# Increasing the Insertion Loss of Sonic Crystal Noise Barriers with Helmholtz Resonators
## 1 Introduction  
A major problem in today's industrialized world is noise and its effects on people's health. Various solutions have been proposed to solve this problem using noise reduction techniques, which are applied at three different stages: near the source, at the transmission, and at the receiver. The most common solution, and the most widely used because of its simplicity, is the use of devices at the transmission level or, more precisely, the installation of noise barriers (NBs) between the source and the receiver [1]. The insulation performance of NB can be improved by a suitable choice of shape: the geometry of the edge or the curvature of the main flat surface can be designed to modify the diffraction of the upper part [2]. Incorporating porous materials into them also helps to improve their acoustic perfor-mance, especially with regard to the reflection of sound waves between the source and body of the NB [3].  
In this field, a new method has emerged in the last decades, which has new properties in contrast to conventional flat walls for noise reduction: sonic crystal noise barriers (SCNB) [4; 5; 6; 7; 8]. SCNBs are made of periodic materials called sonic crystals, which are arrays of scatterers embedded in air [9]. While conventional barriers do not allow air or water to pass through, SCNBs are permeable to fluids. Moreover, they can be arranged to attenuate noise in specific target frequency ranges, which is one of their advantages over conventional barriers that cannot distinguish between different noises. SCNBs exhibit the property of Bragg bandgaps (Bragg-BGs), with frequency ranges for which acoustic waves are not able to propagate due to the interference generated by the multiple scattering of waves [10; 11; 12]. The central frequency of the first Bragg-BG is \(\mathrm{f_{c}}=\mathrm{c}/2\mathrm{a}\), where \(\mathrm{c}\) is the speed of sound in the fluid (\(\mathrm{m}/\mathrm{s}\)), and the lattice constant (\(\mathrm{a}\)) is the distance between scatterers and ischaracteristic of the unit cell of a crystal lattice [13]. Bragg-BGs have improved the ability to control wave propagation through many additional applications, such as waveguides [14] or diffusers [15; 16]. Some approaches have been developed to improve the insulation level and range of Bragg-BGs, e.g., by using fractal geometries [17] or different radii of the scatterers that make up the SCs [18].  
In addition to Bragg scattering, BGs are also produced by local resonances. In particular, they are used to induce low-frequency BGs in SCs and have been studied in one- two-dimensional periodic arrays [19; 20; 21; 22; 23]. In acoustic systems, these resonances are usually generated by Helmholtz Resonators (HRs) [24]. The BGs based on local resonances of HRs (HR-BGs) depend on the alignment of the mouth of the resonator with respect to the incident sound wave. Some studies have shown that an opposite arrangement of the HRs mouths can help to achieve sound insulation and air ventilation simultaneously [25], and that the maximum of the absorption coefficient is extremely angle dependent [26]. The HR-BG has a smaller frequency range than the Bragg-BG, but several authors have proposed to extend the BGs of SCs by multiresonant scatterers [27; 28; 29; 30; 31; 32]. HR-BG and Bragg-BG influence each other [33; 34] and sometimes the interaction between the two wave propagation phenomena does not improve the insulation capacity of the SCNB. [35]. As far as we know, there are no studies on the effect of this interaction on the transmission of a SCNB and on the specific influence of the resonator on the Bragg-BG. The main purpose of this work is to study the effect of the interaction between the local resonance and the Bragg interference in order to increase the noise insulation produced by SCNBs.