# Recent Advances in Non-Traditional Elastic Wave Manipulation by Macroscopic Artificial Structures
## 2 Elastic Metamaterials  
### Representative Studies of Locally Resonant Elastic Metamaterials  
Locally resonant EMTM uses motions of micro-structures exploiting the coupling effect of local oscillators. The most widely used method is the effective medium theory (EMT) in order to directly estimate the behavior of them. EMT in a homogenized scheme has been achieved by analyzing scattering coefficients, so that effective parameters such as effective mass density, \(\rho\), bulk modulus, \(\kappa\), and shear modulus, \(\mu\) can be obtained solely possessing reliance on the filling ratio with respect to the host and inclusion in a matrix at the long wavelength limit within EMTMs [52; 53]. The other method is the \(S\)-parameter retrieval method (SPRM) that numerically extracts effective parameters through scattering components by a toy model. Notice that only a handful of studies on SPRM in EMTMs exists, and one can be found that effective mass density and stiffness have been investigated considering full anisotropy with intrinsic nonlocal responses in the system [54].  
One hybrid EMTM that exhibits two different characteristics depends on operation frequency has been proposed [55]. The proposed EMTM consists of multi-mass locally resonant units which can generate monopolar, dipolar and quadrupolar resonances. In the higher frequency negative dispersion band, only pressure waves can be transferred. In the lower frequency negative dispersion band, compressional waves and shear waves can only propagate in different directions. These novel properties greatly extend the ability to control elastic waves. A two-dimensional EMTM, including fluid-solid composites, which can have negative shear modulus and negative mass density over a large frequency domain has been proposed [56]. The unit cell of proposed EMTM consists of a lattice of rubber-coated water cylinders (Figure 1a). This composite can generate both dipolar and quadrupolar resonances in its structures, so negative shear modulus and negative mass density can be realized. Negative effective refractive index appears in the regime (Figure 1b A to B) where both mass density and shear modulus have negative value. Negative refraction caused by a negative effective refractive index has been demonstrated by multiple-scattering theory simulations (Figure 1c). In addition, mode conversion from transverse waves to longitudinal waves was observed under negative refraction conditions. An EMTM with chiral micro-structures has been also demonstrated to achieve subwavelength negative refraction [57]. Three ribs of the unit cell induce translational resonance as well as rotational resonance about the central piece. Therefore, by selecting the dimension of the chirally distributed ribs, simultaneous translational and rotational resonances and negative refraction at a low-frequency range can be achieved. Negative refraction of the longitudinal elastic wave at the deep-subwavelength scale was experimentally and numerically demonstrated. This EMTM can be used for elastic wave focusing. One locally resonant EMTM that exhibits polarization bandgaps has been designed [58]; it consists of three-dimensional anisotropic locally resonant units (Figure 1d). The polarization bandgaps allow selective suppression of vibration modes. In particular, a unique property of fluids was observed; the frequency region where all flexural vibrations are suppressed, but longitudinal vibration is allowed (Figure 1e). In addition, the frequency region where torsional vibration can be significantly suppressed was observed (Figure 1f). This EMTM can be used for separating and controlling elastic waves of different polarizations.