# Noise Control in Air Mechanical Ventilation Systems with Three-Dimensional Metamaterials
## 2 Literature review on 3D metamaterials  
The term metamaterial was first introduced the mid-20 th century to refer to a periodic cellular structure not found in nature [4; 5]. At the atomic level, scientists identified metamaterials with lattice-type structures/architectures [6]. In the field of electromagnetism and thermal radiation, the metamaterials have been studied to control the electromagnetic waves; as such, the particles of the metamaterials immersed in a homogeneous fluid have been been compared with the thermal radiation between spheres and the convection motions generated in the cavities between composites [7]. The first research studies have been carried out by [PERSON] and [PERSON] based on the investigations of possible materials with negative electric permeability. Unfortunately, the theories about this negative refractive index did not arouse the interest of researchers at that time. Considering the multifunctionality of composite materials, metamaterials can be classified into three main sectors, depending on the types of waves they influence. The authors of this manuscript consider only the acoustic metamaterials and the change in sound pressure alteration within a fluid (i.e., air), although the distinction of the classes is often mixed due to the analogies of application in different fields [6]. When [PERSON] in 2000 proposed the theory on electromagnetic metamaterials, these have been applied to acoustics and noise control fields with the name of sonic crystals [8; 9; 10; 11; 12; 13; 14]. The structural configuration as shown in Figure 1 can be described as follows:  
* Type one-dimensional refers to planes elements, characterized by one periodic direction;
* Type two-dimensional refers to bars elements, characterized by two periodic directions;
* Type three-dimensional refers to spherical elements, characterized by three periodic directions. The baseline concept of the metamaterials is that the control of sound propagation is obtained by the interaction between regular geometric components and incident sound-wave. The sound attenuation is obtained by the destructive interference created while the soundwave propagates through the period disposition of the structures. This phenomenon is known as [PERSON]'s law, based on which the specific frequency attenuation is in function of the soundwave length and the dimensions of the periodic structure, including the space between each other. Figure 1: Different configurations of scatterers for metamaterials: one-dimensional (**a**), two-dimensional (**b**), and three-dimensional (**c**). This concept follows the regime of [PERSON]'s law, such that the band gap frequency (\(f_{BG}\)) depends on the incident angle of the wave (\(\varphi\)), the lattice constant (\(\alpha\)), and the sound speed in the medium (\(c\)), as summarized in Equation (1). \[f_{BG}=\frac{c}{2\alpha\ \sin(\Phi)} \tag{1}\]  
On this basis, when a plane wave affects a structure composed of elements arranged with a regular geometry, the distance between the wavefronts can be described as a \(\sin(\varphi)\), where \(\varphi\) is the angle of incidence of the plane wave, \(a\) is the distance between two consecutive rows of scatterers, and \(\lambda\) is the wavelength of the incident wave. The destructive interference accusses at \(\lambda/2=a\ sin\) (\(\varphi\)). At normal incidence, this implies that the stop band or the Bragg's pass band frequency is equal to \(f_{BG}=c/2a\). Figure 2 shows the scattering of a plane sound wave when hitting an array composed of rows of scatterers. Research studies state that the higher the distance between scatterers, the lower is the frequency band where the sound attenuation is more accentuated. Based on the disposition of the scatterers, it is possible to achieve the sound attenuation at any desired frequency range, which is an advantage compared to the standard sound absorbing panels. The most representative example of metamaterial structure applied to the acoustics is the sculpture realized by [PERSON].