The sound radiation from a periodic array of railway sleepers using a wavenumber domain method

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Rolling noise, radiated by the vibration of the wheels and the track, is an important source of noise from railway operations. In ballasted track the sleepers supporting the rails form an important source of noise at low frequencies. There are difficulties to calculate their sound radiation using numerical models due to their discrete periodic nature and the infinite extent of the track. For the rail, which has invariant geometry in the axial direction, the wavenumber domain (2.5D) finite element or boundary element method can be used to calculate the vibration and noise radiation. However, the 2.5D method cannot be used directly to predict the vibration and sound radiation of railway sleepers due to their discrete spacing. In this work, the discrete spacing of the sleepers in the spatial domain is introduced into the 2.5D method by applying a series of rectangular windows according to the sleeper width and spacing. The vibration of the sleepers in the wavenumber domain is obtained by applying a spatial Fourier transform to the product of these windows and the rail transfer mobility, also allowing for the ratio between the sleeper vibration and the rail vibration. The sleeper radiation obtained from the proposed approach is compared with the result obtained from the Rayleigh integral method, showing good agreement. The mobilities of the sleepers in the wavenumber domain are used to explain the effect of the sleeper spacing on the sound radiation. The spectrum of response in the wavenumber domain contains multiple waves, shifted in wavenumber by multiples of 2π/L, where L is the sleeper spacing. At low frequency, only the fundamental wave corresponding to the rail wavenumber radiates power to the far field; the ratio of the sound power of discrete sleepers to that of continuous sleepers is shown to be equal to the square of the width-to-spacing ratio. As the frequency increases, new wave branches enter the acoustic wavenumber range, leading to peaks in the radiation efficiency. At high frequency, the ratio of the sound powers converges to a constant value equal to the ratio of areas.