# An Ultrasonic Guided Wave Mode Selection and Excitation Method in Rail Defect Detection
## 5 Mode Enhancement and Defect Detection Simulation
### Defect Detection Simulation  
At present, rail head transverse defects are the most dangerous damage for rail operation. Figure 13a presents the simulation of detecting an internal transverse defect. The shape of the defect on the rail cross section is a circle with a diameter of 23 mm and a length of 5 mm in x direction. In Figure 13b, the rail web defect is assumed as a vertical crack. The depth of the crack in z direction is 30 mm and 5 mm in x direction. In Figure 13c, the rail base defect is a transverse defect, the length of which is 25 mm in the y direction and 10 mm in x direction. In defect detection simulation, the modes excitation and data acquisition method are the same as Section 5.2 and the rail defects are 3 m from the left end of the rail. Figure 12: Mode simulation results. (**a**) Mode 7 simulation result; (**b**) Mode 3 simulation result; (**c**) Mode 1 simulation result. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode**} & **Simulated Group** & **Theoretical Group** & **Group Velocity Error** \\  & **Velocity Value (m/s)** & **Velocity Value (m/s)** & **Group Velocity Error** \\ \hline
7 & 3150.4 & 3215.1 & 2.01\% \\
3 & 2994.0 & 3019.9 & 0.86\% \\
1 & 2818.2 & 2852.7 & 1.21\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Group velocities of modes. Figure 13: Defect detection simulation diagram. (**a**) diagram of array excitation mode 7 defect detection; (**b**) diagram of array excitation mode 3 defect detection; (**c**) diagram of array excitation mode 1 defect detection. It is necessary to verify whether the received defect echo at the receiving nodes has occurred in mode conversion before calculating the defect location. The defect echoes extracted separately are transformed by 2D-FFT as shown in Figure 14.  
Similarly, the phase velocity of each mode can be calculated from the frequency wavenumber curve. Table 6 shows the phase velocities of reflected echo modes. Figure 14: Frequency wavenumber curves. (**a**) mode 1 defect reflected echo 2D-FFT results; (**b**) mode 3 defect reflected echo 2D-FFT results; (**c**) mode 7 defect reflected echo 2D-FFT results. As can be seen from Table 6 that the error between the simulated phase velocity and the theoretical phase velocity of each defect echo can be accepted for application. The reflected echoes received at the corresponding nodes have not occurred mode conversion. During the process of simulation, the receiving node is 1 m away from the defect. The position of the rail defect can be obtained by calculating the time difference between the received excitation signal and the reflected echo at the receiving node. As shown in Figure 15, the defect is 3 m from the left end of the rail. The receiving node is 1 m from the defect and 2 m from the excitation signal. When the defect is in the rail head, the detection results of mode 7, mode 3 and mode 1 are as shown in Figure 16. Only mode 7 has reflected echo. The time difference between the received excitation signal and the reflected echo is 0.6137 ms. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode**} & **Simulated Phase** & **Theoretical Phase** & **Phase Velocity Error** \\  & **Velocity Value (m/s)** & **Velocity Value (m/s)** & **Phase Velocity Error** \\ \hline
7 & 2721.0 & 2737.6 & 0.61\% \\
3 & 2262.5 & 2286.1 & 1.03\% \\
1 & 1994.6 & 1983.8 & 0.54\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Phase velocities of reflected echo modes. Figure 16: _Cont._  
Figure 15: Diagram of the rail head defect detection. When the defect is in the rail web, only mode 3 has reflected echo as shown in Figure 17 and the time difference is 0.6478 ms.  
Figure 16: Rail head defect detection. (**a**) Mode 7 rail head defect detection; (**b**) mode 3 rail head defect detection; (**c**) mode 1 rail head defect detection. Figure 17: _Cont._  
Similarly, when the defect is in the rail base, only mode 1 has reflected echo as shown in Figure 18 and the time difference between is 0.6913 ms.  
Figure 17: Rail web defect detection.