# Path Loss Model for 3.5 GHz and 5.6 GHz Bands in Cascaded Tunnel Environments
## 2 Path Loss Model for Cascaded Tunnels in Far-Field Region of Propagation
### The Joint Channel Model Based on Waveguide and SBR  
Basically, SBR is used to calculate attenuation by considering the number of reflections and reflection coefficients in the tunnel. For the ray with initial power \(P_{0}\), the power \(P\) after propagating at a certain distance \(z\) can be expressed as [38]:  
\[\frac{P_{0}}{P}=\frac{1}{\left|\Gamma_{1}\right|^{2N_{1}z}\cdot\left|\Gamma_{2 }\right|^{2N_{2}z}} \tag{6}\]  
where \(N_{1}\) and \(N_{2}\) represent the number of reflections per meter in the vertical and horizontal walls, and \(\Gamma_{1}\) and \(\Gamma_{2}\) represent the reflection coefficients in the vertical and horizontal walls. Since the tunnels have curvature along their vertical sidewalls, this study only focuses on the path-loss difference caused by the curvature of the vertical tunnel sidewalls. For vertical polarization, \(\Gamma_{1}\) can be expressed as [39]:  
\[\Gamma_{1}=\frac{\cos\ \phi_{1}-\sqrt{\cos^{2}\phi_{1}+e_{r}^{*}-1}}{\cos\ \phi_{1}+\sqrt{\cos^{2}\phi_{1}+e_{r}^{*}-1}} \tag{7}\]  
where \(e_{r}^{*}\) represents the complex permittivity of the wall, and \(\phi_{1}\) represents the incident angle of the ray with the vertical wall. In a straight tunnel, \(\phi_{1}\) and \(N_{1}\) are defined as [30]:  
\[\phi_{1}=\frac{\pi}{2}-\frac{m\lambda}{2w},\quad m=1,\ 2,\ 3,\ \ldots \tag{8}\]  
\[N_{1}\ \approx\ \frac{m\lambda}{2\text{w}^{2}},\ m=1,\ 2,\ 3,\ \ldots \tag{9}\]  
where \(m\) represents different modes in the near-field region. Because high-order modes fade rapidly after BP, it is assumed that only the basic mode (\(m=1\)) exists in the far-field region. In the curved tunnel as shown in Figure 2, \(y\) represents the distance between two reflection points (\(P_{1}\) and \(P_{2}\)) in the tunnel with the radius of curvature (\(R\)), and \(\phi_{1}\) can be defined as:  
\[\phi_{1}=\cos^{-1}\left(\frac{y}{2R}\right) \tag{10}\]  
In comparison with the incident angle in the straight tunnel (\(\phi_{s}\)), as shown in Figure 3, the decrease in the incident angle in the curvature (\(\phi_{c}\)) leads to an increase in reflection times and a decrease in the reflection coefficient, which generates EL in the curved tunnel, as shown in Figure 4. The EL can be expressed as:  
\[\mathrm{EL}=10\;log\left(\frac{P_{straight}}{P_{curve}}\right)[\mathrm{dB}] \tag{11}\]  
Figure 4: The relationship between incident angle and reflection coefficient. Figure 3: Schematic diagram of reflection in cascaded tunnel. Figure 2: Schematic diagram of reflection in cascaded tunnel. According to (6), the EL can be rewritten as a function of the propagation distance of curvature (2). \[\Gamma_{1}=\frac{\cos\ \phi_{1}-\sqrt{\cos^{2}\phi_{1}+e_{r}^{*}-1}}{\cos\ \phi_{1}+\sqrt{\cos^{2}\phi_{1}+e_{r}^{*}-1}} \tag{7}\]0  
In (12), the ELC represents the extra loss coefficient in the curved tunnel, and its expression is as follows:  
\[\Gamma_{1}=\frac{\cos\ \phi_{1}-\sqrt{\cos^{2}\phi_{1}+e_{r}^{*}-1}}{\cos\ \phi_{1}+\sqrt{\cos^{2}\phi_{1}+e_{r}^{*}-1}} \tag{7}\]1  
where \(\Gamma_{1s}\) and \(\Gamma_{1c}\) represent the reflection coefficients along the vertical walls of the straight and curved tunnels, respectively, and \(N_{1s}\) and \(N_{1c}\) represent the number of reflections per meter along the vertical walls of the straight and curved tunnels, respectively. It is assumed that the fundamental mode is reflected \(M\) times on the vertical wall of the curved tunnel with a length of \(d^{\prime}\), and \(\phi_{i}\) is the incident angle of the \(i\)-th reflection.