# Wave Polarization Control in Anisotropic Locally Resonant Materials
## 3 Results
### Dispersion Properties: Band Gaps and Polarization Bands  
To obtain the dispersion relation of the homogenized media, we consider the propagation of the wave:  
\[\mathbf{U}^{0}(\mathbf{x})=\mathbf{p}e^{i\mathbf{k}\cdot\mathbf{x}}, \tag{17}\]  
where \(\mathbf{p}\) is the polarization vector, \(\mathbf{k}=k\mathbf{n}\) is the wavevector, \(k\) is the wavenumber and \(\mathbf{n}\) is the direction of propagation. Replacing (17) in the homogenized Helmholtz Equation (13), one obtains  
\[\left[\mathbf{Q}^{0}(\mathbf{n})-c^{2}\boldsymbol{\rho}^{0}(\omega)\right] \cdot\mathbf{p}=\mathbf{0}, \tag{18}\]  
where \(c=\omega/k\) is the phase velocity and \(\mathbf{Q}^{0}(\mathbf{n})\) is the effective acoustic tensor of the media defined as  
\[\mathbf{Q}^{0}(\mathbf{n})=\mathbf{n}\cdot\mathbb{D}^{0}\cdot\mathbf{n}. \tag{19}\]  
The latter is a second-order symmetric tensor due to the symmetries of \(\mathbb{D}^{0}\), and it is also positive definite. In fact, due to the strong ellipticity condition (8), one has  
\[\mathbf{v}\cdot\mathbf{Q}^{0}\cdot\mathbf{v}=\mathbf{n}\otimes\mathbf{v}: \mathbb{D}^{0}:\mathbf{n}\otimes\mathbf{v}\geq\alpha\|\mathbf{v}\|^{2}>0 \qquad\forall\mathbf{v}\
eq\mathbf{0}. \tag{20}\]  
Figure 3: Contours of the first four eigenmodes of the inclusion fixed at its boundary for \(R_{f}/\ell=0.24\), \(R_{\mathrm{c}}/\ell=0.36\), \(\epsilon/\ell=0.1\) and \(\theta=0\). Top row: horizontal displacement; second row: vertical displacement. (**a**) \(f^{*}=6.08\) m/s; (**b**) \(f^{*}=8.78\) m/s; (**c**) \(f^{*}=12.22\) m/s; (**d**) \(f^{*}=18.11\) m/s. Equation (18) can be recognized as a generalized eigenvalue problem. Since \(\mathbf{Q}^{0}\) and \(\boldsymbol{\rho}\) are real and symmetric, Equation (18) defines two real eigenvectors \(\mathbf{p}_{\alpha}\) and \(\mathbf{p}_{\beta}\) associated with the real eigenvalues \(c_{\alpha}^{2}\) and \(c_{\beta}^{2}\), which solve the dispersion relation:  
\[\det[\mathbf{Q}^{0}(\mathbf{n})-c^{2}\boldsymbol{\rho}^{0}(\omega)]=0. \tag{21}\]  
The eigenvectors \(\mathbf{p}_{\alpha}\) and \(\mathbf{p}_{\beta}\), which in general do not coincide with \(\mathbf{e}_{\text{I}}\) and \(\mathbf{e}_{\text{II}}\), satisfy the orthogonality conditions:  
\[\mathbf{p}_{\alpha}\cdot\mathbf{Q}^{0}\cdot\mathbf{p}_{\beta}=\mathbf{p}_{ \alpha}\cdot\boldsymbol{\rho}^{0}\cdot\mathbf{p}_{\beta}=0, \tag{22}\]  
while the eigenvalues \(c_{\alpha}^{2}\) and \(c_{\beta}^{2}\) can be expressed through the Rayleigh quotients:  
\[c_{\alpha}^{2}=\frac{\mathbf{p}_{\alpha}\cdot\mathbf{Q}^{0}\cdot\mathbf{p}_{ \alpha}}{\mathbf{p}_{\alpha}\cdot\boldsymbol{\rho}^{0}\cdot\mathbf{p}_{\alpha }}\qquad\text{and}\qquad c_{\beta}^{2}=\frac{\mathbf{p}_{\beta}\cdot\mathbf{ Q}^{0}\cdot\mathbf{p}_{\beta}}{\mathbf{p}_{\beta}\cdot\boldsymbol{\rho}^{0}\cdot \mathbf{p}_{\beta}}. \tag{23}\]  
Since \(\mathbf{Q}^{0}\) is positive definite, the numerators of Equation (23) are always positive, which means that the sign of \(c_{\alpha}^{2}\) and \(c_{\beta}^{2}\) is determined by the spectral properties of the effective mass tensor. In our two-dimensional problem, three different cases can be distinguished, which are indicated in Figure 2b with circled numbers. **Case 1**\(\rho^{0}\) is positive definite (i.e., \(\rho_{1}^{0}\) and \(\rho_{\text{II}}^{0}\) are positive). In this case, one has \(\mathbf{p}\cdot\boldsymbol{\rho}^{0}\cdot\mathbf{p}>0\) for every \(\mathbf{p}\
eq\boldsymbol{0}\), which means, from (23), that both \(c_{\alpha}^{2}\) and \(c_{\beta}^{2}\) are positive. In the frequency ranges where the effective mass is positive definite, called _pass bands_, Equation (21) admits two positive real wave velocities \(c_{\alpha}\) and \(c_{\beta}\), i.e., elastic waves can propagate within the metamaterial with polarization vectors \(\mathbf{p}_{\alpha}\) and \(\mathbf{p}_{\beta}\). **Case 2**\(\rho^{0}\) is negative definite (i.e., \(\rho_{1}^{0}\) and \(\rho_{\text{II}}^{0}\) are negative).