## 2 Potential Applications of Aeroacoustic Metamaterials
### Reflection  
A smart way to reduce the incidence of noise pollution from an aircraft is to deflect the sound waves in order to redirect them away from the ground and community (a qualitative directivity pattern of noise emitted by turbofan components can be found in Figure 6). Upward redirection of part of the emitted acoustic energy has been tested in the past using negatively scarfed nacelle inlets. The main characteristic of such engine intakes is to have the bottom part longer than the upper, causing greater reflection of acoustic waves to the upper direction. Despite the promising community noise reduction achievable, aerodynamic side effects of such designs have been found to be too severe, affecting the overall performance of the aircraft, and thus making its usage unsuitable [52]. Also, here, metamaterials that would be able to effectively modify the reflection of acoustic waves could be candidates for engineered \"virtually scarfed nacelles\", obtaining the desired acoustic effect without compromising the inlet shape, which is still be optimized for aerodynamic and engine efficiency. Figure 5: Image from [48]. An absorbing metamaterial composed of various different Helmholtz resonators separated by microporous septa. The reference values for parameters are \(d_{1}\), \(t_{2}\), \(d_{2}\), \(T=1\) mm, \(t_{1}=2\) mm, \(b_{1}=3.6\) mm, \(b_{2}=4\) mm, \(H=20\) mm, and these can be tuned to obtain the maximum absorption at the desired frequency (e.g., an absorption peak at 580 Hz is obtained with \(H=60\) mm \(d_{2}=0.2\) mm and \(d_{1}=1.3\) mm). Many of the space-coiling metamaterials studied in the papers mentioned in the introduction have shown intriguing and abnormal reflection capabilities. In the authors' opinion, one of the most interesting contribution was presented in the works by [PERSON] et al. both numerically and experimentally [22; 23]. The main idea was to impose a phase shift profile to the reflected field by properly selecting geometrical parameters of an ultrathin, planar metasurface, whose elementary unit was made of two stiff \"corrugated beams\" with a hard back-end (Figure 7). A reduced wall thickness and overall depth (in the order of 1/20 of wavelength) are strengths of the proposed approach, together with the ease of tuning of the design to work at a desired frequency. Clearly, a continuous phase profile is not achievable with the proposed design due to the necessary width of the unit cell. The cited works [22; 23] successfully adopted a \(\pi/4\) discretization step over the \(2\pi\) range. The ideal theoretical phase profile was hence approximated using eight unit cells to provide discrete phase shifts. When a sound wave enters a cell, it is forced to follow a longer path with respect to the thickness of the cell itself, coming out with a phase delay directly connected to the equivalent straight path of the considered cell. Figure 6: Typical noise directivity from components of a modern turbofan. Figure 7: Shaping the reflection field, from [22; 23]. The unit cell is easily parametrized (**a**) to obtain a phase shift profile (**b**, colors indicate the phase of the reflected field when a plane wave is incoming) combining different cells in an array (**c**). An arbitrary shape can be obtained, imposing different phase shift profiles, \(\phi(y)\), to the reflected field (**d**). When considering an incident wave normal to the metasurface, the propagation angle of the reflected wave can be evaluated as \(\theta_{r}=\arcsin\left(\frac{\lambda}{2\pi}\frac{d\phi(y)}{dy}\right)\), where \(\lambda\) is the wavelength of the incoming noise disturbance and \(\phi\) is the acoustic phase shift. If the actual interest is not to achieve an exact reflection angle, but, as in a virtually scarfed nacelle application, an approximate value within a range, then the metasurface can be considered to be effectively broadband, and for multiples of the working frequency, it would still have the expected behavior.