## 2 Potential Applications of Aeroacoustic Metamaterials
### Scattering Abatement (Cloaking)  
The term (acoustic) cloaking is commonly used to describe the ability of a device, the \"cloak\", to \"hide\" an obstacle from the incident noise perturbation field, thus preventing scattering from objects that may be embedded in such a domain. The first examples of cloaking were published for electromagnetic waves in 2006 by [PERSON] [64] and [PERSON] [65]. Successively, the exploitation of the background idea of coordinate-transformation invariance of the governing equations has been transported to the acoustic domain by [PERSON] and [PERSON] [66]. This idea implies that any coordinate transformation-based deformation of the space can be physically created in real space with appropriate and typically complex distributions of acoustic properties of the propagation medium. [PERSON] [67; 68] gave a fundamental contribution to the theoretical development of the acoustic cloaking theory, showing that the first path discovered to obtain this effect was called \"inertial cloaking\" because it required a material to exhibit anisotropic density and led to an infinitely massive device. This a subcase of a more general theory--anisotropy can be distributed between the density and bulk modulus, the extremes defining inertial and pentamode metafluids, which both possess fluid-like properties that are not found in simple fluids. The first is defined by a tensorial anisotropic mass density and a scalar bulk modulus. The latter represents a solid structure, ideally behaving like a fluid, possessing five compliant, soft modes of deformation (connected to vanishing shear stiffness) and only one rigid mode (linked to the only non-null eigenvalue of the elasticity tensor representing compressibility). Pentamode metafluids may be engineered with a lattice of carefully interconnected solid struts, creating a structure that exhibits a quasi-vanishing shear stiffness [69; 70] (Figure 9). Figure 9: A pentamode material as realized by [PERSON] [69] with 3D laser lithography. The unit elements are linked to each other with connections that are 0.55 μm thick. During the last decade, the scientific community produced a large number of papers on the topic, with numerical and theoretical papers and practical realizations of acoustic cloaking [71; 72; 73; 74; 75; 76; 77; 78]. Alternative approaches exploiting coordinate transformations have been explored, like the so-called carpet cloaking [79; 80; 81; 82; 83; 84; 85; 86; 87], i.e., making an arbitrary-shaped surface act like a flat wall. The transformation of the acoustic cloaking theory is valid only for quiescent media, hence when a moving fluid is present, the convective terms cause the governing equation to no longer be formally invariant under a coordinate transformation. Recent studies have tried to extend the existing theory of acoustic cloaking to the aeroacoustic domain, i.e., in the presence of flow [88; 89; 90; 91; 92] with interesting results (an example is given in Figure 10), even addressing complex turbulent flows via numerical optimization of the cloak design [93] and also exploiting a reinterpretation of the acoustics governing equations in four-dimensional space-time that exposes the relativistic structure of the phenomenon [94; 95; 96; 97; 98]. So far, however, practical realizations have been limited to static acoustics and, for the most part, intended for carpet cloaking of small objects [83; 87] or based on large and/or massive devices [99; 100]. Possibly the most interesting work has come from [PERSON] [78], who designed a feasible pentamode material for acoustic cloaking by topology optimization, starting from the desired metafluid acoustic characteristics, and numerically tested it. Although the technology is not yet ready, one can think about the potential applications of cloaking an aircraft to reduce community noise. For instance, a virtually scarfed nacelle device could take advantage of the adoption of cloaking treatment of the upper side of the nacelle.