## 1 Introduction
The third method employs longitudinal critically refracted (LCR) waves to measure stress. The LCR waves, in particular, exhibit a relatively great change in TOF according to the change in stress and are least affected by material texture, so the technique using them is the most suitable for stress measurement using other ultrasonic waves [22; 23]. In addition, because the LCR waves are bulk longitudinal waves traveling down the surface of the material, the technique using them does not impose a geometric limitation that the subject must consist of two parallel surfaces [23]. In recent years, LCR waves have more commonly been used for ultrasonic stress measurement, so most studies use these rather than other ultrasonic waves. [PERSON] and [PERSON] [24; 25] employed LCR waves to evaluate bending stress in steel bars and plates. They also used LCR waves to detect residual stresses in steel and aluminum weld seams to determine the relaxation phenomenon of the weld residual stress [26]. [PERSON] et al. [27; 28; 29] employed LCR waves to measure and evaluate the residual stress of a stainless-steel welded plate and welded stainless-steel pipe of different thicknesses and at different wave frequencies, respectively. They compared immersion and contact ultrasonic methods to evaluate welding residual stress of dissimilar joints using LCR ultrasonic waves and finite element welding simulation [30]. In later work, the same group employed an ultrasonic stress measurement method using LCR waves with a finite element (FE) model and hole drilling to investigate the potential for residual stress measurement in stainless-steel welded plates [31]. They also used the Taguchi method with LCR waves to optimize the residual stress produced by the friction stir welding (FSW) of aluminum plates [32]. [PERSON] et al. [33] and [PERSON] et al. [34] confirmed that the ultrasonic stress measurement method could be used to measure through-thickness stress. On the other hand, the contact characteristics between the rail and the wheel of the train are greatly influenced by their geometric shape. The distribution of residual stress in the rail due to the lateral displacement motion of the wheel tends to be non-uniform, depending on the geometrical contact characteristics according to the shape combination of the asymmetric wheel surface and the curvature rail surface. In addition, because the shapes of the wheels are different, each rail in use exhibits a different stress distribution. Although analysis techniques for the geometrical contact characteristics of wheel-rail have already been developed, studies to confirm the distribution of residual stress in the head part of rails using LCR waves have not yet been conducted. In this study, an ultrasonic probe was designed and fabricated to develop a stress measurement technique using ultrasonic longitudinal critical refraction waves. Based on the applied result, the correlation was verified by calculating the coefficient of determination between the ultrasonic parameter and the tensile stress. In addition, by measuring the residual stresses in used rails using the acoustoelastic coefficients obtained from the tensile testing, the distribution of residual stresses in the head of the rails was confirmed.