# Sonic Crystal Noise Barrier with Resonant Cavities for Train Brake Noise Mitigation
## 1 Introduction  
In today's industrialized world, noise pollution is a pervasive issue with significant implications for public health and well-being [1]. Road traffic noise, especially mid-to-high-frequency components (1 kHz to 5 kHz), has produced several health-related depression issues [2]. Among all different transportation vehicles, trains are the ones with the highest levels of noise at those frequencies, where human hearing sensitivity has a low audibility threshold and the annoyance is greater [3]. In particular, the most annoying noise produced is the train brake noise (TBN), an unpleasant noise that typically ranges from 2.0 to 4.5 kHz as several studies have proved [4; 5; 6; 7; 8]. More precisely, [PERSON] et al. [4] and [PERSON] et al. [5] analysed different models of disc brakes in numerical and laboratory studies, while [PERSON] et al. [6] and [PERSON] et al. [7] measured some trains braking in railways at different velocities. For effective noise-reduction strategies, extensive research has been carried out at three key stages: near the noise source, during transmission and at the receiver. A widely adopted approach, primarily due to its simplicity, is to use devices at the transmission level, in particular noise barriers (NB), to mitigate noise propagation [9; 10]. The performance of these NB structures can be enhanced by carefully tailoring their geometrical properties, including the shape of their edges and the curvature of their primary surfaces, to control sound diffraction [11; 12]. In addition, the incorporation of porous materials into NBs further improves their acoustic efficacy, particularly with respect to the reflection of sound waves between the source and the barrier structure [13; 14]. In this scope of noise mitigation, a novel methodology has gained prominence in recent decades, offering some advantages compared to conventional noise barriers: soniccrystal noise barriers (SCNBs) [15; 16; 17; 18]. SCNBs incorporate periodic materials known as sonic crystals, which consist of arrays of scatterers embedded in air [19]. SCNBs can be strategically arranged to attenuate noise within specific target frequency ranges and this capability sets them apart from traditional barriers incapable of distinguishing between different types of noise. SCNBs are also permeable to fluids, and in the case of train noise mitigation, open barriers have tested its benefits compared with closed ones [20; 21; 22; 23]. Notably, SCNBs exhibit a characteristic feature called Bragg bandgaps (Bragg-BGs), which are frequency ranges where acoustic waves are prohibited from propagating due to the interference generated by multiple wave scattering [24; 25]. The central frequency of the first Bragg-BG is determined by \(\mathrm{f}_{\mathrm{ Bragg}}=\mathrm{c}/2\mathrm{a}\), with 'c' representing the speed of sound in the fluid (m/s), air in this case, and 'a' denoting the lattice constant, which is the distance between scatterers, characteristic of the unit cell in a crystal lattice [26]. Bragg-BGs have revolutionized the control of wave propagation and found applications in diverse fields different from NBs such as waveguides [27; 28], diffusers [29] and windows [30; 31]. In addition to Bragg scattering, bandgaps can also be generated through local resonances (LRs) [32; 33; 34; 35; 36]. The resulting bandgaps based on local resonances of LRs (LR-BGs) depend on the alignment of the resonator's mouth with respect to the incident sound wave. It is worth noting that LR-BGs have a more limited frequency range compared to Bragg-BGs. The interaction between LR-BGs and Bragg-BGs has been explored [37; 38; 39] and at times, the interaction between these two wave propagation phenomena fails to yield improvements in the sound insulation capacity of SCNBs [40; 41].