# Reflection Propagation Law of Electromagnetic Waves in U-Shaped Roadway
## 4 Reflected Field Strength Calculation
### Reflected Field Strength  
A half-wave dipole antenna was used as a transmitter, and its electric field is given by  
\[E_{\theta}=\frac{j60I_{m}}{d}\cdot e^{-j\omega t}\cdot\frac{\cos\left(\frac{ \pi}{2}\cos\theta\right)}{\sin\theta} \tag{14}\]  
In (14), \(d\) is the distance from the reflection point to the relay node, \(I_{m}\) is the maximum current in the antenna, \(k\) is the propagation constant, and \(\theta\) is the angle of incidence. Then, the incident wave electric field vector is  
\[\overrightarrow{E^{i}}=E_{\theta}\cos\alpha\overrightarrow{e_{\parallel}} +E_{\theta}\cos\beta\overrightarrow{e_{\perp}} \tag{15}\]  
According to [PERSON]'s law of reflection, the reflected wave electric field vector is  
\[\overrightarrow{E^{\prime}}=H_{\parallel}E_{\theta}\cos\alpha\overrightarrow {e_{\parallel}}+H_{\perp}E_{\theta}\cos\beta\overrightarrow{e_{\perp}} \tag{16}\]  
In (15) and (16), \(\alpha\) is the angle between the incident wave and the horizontally polarized wave; \(\beta\) is the angle between the incident wave and the vertically polarized wave; \(\overrightarrow{e_{\parallel}}\) and \(\overrightarrow{e_{\perp}}\) are the unit  
\[\stackrel{{\rightarrow}}{{e}}_{\perp}^{\rightarrow}=\frac{\stackrel{{ \rightarrow}}{{n}}\times\stackrel{{\rightarrow}}{{r}}}{\left| \stackrel{{\rightarrow}}{{n}}\times\stackrel{{ \rightarrow}}{{r}}\right|};\quad\stackrel{{ \rightarrow}}{{e}}_{\parallel}^{\rightarrow}=\frac{\stackrel{{ \rightarrow}}{{e}}_{\perp}^{\rightarrow}\times\stackrel{{ \rightarrow}}{{r}}}{\left|\stackrel{{\rightarrow}}{{e}}_{ \perp}^{\rightarrow}\times\stackrel{{\rightarrow}}{{r}}\right|} \tag{18}\]  
\[\stackrel{{\rightarrow}}{{e}}_{\parallel}^{\rightarrow}=2( \stackrel{{\rightarrow}}{{n}}\cdot\stackrel{{ \rightarrow}}{{e}}_{\parallel}^{\rightarrow})\stackrel{{ \rightarrow}}{{n}}-\stackrel{{\rightarrow}}{{e}}_{\parallel}^{ \downarrow};\quad\stackrel{{\rightarrow}}{{e}}_{\perp}^{ \rightarrow}=\stackrel{{\rightarrow}}{{e}}_{\perp}^{\rightarrow} \tag{19}\]  
For a single reflection, the field strength [24] for each primary reflection is  
\[\stackrel{{\rightarrow}}{{E}}_{1}^{\rightarrow}=H_{11}[(\stackrel{{ \rightarrow}}{{E}}_{1}^{\rightarrow}\cdot\stackrel{{ \rightarrow}}{{e}}_{\parallel}^{\rightarrow})\cdot C(q)]\stackrel{{ \rightarrow}}{{e}}_{\parallel}^{\rightarrow}+H_{1\perp}[(\stackrel{{ \rightarrow}}{{E}}_{1}^{\rightarrow}\cdot\stackrel{{ \rightarrow}}{{e}}_{1\perp}^{\rightarrow})\cdot C(q)]\stackrel{{ \rightarrow}}{{e}}_{1\perp}^{\rightarrow} \tag{20}\]  
\[C(q)=\frac{q}{q+q^{\prime}} \tag{21}\]  
In (20) and (21), \(H_{1\perp}\) and \(H_{1\parallel}\) are the vertical and horizontal polarized wave reflection coefficients of the primary reflection, respectively; \(\stackrel{{\rightarrow}}{{e}}_{\parallel\parallel}^{\rightarrow}\) and \(\stackrel{{\rightarrow}}{{e}}_{1\perp}^{\rightarrow}\) are the unit vectors of the horizontal and vertical polarization before the primary reflection, respectively; \(\stackrel{{\rightarrow}}{{e}}_{\parallel}^{\rightarrow}\) and \(\stackrel{{\rightarrow}}{{e}}_{1\perp}^{\rightarrow}\) are the unit vectors of the horizontal and vertical polarization after one reflection, respectively; \(q\) is the distance from the relay node to the reflection point; and \(q^The simulation results in Figure 10 show that the number of effective reflection lines for each number of times shows an overall decreasing trend. This is because the distance from the launch point to the corner increases, the initial incidence angle of the rays decreases, and most of the rays are blocked by the obstacle plane, which makes it difficult for them to reach the receiving point. When the emission point was farthest from the corner, the lowest number of valid reflection lines was available.