# Recent Advances in Non-Traditional Elastic Wave Manipulation by Macroscopic Artificial Structures
## 2 Elastic Metamaterials
### Elastic Hyperlens
The first hyperbolic EFC elastic hyperlens [91] used different mechanical deformation modes in two orthogonal directions to make the wave speed much slower in the \(\theta\) direction than in the \(r\) direction. Because of these characteristics, the wave should not propagate along the \(\theta\) direction but only in the \(r\) direction. This hyperlens (Figure 6a) was designed without local resonance structures to realize broadband operation frequency spectrum and to reduce losses. In simulations of hyperbolic EFCs for varying frequencies and width (Figure 6b), this hyperbolic hyperlens had better resolution than the previously proposed [90] elliptical hyperlens. The simulation results for their comparison were valid with a fine resolution (Figure 6c-e). Figure 5: Schematics of spherical EFCs (red solid line), elliptic EFCs (blue solid line) and hyperbolic EFCs (black solid line). Red broken line: an unbounded area of spherical EFCs. Blue broken line: an unbounded area of elliptic EFCs. Red \(k_{c}\) and \(-k_{c}\): cutoff wavevectors of spherical EFCs. Blue \(k_{c}\) and \(-k_{c}\): cutoff wavevectors of elliptic EFCs. Since then, due to the convenience of making a bandgap at an extremely low frequency, many studies have used internal resonance structures rather than PCs. As we mentioned in the introduction, the gap of micro-structures in PCs must be comparable to the wavelength. For this reason, it is impossible to miniaturize the structure smaller than the wavelength while making a bandgap for a low frequency (long wavelength) wave. However, in locally resonant hyper MTMs which have extreme anisotropy [92], super-resolution results from the nearly flat branch that occur before the resonance. A nearly flat dispersion curve directly below the bandgap occurred in hyperbolic MTMs implies that the magnitude of wave vectors can be significantly increased without changing the frequency. Therefore, the use of internal resonance structures can effectively reduce its size regardless of the wavelength. One EMTM lens (Figure 7a) having hyperbolic EFCs (Figure 7b) exploits anisotropic dynamic mass densities by using a resonance mechanism in the horizontal translational motion [93]. Extreme stiffness induced by the translational resonance of its local resonators compensates for the decreased effective mass density and permits total transmission and subwavelength resolution (Figure 7c). Another elastic hyperlens for a longitudinal elastic wave uses two decoupled resonators as a unit cell (Figure 7d); in numerical simulations, it achieved super-resolution of \(\lambda/3\)[94]. A field that includes an elastic lens having hyperbolic EFCs (Figure 7e) induces two-directional longitudinal wave propagation (Figure 7f, top), whereas a field that does not include a hyperlens (Figure 7f, bottom) yields an omnidirectional wave [94]. A single-phase hyperbolic EMTM (Figure 7g) that exploits the anisotropy of effective mass density was optimized using a topology strategy [95]. This topology optimization realized a hyperbolic EMTM exhibiting hyperbolic EFCs (Figure 7h) that has the finest super-resolution (\(\lambda/64\)) to date (Figure 7i). Figure 6: (**a**) Elastic hyperlens (top), close-up view (lower left) and geometry of the unit cell (lower right); (**b**) hyperbolic EFCs at 15 kHz for varying the width \(w\) of unit cell (upper). Hyperbolic EFCs for varying frequencies with fixed width 14.4 mm (lower); (**c**) Comparison of the hyperlenses having hyperbolic (upper) and elliptic EFCs (lower) with which the distance between two sources is 0.48\(\lambda\); (**d**) 0.45\(\lambda\); (**e**) 0.41\(\lambda\). (**a–e**) adapted from [91]. has also been applied in acoustics [14; 100; 101]. However, the equations of motions for elastic medium do not guarantee their form-invariant and are mapped to a more general system with non-scalar density according to geometric changes [99], so cloaking of the elastic acoustic waves has been difficult to achieve with conventional methods.