## 2 Numerical Simulation
### Wavefield Characteristics Analysis  
In the three-layer isotropic coal seam model, after the source is excited, three-dimensional snapshots of the x, y, and z wavefield components at different times in the x-z plane at y = 250 m are displayed in (Figures 3-5). Figure 3 shows snapshots of the x wavefield component in the model at 20, 40, 60, and 80 ms. Compared with the P- and S-waves propagating through the surrounding rock, the channel waves have relatively longer wavelengths, lower velocities, and higher energy. From these snapshots of the wavefield, for the dispersion region at 20 ms, at the initial stage of source excitation, the P-waves, S-waves, and converted waves are superimposed on each other, and multiple groups of these waves interfere constructively (as shown in Figure 3a). The phase interference is weakened. As the propagation distance increases, the wave front of critically refracted waves is indicated by lines perpendicular to the interfaces. Over time, the total reflection characteristics at the critical angle become increasingly apparent (Figure 3b-d). As the propagation distance increases, the wave group interval between phases changes, and the roof, floor, and coal seam layers display the total reflection feature in the coal seam. The wave group propagation path is clear, the interfaces separating layers at z = 25 m and z = 50 m are distinguishable, and the wavefront of the channel wave experiences total reflection at the critical angle, which serves as an identifying feature for extra-thick coal seams.  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Layer** & **Thickness/Radius R/(m)** & **P/(m\(\cdot\)s\({}^{-1}\))** & **S/(m\(\cdot\)s\({}^{-1}\))** & **\(\rho\)/(kg\(\cdot\)m\({}^{-3}\))** \\ \hline Roof & 25 & 4200 & 2300 & 2650 \\ Coal seam & 25 & 2200 & 1400 & 1450 \\ Floor & 25 & 4200 & 2300 & 2650 \\ Collapse column & 25 & 3600 & 2000 & 2200 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the model strata and collapse column.  
Figure 2: Three-dimensional model of the extra-thick coal seam: (**a**) model of a stable extra-thick coal seam; (**b**) model of an extra-thick coal seam with a collapse column.  
Comparing the x, y, and z components, we find that the energy confined to the coal seam of the z component is more than other components. The critical angle total reflection is the most obvious in the weak dispersion region, and the characteristics of other coal seams are different. As seen in the wavefield snapshot at 80 ms, the phases of the x and y components have good sym7 metry with the critical angle (Figures 3d and 4a), the propagation characteristics of the z component are consistent with those of the x and y components, and the energy of the z component seems to be concentrated in the coal seam, in which the amplitude is relatively strong, the vertical interference of wave phases is obvious, and the occurrence of total reflection is prominent (Figure 4b).  
Next, the wavefield snapshots in the models with and without a collapsed column to be compared (Figure 3d) show that the waves reflect from the collapsed column (Figure 5). This suggests that the formation of a collapse column in a coal seam changes the coal seam waveguide properties and the formation conditions of channel waves.  
Figure 4: Wavefield snapshots of the y and z components at 80 ms: (**a**) y component; (**b**) z component.  
Figure 5: Wavefield snapshot of the x component at 80 ms in the presence of a collapse column.