## 4 Results and Discussion  
First, an experiment using a single transducer was performed to obtain the elasticity modulus of the specimens. The transverse and longitudinal velocities obtained through the ultrasonic experiments were 3217 and 5791 m/s, respectively. [PERSON]'s ratio (0.277) was obtained through Equation (6), and the elasticity modulus (207,199 MPa) was calculated by substituting the density of the rail (7840 kg/m\({}^{3}\)) in Equation (5). The value of the elasticity modulus obtained was used in place of a calculated acoustoelastic coefficient. Subsequently, experiments using LCR waves were performed to confirm the difference in ultrasonic TOF according to the presence of stresses. Although the amplitudes of the LCR wave signals were relatively small compared to other types of waves, the LCR wave propagated faster than other wave modes, so it could be identified as the first burst among the received signals. The time values of the maximum peak points for the signals received by the two receivers were used as the travel times of each ultrasonic signal. Then, the values of \(\Delta\)TOF, which are the difference between respective travel times (TOF\({}_{1}\) and TOF\({}_{2}\)), were calculated. Figure 8 shows LCR wave signals from the receivers. Figure 8: Received signals used for calculating travel times. Figure 7: Photograph of the experimental setup for measuring residual stress using LCR waves. The LCR wave signals received from each receiver were acquired while the load applied to the test piece was changing. This was done to confirm that a difference in travel time of the ultrasonic waves appeared according to the stresses. Moreover, according to Equation (4), the acoustoelastic coefficient could be measured using the slope of the stress vs. \(E(t-t_{0})/t_{0}\) graph for the tensile test results. Figure 9 shows the respective results obtained in tensile tests on two specimens. To perform an uncertainty analysis of the results of the tensile test, corrections through linear regression were performed using a least-squares method. The regression lines obtained through the experimental results were \(\text{y}=2.798\text{x}+61.382\) and \(\text{y}=2.823\text{x}+47.669\), respectively. In other words, the slopes of these lines, 2.798 and 2.823, are the acoustoelastic coefficients of each rail specimen used in the experiments. The coefficients of determination (R\({}^{2}\)) for these regression lines were 0.990 and 0.993; that is, the correlation coefficients (R) were 0.995 and 0.997. Each residual standard error was 68.201 and 57.205. That is, each result was proven to be very reliable because they almost converged to the value of 1. In addition, the values of the acoustoelastic coefficients obtained in two results did not differ significantly. Therefore, it was demonstrated that the acoustoelastic coefficients obtained through constant tensile load values can be used to measure the residual stress of rails while they are in use in the field. Through this test method, in order to verify that the time differences are clearly indicated according to the stress differences, the \(dt/t_{0}\) vs. stress graph for the tensile load is shown in Figure 10.  
The regression lines of these graphs were \(\text{y}=7.367\times 10^{10}\text{x}-20.360\) and \(\text{y}=7.303\times 10^{10}\text{x}-16.009\). The coefficients of determination for these regression lines were 0.990 and 0.993, that is, the correlation coefficients were 0.995 and 0.997, respectively. The residual standard errors were 24.412 and 20.308. The evidence of this approach is that the results calculated through this experiment have proven to be very reliable. Therefore, it was also confirmed that the ultrasonic technique using LCR waves can be used to measure sensitively the tensile stress. Figure 11 shows the results of the experiment to measure the residual stresses of the head part on the rail samples using the average value of the obtained acoustoelastic coefficients.