# Wave Polarization Control in Anisotropic Locally Resonant Materials
## 3 Results
### Anisotropic Effective Mass  
Through (14), one can compute the components of the homogenized mass density tensor \(\mathbf{\rho}^{0}\), which is symmetric and, in general, anisotropic. Due to the spectral theorem, \(\mathbf{\rho}^{0}\) admits two eigenvectors \(\mathbf{e}_{\mathrm{I}}\) and \(\mathbf{e}_{\mathrm{II}}\), i.e., the principal mass directions, such that  
\[\mathbf{\rho}^{0}=\rho_{\mathrm{I}}^{0}\ \mathbf{e}_{\mathrm{I}}\otimes\mathbf{e}_{ \mathrm{I}}+\rho_{\mathrm{II}}^{0}\ \mathbf{e}_{\mathrm{II}}\otimes\mathbf{e}_{\mathrm{II}}, \tag{16}\]  
where \(\rho_{\mathrm{I}}^{0}\) and \(\rho_{\mathrm{III}}^{0}\) are the corresponding eigenvalues, i.e., the principal mass densities. As it can be observed by looking at Equations (14) and (15), the contribution that the matrix provides to the mass tensor is always isotropic. This means that the source of anisotropy is related to the inclusion only. For the assumed geometry, the line connecting the centers of the fiber and of the coating (red dotted line in Figure 2a) is a symmetry axis of the coated  
Figure 2: (**a**) Geometry of the square unit cell with eccentric fiber; (**b**) principal values of the homogenized mass as a function of the reduced frequency for \(R_{f}/\ell=0.24\), \(R_{c}/\ell=0.36\) and \(e/\ell=0.1\). Circled numbers represent the three cases discussed in Section 3.2.  
inclusion, i.e., it is a principal direction of mass that we identify with \(\mathbf{e_{I}}\). The other one, \(\mathbf{e_{II}}\), is defined by the direction orthogonal to \(\mathbf{e_{I}}\).  
Figure 2b shows the typical behavior of \(\rho_{1}^{0}\) (blue) and \(\rho_{\Pi}^{0}\) (red), which are piecewise monotonously increasing functions of the frequency. Dashed lines represent vertical asymptotes which arise in correspondence to the resonant frequencies of the inclusion, fixed at its boundary, whose eigenmodes are not orthogonal with respect to \(\mathbf{e_{I}}\) (for \(\rho_{1}^{0}\)) or \(\mathbf{e_{II}}\) (for \(\rho_{1}^{0}\)); see [15] for further details.  
The eigenmodes are shown in Figure 3 in the case of the horizontally eccentric fiber (\(\theta=0\)), which means \(\mathbf{e_{I}}\) is parallel to \(\mathbf{e_{I}}\) and \(\mathbf{e_{II}}\) is parallel to \(\mathbf{e_{2}}\). The eigenmodes shown in Figure 3a,b,d are orthogonal with respect to \(\mathbf{e_{I}}\) but not with respect to \(\mathbf{e_{II}}\). This is why only \(\rho_{\Pi}^{0}\) has three vertical asymptotes at the reduced frequencies \(f^{*}=\omega\ell/2\pi=6.08\), \(8.78\) and \(18.11\) m/s. Conversely, the eigenmode shown in Figure 3c is orthogonal with respect to \(\mathbf{e_{II}}\) but not with respect to \(\mathbf{e_{I}}\), which implies that only \(\rho_{1}^{0}\) has a vertical asymptote for \(f^{*}=12.22\) m/s.  
Due to the presence of such vertical asymptotes, one or both principal values of the effective mass can become negative in some frequency intervals; see [15] for further details. These ranges are shaded in Figure 2b in blue (\(\rho_{1}^{0}<0\)) and red (\(\rho_{\Pi}^{0}<0\)). Their superpositions, i.e., the purple-shaded regions, define the intervals where both principal mass densities are negative.