{p}}^{r}\), where \(\chi\) is the P-wave wavelength.  
Figure 10 shows transmitted waveforms (P-wave component) after a P wave obliquely propagates through four parallel filled joints with an identical filled thickness \(h_{\mathrm{f}}\) of 40 mm and joint spacing \(h_{r}\) of 50 m, considering different water contents. It displays that each transmitted wave is made up of several cycles of waveforms arriving at different times induced by multiple reflections at joints. We can clearly see the wave superposition presenting multiple peaks and valleys in turn. As the water content increases, the transmitted wave decreases in the amplitude and the component of the later-arriving wave gradually fades away. When the water content is 15%, it is difficult to observe the later-arriving transmitted wave (after 0.18 s).  
Figure 9: The energy loss \(E_{\mathrm{loss}}\) versus the quality factor \(Q\) corresponding to Figure 8.  
Figure 11 shows that variation of transmission coefficients \(T_{\mathrm{P}\rightarrow\mathrm{P}}\) and \(T_{\mathrm{P}\rightarrow\mathrm{sv}}\) versus \(\xi\) after the P wave obliquely propagates through four parallel joints considering different water contents. It shows that there exists two threshold lines \(\xi_{1}\) and \(\xi_{2}\) dividing each curve into three parts, which is similar to results in published papers [21; 33; 34; 35; 44]. \(T_{\mathrm{P}\rightarrow\mathrm{P}}\) first decreases with increasing \(\xi\) (\(\xi<\xi_{1}\)) and then it decreases instead (\(\xi_{1}<\xi<\xi_{2}\)) and at last it basically keeps invariant (\(\xi>\xi_{2}\)). With the water content increasing, \(T_{\mathrm{P}\rightarrow\mathrm{P}}\) decreases and the position for the first threshold point moves to the left while the second one moves to the right. The increasing part of the curve (\(\xi<\xi_{1}\)) shortens while the decreasing part (\(\xi_{1}<\xi<\xi_{2}\)) widens. \(T_{\mathrm{P}\rightarrow\mathrm{sv}}\) first increases with increasing \(\xi\), then it decreases after achieving a peak in general. When the water content is low, there exist fluctuations on the curve. As the water content becomes higher, these fluctuations gradually fade away and the curve becomes more and more flat. From Figures 10 and 11, it indicates that the effects of multiple reflections at joints on the transmitted wave and transmission coefficient were significantly reduced when the water content increased.  
Figure 10: The incident P wave and transmitted waves (P-wave component) after propagating through four parallel filled joints with an identical spacing of 50 m considering different water contents. The incident P wave has the frequency of 50 Hz and the incident angle of 20\({}^{\circ}\).