## 2 Potential Applications of Aeroacoustic Metamaterials
### Noise Trapping  
Noise trapping is one of the most intriguing metamaterial applications. It involves allowing a sound wave to enter into a domain where it is contained and forbidden to exit, creating a sort of insulation of the exterior domain from the trapped sound. The most interesting application, in the authors' opinion, of sound trapping [63] uses a one-way metasurface, made by a layer of Helmholtz resonators with different depths attached to a near-zero index space-coiling based metamaterial (in a zero index acoustic metamaterial, the sound wave is infinitely stretched in space, and it travels with extremely high phase velocity). The functioning principle is depicted in Figure 8. A normal incident wave with a uniform phase profile is allowed to pass through the zero index metamaterial. The phase array steers the wave phase profile, and once the wave is reflected it is no longer allowed to pass through the zero index layer due to its high incident angle selectivity. The described effect could very well turn out to be useful in aeronautical applications. It could be, for example, exploited as a way to enhance absorption and to be used in synergy with focusing devices or negatively refractive surfaces and absorbers. In a box delimited on one side by a suitable absorbing layer and on the other side by this metasurface, oriented such that the layer of resonator points is inside the considered domain, a sound wave can enter the domain through the zero index material, experiencing a redirection due to the phase shift obtained by tuning the depths of the resonators. Then, the wave can be partially absorbed by the absorbing layer. The reflected wave, however, is not allowed to exit the box due to the presence of the one-way metasurface. On the contrary, it is reflected back towards the absorber, enhancing its effect. The effectiveness of a non-perfect absorber with a non-negligible reflection coefficient can be improved, forcing the acoustic noise perturbation to repeatedly impinge on the absorbing device. Nowadays, acoustic liners are used to reduce engine noise. Liners are applied on the internal walls of the engine nacelle, both in the intake and bypass ducts and use the Helmholtz resonance principle for the dissipation of incident acoustic energy. Placing a one-way metasurface at the interface between a liner and the internal side of the nacelle can result in an enhanced effect of the liner itself, especially at low frequencies, provided that sound waves cannot escape the domain occupied by the absorber. Obviously, one could imagine that to enhance the overall effect by both optimizing the metasurface and the liner positioning, and a totally new generation of liners needs to be designed, using, for example, a metamaterial absorber such as those described in the previous Section 2.1. The phase shift can also be tuned to maximize the number of possible reflections for a wide range of incident angles in the primary acoustic field. Note also that the phenomenon of the near-zero index is  
Figure 8: Noise trapping unit cell, from [63] and the functioning scheme of a noise trapping concept. The incident wave is allowed to pass through the metamaterial, and it is steered by the phase array into an absorbing material. The reflected fraction of the sound cannot escape and multiple reflections and absorption occur. frequency-dependent, which means that the asymmetric acoustic behavior can be realized within a relatively narrow band, allowing for a shaping of the noise signature from the aircraft, as mentioned in Section 2.1, as multilayered structures can easily extend the working range of the concept. A limitation of the design presented in [63] may be the resulting thickness, which is near to one half of the wavelength of interest, which would lead to the use of excessive thicknesses to be effective at the typical BPF (Blade Pass Frequency) of current turbofan engines and would thus not fit in the nacelle, especially if multiple layers need to be stacked.