# Complex-Eigenfrequency Band Structure of Viscoelastic Phononic Crystals
## 4 Discussion  
In all numerical results presented in Section 3, the imaginary parts of the viscoelastic constants were much smaller than their real parts. These results are thus mainly relevant to the case where one is interested in engineering wave propagation properties through a sonic or phononic crystal with small damping. Of course, the equations defining the complex-eigenfrequency band structure also apply beyond this small attenuation regime, as could be interesting for applications to the mitigation of sound or vibrations. In the small damping limit, the first-order perturbation theory of Ref. [11]  
Figure 3: Two-dimensional phononic crystals composed of holes in silicon. The lattice constant is \(a=1\) μm. (**a**) Unit cell of a square lattice phononic crystal with \(d/a=0.833\) and (**b**) the corresponding complex-eigenfrequency band structure. (**c**) Unit cell of a hexagonal lattice sonic crystal with \(d/a=0.7\) and (**d**) the corresponding complex-eigenfrequency band structure. The color scale of the band diagrams correspond to the quality factor \(Q\) of each Bloch eigenstate. applies. The main result of this theory in our context is that the relative variation of the frequency of a band at a fixed value of the Bloch wavevector is given by  
\[\frac{\delta\omega|_{\mathbf{k}}}{\omega}=\frac{\imath}{2}\omega\frac{\langle \
abla_{\mathbf{k}}\mathbf{u}|\eta|\
abla_{\mathbf{k}}\mathbf{u}\rangle}{ \langle\
abla_{\mathbf{k}}\mathbf{u}|c|\
abla_{\mathbf{k}}\mathbf{u}\rangle}, \tag{19}\]  
where  
\[\langle\
abla_{\mathbf{k}}\mathbf{u}|c|\
abla_{\mathbf{k}}\mathbf{u}\rangle =\int_{\Omega}(u_{i,j}^{*}+\imath k_{j}u_{i}^{*})c_{ijkl}(u_{k,l}- \imath k_{l}u_{k}), \tag{20}\] \[\langle\
abla_{\mathbf{k}}\mathbf{u}|\eta|\
abla_{\mathbf{k}} \mathbf{u}\rangle =\int_{\Omega}(u_{i,j}^{*}+\imath k_{j}u_{i}^{*})\eta_{ijkl}(u_{ k,l}-\imath k_{l}u_{k}). \tag{21}\]  
The latter equations are the spatial averages of the real and imaginary part of the viscoelastic tensor taken with respect to the particular Bloch wave considered. They include all the influence of the details of the crystal on the appearance of an imaginary part for the eigenfrequency. The quality factor for band \(m\) can then be estimated as  
\[Q_{m}(\mathbf{k})=\frac{\langle\
abla_{\mathbf{k}}\mathbf{u}|c|\
abla_{ \mathbf{k}}\mathbf{u}\rangle}{\omega_{m}(\mathbf{k})\langle\
abla_{\mathbf{k} }\mathbf{u}|\eta|\
abla_{\mathbf{k}}\mathbf{u}\rangle}. \tag{22}\]  
A practical algorithm, and an alternative to the theory of Section 3 in the limit of small viscoelasticity, is to solve for the lossless band structure first, thus obtaining the real eigenvalues \(\omega_{m}(\mathbf{k})\) and their corresponding Bloch waves. Equation (22) then gives a first-order perturbation theory estimate of the quality factor for any band. When there is a single homogeneous material in a sonic crystal, e.g., as is the case for the air crystal with rigid inclusions of Figure 1, the relative frequency variation simplifies to  
\[\frac{\delta\omega|_{\mathbf{k}}}{\omega}=\frac{\imath}{2}\omega\frac{\eta}{B}, \tag{23}\]  
i.e., the distribution of the Bloch wave has no incidence on the result and the quality factor has the same expression as for a plane wave in an homogeneous medium, \(Q=\frac{B}{\omega\eta}\). This is the reason why the sonic crystal and the empty lattice model show the same variation of the quality factor with frequency in Figure 1. Note that the quality factor is also independent of the filling fraction of the crystal in this scalar homogeneous case. When there is a single homogeneous but anisotropic material in a phononic crystal, as is the case for the holey silicon crystal of Figure 3, similar conclusions can be drawn but the quality factor is not independent of the filling fraction and it further depends on the direction of propagation and on the polarization of the particular Bloch wave, i.e., the ratio of the material averages \(\langle\mathrm{C}\rangle/\langle\eta\rangle\) is different for quasi longitudinal and quasi shear Bloch waves.