## 1 Introduction  
Rail corrugation refers to surface roughness in the longitudinal direction of rail with a regular shape generated by friction and sliding between train wheel and rail. It can be generated in forms that vary by cause and also occurs in straight sections but mainly on low rails of curved sections. Applying cant, which is a gap between the inner and outer rails of an appropriate curved track according to train operating speed, can reduce corrugation or damage to the rails. Rail corrugation is in the form of wear with short wavelengths (30-100 mm) and long wavelengths (100-300 mm) depending on train weight, speed, and frequency, track curvature radius, and rail hardness. Continuous generation and development of corrugation increases excitation of the wheels and train rails, causing rail defects and increasing track maintenance costs and risks to safety. It is also a major source of environmental noise. Furthermore, rail corrugation can not only affect the passenger comfort [1], but also deteriorate the vehicle-infrastructure interaction performance [2]. Together, these factors make it necessary to monitor for rail surface corrugation through continuous inspection and accurate detection [3]. Rail surface roughness is measured to determine the degree of corrugation on the track's rail surface, with the measured surface roughness level expressed as a function of rail wavelength. The rail wavelength is associated with noise-generating frequencies related to speed, with the wavelength of acoustic roughness associated with the rolling noise increasing as the speed of the train running on the rail increases. The rail surface roughness with a long wavelength (or associated frequency about 0 Hz-100 Hz) induces dynamic stress and causes structure-born noise, and the relatively short wavelength(or associated frequency about 30 Hz-5 kHz) on the rail surface irregularity is the main cause of airborne noise [4; 5; 6; 7]. This paper measured acoustic roughness and rail corrugation in the rail wavelength (\(\lambda\) = 7-70 mm) domain related to 500-2000 Hz, which is the frequency band in urban train operating speed (50-120 km/h). Although the wavelength range used in measurement of acoustic surface roughness and the prediction of rolling noise varies by track and train speed, international standards generally specify 3.15-250 mm (EN 15610), 3.15-630 mm (EN 3095), and 10-1000 mm (EN 13231-3) [8; 9; 10]. A variety of portable devices have been developed to measure rail surface roughness. These include ones that use an acceleration sensor on a moving platform [11] and trolley-type devices that use a displacement sensor fixed to the platform [12]. Other types include non-contact sensors, such as those that involve attaching a laser displacement sensor to a device fixed to the rail, to measure the surface along the guide rail reciprocally. A recently developed measurement method using displacement sensors measures the relative displacement between upper platform and rail surface. In particular, the displacement sensor such as LVDT is in direct contact with the rail surface and observes displacement changes caused by surface irregularity when the platform of the measurement system moves while maintaining a constant height from the surface and converts the collected data into the rail surface roughness. However, the mobile measurement platform does not have a fixed reference for sensor measurement and the sensor itself may be tilted, causing potential for distortion in the size and phase of the measured value. Therefore, new ways have been proposed, such as the chord-based measurement method, which assumes that the longitudinal rail surface is a sinusoidal wave function consisting of various wavelengths and uses more than one sensor to correct the measured value mutually, and have been used to complement the structural limitations of mobile measuring systems with displacement sensors [13; 14].