## 1 Introduction  
With the continuous increase in the speeds of high-speed train operation, the noise problem caused by high-speed railways is becoming increasingly prominent, aggravating the overall noise-related issue in various parts of the line. The noise of elevated high-speed railway lines covers the full frequency band; for example, wheel-track system noise, collector system noise, and aerodynamic noise are mainly concentrated in the middle and high-frequency bands, while the noise related to elevated concrete bridge sound radiation is mainly low-frequency. Therefore, elevated lines compared to other lines encounter a more prominent noise problem. In addition, due to the characteristics of low-frequency sound waves, acoustic waves will have a larger impact on human health and the ecological environment [1; 2; 3; 4; 5], so the problem of low-frequency acoustic radiation as associated with high-speed railway bridges is now a close concern of scholars from various countries. Predicting the low-frequency acoustic radiation associated with high-speed railway bridges is difficult using analytical methods, so numerical calculation methods are widely used [6; 7; 8; 9; 10; 11; 12]. The main purpose of predicting the low-frequency acoustic radiation characteristics of elevated high-speed railway concrete bridges is to evaluate whether these acoustic waves are harmful to people and the ecological environment, investigate the key mechanisms of acoustic radiation, provide appropriate control targets for subsequent noise reduction designs, and finally realize the green and sustainable development of rail transportation. The authors here consider that noise reduction designs for high-speed railway bridges can be achieved in the following two ways. First, developing low-noise bridge types, i.e., low-noise-radiating bridges, in the early design stage, will reduce the capacity of bridges to radiate sound waves at the source. Second, designing and adopting suitable noise reduction measures for existing bridges, will limit the capacity for existing bridges to radiate sound. Both methods can to some extent reduce the low-frequency noise radiated by the bridge structure, helping to alleviate the noise impact. At present, most scholars tend to focus on the development of vibration-reducing tracks [13; 14; 15; 16; 17; 18] as a way to achieve noise reduction. Via controlling the vibration of the bridge, this design conforms to the general law of vibration-induced noise but can only affect the acoustic radiation caused by severe vibration in the bridge. In contrast, it is difficult to achieve a true noise-reducing design using vibration reduction in the low-frequency, small-amplitude vibration band, where the bridge's characteristics determine its acoustic radiation capability [11; 19]. This shows that the effect of using a vibration reduction track to control the low-frequency structural noise of the bridge will not be satisfactory but will cause the vibration in the vibration reduction track itself to be amplified and will also cause rail corrugation and many other problems [20]. Therefore, the use of a vibration reduction track in the control of bridge structural noise has severe limitations and will not achieve the same effect as vibration reduction. The most common noise-reducing bridge designs can be broadly divided into the following cases. First, the thickening of various parts of the bridge can achieve good results in both vibration and noise reduction [21; 22; 23; 24; 25]; however, the theoretical basis for the selection of the optimal thickness of each part is lacking. Second, installing tuned mass dampers (TMDs) or multi-tuned mass dampers (MTMDs) with enhanced robustness on bridges can effectively attenuate the vibration problems encountered in BGBs [26; 27; 28; 29; 30], and thus many scholars have conducted noise control studies related to this issue [31; 32].