# Wave Polarization Control in Anisotropic Locally Resonant Materials
## 3 Results
### Dispersion Properties: Band Gaps and Polarization Bands
Unlike the previous case, now \(\mathbf{p}\cdot\boldsymbol{\rho}^{0}\cdot\mathbf{p}<0\) for every \(\mathbf{p}\
eq\boldsymbol{0}\). Therefore, from Equation (23) one has that \(c_{\alpha}^{2}\) and \(c_{\beta}^{2}\) are both negative, i.e., Equation (21) has no real solution. In frequency intervals where the effective mass is negative definite, called _band gaps_, elastic waves cannot propagate without being attenuated. The purple-shaded regions of Figure 2b, i.e., the superposition of the blue and red ones, correspond to the band gaps of the locally resonant metamaterial. To validate this result, we performed a Bloch-Floquet numerical analysis on the same metamaterial considered in Figure 2. The resulting dispersion surfaces are shown in Figure 4b on the whole First Brillouin Zone of the lattice, which in this case is a square of side \(2\pi/\ell\). Figure 4b shows, on the left, a side view of the dispersion surfaces of Figure 4a. In purple, the band gaps obtained through Bloch-Floquet -analysis are shaded: they are in  
Figure 4: (**a**) Dispersion surfaces on the whole First Brillouin Zone; (**b**) comparison between the band gaps obtained through Bloch–Floquet analysis (left) with those predicted by homogenization (right). good agreement with those predicted by the proposed homogenization approach as the interval of negative definition of the effective mass tensor (Figure 4b, on the right). **Case 3**\(\rho^{0}\) is indefinite (i.e., \(\rho^{0}_{\rm I}\) and \(\rho^{0}_{\rm II}\) have opposite signs). In this situation, the sign of \(\mathbf{p}\cdot\mathbf{\rho}^{0}\cdot\mathbf{p}\) cannot be established a priori. Introducing the tensor  
\[\left[\mathbf{Q}^{0}(\mathbf{n})-c^{2}\boldsymbol{\rho}^{0}(\omega)\right] \cdot\mathbf{p}=\mathbf{0}, \tag{18}\]0  
and making use of the orthogonality conditions (22), it is possible to show that  
\[\left[\mathbf{Q}^{0}(\mathbf{n})-c^{2}\boldsymbol{\rho}^{0}(\omega)\right] \cdot\mathbf{p}=\mathbf{0}, \tag{18}\]1  
Since \(\mathbf{\rho}^{0}\) and \(\mathbf{P}^{\top}\cdot\mathbf{\rho}^{0}\cdot\mathbf{P}\) are congruent tensors, due to Sylvester's law of inertia, they share the same signature. Thus, \(\mathbf{p}_{\alpha}\cdot\mathbf{\rho}^{0}\cdot\mathbf{p}_{\alpha}\) and \(\mathbf{p}_{\beta}\cdot\mathbf{\rho}^{0}\cdot\mathbf{p}_{\beta}\) (which are the eigenvalues of \(\mathbf{P}^{\top}\cdot\mathbf{\rho}^{0}\cdot\mathbf{P}\)) have opposite sign as the eigenvalues \(\rho^{0}_{\rm I}\) and \(\rho^{0}_{\rm II}\) of the effective mass tensor. From (23), one can deduce that also \(c^{2}_{\alpha}\) and \(c^{2}_{\beta}\) have different signs and, therefore, Equation (21) admits only one real solution. Elastic waves hence propagate with a unique phase velocity (i.e., the positive eigenvalue) and a unique polarization vector (i.e., the associated eigenvector), and the corresponding frequency intervals are called _polarization bands_. Without loss of generality, let us suppose that \(\rho^{0}_{\rm I}>0\), \(\rho^{0}_{\rm II}<0\) and let \(c^{2}_{\alpha}\) be the positive eigenvalue. This implies, from (23), that  
\[\left[\mathbf{Q}^{0}(\mathbf{n})-c^{2}\boldsymbol{\rho}^{0}(\omega)\right] \cdot\mathbf{p}=\mathbf{0}, \tag{18}\]2  
Making use of (16) and expressing the polarization vector as \(\mathbf{p}_{\alpha}=\cos\varphi\ \mathbf{e}_{\rm I}+\sin\varphi\ \mathbf{e}_{\rm II}\), being that \(\varphi\) is the angle that \(\mathbf{p}_{\alpha}\) forms with \(\mathbf{e}_{\rm I}\), the previous inequality can be rewritten as  
\[\left[\mathbf{Q}^{0}(\mathbf{n})-c^{2}\boldsymbol{\rho}^{0}(\omega)\right] \cdot\mathbf{p}=\mathbf{0}, \tag{18}\]3  
which implies:  
\[\left[\mathbf{Q}^{0}(\mathbf{n})-c^{2}\boldsymbol{\rho}^{0}(\omega)\right] \cdot\mathbf{p}=\mathbf{0}, \tag{18}\]4  
Equation (28) provides an upper bound estimation of \(\varphi\), i.e., of the angle between the polarization vector \(\mathbf{p}_{\alpha}\) and the direction of positive effective mass \(\mathbf{e}_{\rm I}\).