## 1 Introduction  
Rail corrugations are periodic defects commonly observed in all types of railway tracks. According to the current understanding of their development mechanisms, namely the damage mechanisms and wavelength-fixing mechanisms [1], rail corrugations can be classified into six groups: heavy haul, light rail, booted sleepers, contact fatigue, rutting and short pitch corrugation. Among these groups, the development mechanisms of short pitch corrugation (hereafter corrugation) are not fully understood. Although corrugation was identified more than a century ago and despite extensive research efforts [1; 2], an effective solution has not been developed to avoid the corrugation problem, which remains an enigma that has puzzled many researchers and engineers [3; 4]. Grinding appears to be the only effective corrective countermeasure; however, it increases maintenance costs and reduces the availability of the railway network. Thus, the development mechanisms of this phenomenon must be elucidated, and an effective solution for its control at an early stage must be identified. Corrugation mainly occurs on straight tracks or gentle curves where contact does not occur between the wheel flange and rail gauge corner (Figure 1), and it usually manifests as shiny ripples and dark valleys. The typical wavelength is in the range of 20-80 mm, with amplitudes up to 100 \(\upmu\)m [1]. Corrugation is one of the most prominent problems for railway infrastructure managers because it increases the vibrations of the vehicle-track system and results in higher frequencies (more than 500 Hz) of the wheel-rail dynamic contact forces, which leads to accelerated degradation of vehicle-track system components and shortened service life [5]. In addition, the noise generated by vibrations is a nuisance to residents living near railway lines [1; 6]. Because of the high level of noise, corrugation is also known as \"roaring rail\". Corrugation can also generate rolling contact fatigue (RCF), such as squats [7] (Figure 1c). The main damage mechanism of corrugation is commonly considered to be wear caused by longitudinal wheel slip [8; 9]. Plastic deformation is another possible damage mechanism [1], and it has been investigated by a numerical approach [10] and metallurgical analyses [11]. The wear and  
Figure 1: Short pitch corrugation and the resulting squats in the Dutch railway network. Wavelength and periodicity are distinguished (wavelength is the distance between two adjacent shining spots of ripples, and periodicity refers to the periodic pattern that contains multiple wavelengths of non-uniform amplitude of corrugation). (**a**) is at a gentle curve, and (**b**,**c**) are on straight tracks. (**a**) Uniform corrugation with a wavelength of approximately 30 mm. Squats have not yet developed, and a ballasted track with mono-block sleepers and fastenings with a W-shaped tension clamp is shown. Photo taken near Assen, the Netherlands. (**b**) Non-uniform corrugation of a constant periodicity of a sleeper span. The corrugation wavelength varies largely within a period. Squats have not yet developed, and a ballasted track with duo-block sleepers and fastenings with Deenik clips is shown. Photo taken near Steenwijk, the Netherlands. (**c**) Non-uniform corrugation with a periodicity shorter than a sleeper span. The corrugation wavelength is approximately 30 mm. The squats were caused by corrugation, and a ballasted track with duo-block sleepers and fastenings with Deenik clips is shown. Photo taken near Steenwijk, The Netherlands. deformation underlying corrugation are differential, i.e., they are selective processes that \"consistently\" occur more at certain locations than at adjacent locations, which results in the accumulation and growth of typical wave patterns [12]. In this case, \"consistently\" indicates that the wear and deformation caused by one-wheel passage repeats the same wavelengths and phase angle of previous wheel passages, which results in the accumulation of wear and deformations and the initiation of corrugation growth.