## 1 Introduction  
Ground-borne vibrations induced by subway trains can have terrible effects on buildings, residents and instruments [1]. As comfortable urban living environments have become great concerns, track vibration-mitigation techniques attract a lot of attention. In order to mitigate ground-borne vibrations, fasteners of low vertical stiffness have been used to weaken the coupling between the rail and the sleeper, which can prevent vibrations transmitted from the rail to the infrastructure. Nevertheless, low stiffness reduces the decay of vibration with distance along the rail and leads to an increase of rolling noise radiated by the rail, causing severe noise problems. Rail sound power reflects total sound energy radiated by the rail per unit time, which does not depend on a receiver location. Therefore, investigating the influence of vertical fastener stiffness on the sound power characteristics of rail rolling noise is of great help to solve the problems of subway vibrations and noises.  
The rail sound power is determined by the vibration velocity amplitude, the decay rate (DR) and other factors [2]. There have been many studies on the rail vibration characteristics which are essential to the insight into the rail sound power. [PERSON] and [PERSON] [3] utilized a frequency-domain analysis to explain the receptance behaviors of a railway track. [PERSON] and [PERSON] [4] used the impact excitation technique to carry out the field investigations for the dynamic properties of the railway track. [PERSON] et al. (2020) built whole- and half-track finite element models to investigate the vertical dynamic behaviors of railway tracks with monoblock sleepers. [PERSON] et al. (2019) used a new frequency-domain numerical method to study the frequency response of a railway ballast track and performed low-frequency receptance measurements.  
The DR reflects the attenuation characteristics of rail vibrations in the longitudinal direction and controls the effective radiating length of the rail. [PERSON] et al. (2017) used simulations and measurements of frequency response functions to investigate DRs of different track structures. [PERSON] et al. (2018) built a Timoshenko beam model to obtain the DR of an infinite beam on the continuous two-layer foundation. [PERSON] et al. (2019) established a finite element model of the ballast track to consider the influences of section deformation and discrete supports on the DR. [PERSON] et al. (2020) used the semi-analytical finite element method to obtain the DRs of a continuously supported rail.  
The rail sound power has also been investigated by the previous works. [PERSON] et al. (2017) used TWINS (Track-Wheel Interaction Noise Software) to calculate the total A-weighted sound power level of rail rolling noise in different calculation cases, and analyzed the influence of vertical fastener stiffness on the total A-weighted sound power level. However, the influence of vertical fastener stiffness on the rail sound power level at different frequencies is not investigated in that literature. [PERSON] et al. (2019) predicted the sound radiation of a railway rail in close proximity to a ground by the boundary element method in two dimensions, paying no attention to the influence of vertical fastener stiffness. Thus, the influence of vertical fastener stiffness on the frequency-domain characteristics of rail sound power requires further studies.  
The spectral element method (SEM) is a wave-based frequency-domain analysis method with high precision. The procedures of the SEM and the conventional finite element method (FEM) are similar. The SEM firstly divides a whole structure into a series of spectral elements and then assembles the spectral stiffness matrix of the whole structure with the spectral element stiffness matrixes (Stein