# Scaled Modeling and Measurement for Studying Radio Wave Propagation in Tunnels
## 3 Scaling the Ray Tracing Model  
The choice of wave propagation model largely depends on the type of tunnel and its dimensions. In our model we assumed a straight concrete rectangular tunnel with smooth walls. We employed the ray-tracing approach for simplicity of calculation and physical explanation for this kind of tunnel. In ray tracing, the propagation of the signal depends on the number of rays arriving at the receiver location. This includes the LOS rays, reflected waves, diffracted waves, and scattered waves [12; 29; 30]. The received power \(P_{r}(Ttunel)\), normalized by the LOS link budget \(P_{r}(LOS)\) for a distance \(P_{r}(0,\ 0)\) between the transmitter and the receiver, is given by the following ratio [4]:  
\[\frac{P_{r}(Ttunel)}{P_{r}(LOS)}=\left|\sum_{m=-\infty}^{+\infty}\sum_{n=- \infty}^{+\infty}\Gamma^{|m|}(\theta_{m})\Gamma^{|n|}(\theta_{n})\frac{G[m,n]} {G[0,\ 0]}\frac{R[0,\ 0]}{R[m,n]}e^{-\frac{2\pi}{\lambda}[R(m,n)-R(0,\ 0)]}\right|^{2}, \tag{1}\]  
where \(\lambda=c/f\) is the wavelength at wave frequency \(f\) (\(c\) is the light in a vacuum), \(m\) and \(n\) are indices of rays traveling and reflected from the walls, ceiling and floor of the tunnel, \(R(m,n)\) is the path propagation length of the \((m,n)\) ray, for which the corresponding antenna gain is \(G[m,n]=\sqrt{G_{t}[m,n]G_{r}[m,n]}\) (\(G_{t}[m,n]\), and \(G_{r}[m,n]\) are the gains of the transmitting and receiving antennas, respectively). \(\theta_{m}\) and \(\theta_{n}\) are the incidence angles between the \((m,n)\) ray and the corresponsive surface in the horizontal (ceiling or floor) and vertical (walls) dimensions respectively. \(\Gamma(\theta_{m})\) and \(\Gamma(\theta_{n})\) are the reflection coefficients from the surface for the \(m\) and \(n\) incident ray. The received power in the LOS path \((m=0,n=0)\) is given by the Friis formula:  
\[P_{r}(LOS)=G_{r}[0,0]\bigg{(}\frac{\lambda}{4\pi R(0,\ 0)}\bigg{)}G_{t}[0,\ 0]P_{t}, \tag{2}\]  
where \(P_{t}\) is the transmitted power. Assuming that the transmitter and the receiver are both located at the center of the tunnel cross-section, the travelling range \(R[m,n]\) of ray \([m,n]\) along the tunnel is described by [31]  
\[R[m,n]=R[0,\ 0]\sqrt{1+\bigg{[}m\frac{W}{R[0,\ 0]}\bigg{]}^{2}+\bigg{[}n\frac{ H}{R[0,\ 0]}\bigg{]}^{2}}, \tag{3}\]  
where \(W\) and \(H\) are the width and height of the rectangular tunnel, respectively. Inspection of Equations (1) and (2) reveals a non-dimensional parameter \(R[0,0]/\lambda\), which can be used for scaling. Equation (3) describes the overall path length along which the \((m,n)\) ray is traveling from the transmitting antenna until it arrives at the antenna of the receiver. Introducing this parameter into Equation (3) results in two additional scaling ratios \(W/\lambda\) and \(H/\lambda\). We further show that by keeping these ratios, a scaling model of a long tunnel can be constructed demonstrating an identical link budget. For all our examined cases, the tunnel surfaces were sufficiently smooth and their roughness depth \(\delta\) satisfied the Rayleigh criterion \(\delta\ll\lambda/[8\sin(\theta)]\). This allowed the utilization of Fresnel equations for calculating the wall's reflection coefficients \(\Gamma(\theta)\) for the vertically or horizontally polarized ray [32]:  
\[\Gamma(\theta)=\left\{ \begin{array}{ll}\frac{c_{r}\sin\theta_{V}-\sqrt{c_{ r}-\cos^{2}\theta_{V}}}{c_{r}\sin\theta_{V}+\sqrt{c_{r}-\cos^{2}\theta_{V}}}&Vertical,\\ \frac{\sin\theta_{H}-\sqrt{c_{r}-\cos^{2}\theta_{H}}}{\sin\theta_{H}+\sqrt{c_ {r}-\cos^{2}\theta_{H}}}&Horizontal,\end{array} \right. \tag{4}\]where \(e_{r}\) is the dielectric coefficient of the building material composing the wall. Using the ray optic simulation [33], the top and bottom ray illustrations of Figure 1 demonstrate the reflection from the side walls and ceiling/floor, respectively, for the case when \(W<H\). When summed, the phase-shift between rays results in constructive and destructive interferences, causing power fluctuations along the tunnel.