# Complex-Eigenfrequency Band Structure of Viscoelastic Phononic Crystals
## 2 Theory  
### Complex Eigenvalue Problem  
We consider a primitive unit-cell of an artificial crystal, such as those depicted in Figures 1-3. Elastic Bloch waves have a displacement vector of the form \(\mathbf{U}(\mathbf{r},t)=\mathbf{u}(\mathbf{r})\exp(\imath(\omega t-\mathbf{ k}\cdot\mathbf{r}))\) where \(\mathbf{u}(\mathbf{r})\) is the periodic part of the Bloch wave, \(\omega\) is the angular frequency, \(t\) is the time variable, and \(\mathbf{k}\) is the Bloch wavevector. In elastic solids, the stress and strain tensors are related by [PERSON]'s law via the order-four elastic tensor \(c_{ijkl}\). Bloch waves of phononic crystals are obtained by solving an eigenfrequency problem under periodic boundary conditions. In viscoelastic solids, [PERSON]'s law is generalized to a complex-valued elastic tensor \(c_{ijkl}+\imath\omega\eta_{ijkl}\), with \(\eta_{ijkl}\) the order-four phonon viscosity tensor. The imaginary part is thus explicitly proportional to the angular frequency. As a result, the eigenfrequency problem becomes non trivial, since the matrix coefficients depend on frequency. A complex eigenfrequency band structure can be obtained anyway using the method described in the following and with the assumption that viscoelastic losses are proportional to \(\omega\). We start with the formal eigenproblem with a viscoelastic term  
\[(\mathbf{K}(\mathbf{k})+\imath\omega\mathbf{V}(\mathbf{k})-\omega^{2}\mathbf{ M})\mathbf{u}=0, \tag{1}\]  
with \(\mathbf{u}\) a vector of the degrees of freedom (d.o.f), \(\mathbf{K}\) a stiffness matrix, \(\mathbf{M}\) a mass matrix, and \(\mathbf{V}\) a viscosity matrix. Note that both the stiffness matrix and the viscosity matrix are a function of the Bloch wavevector \(\mathbf{k}\); we omit the dependence on \(\mathbf{k}\) in the rest of this sub-section for simplicity of the presentation. The mass matrix generally has constant coefficients. All matrices are square and the number of lines equals the number of d.o.f of the system. We set \(\lambda=\imath\omega\) and rewrite Equation (1)  
\[(\mathbf{K}+\lambda\mathbf{V}+\lambda^{2}\mathbf{M})\mathbf{u}=0. \tag{2}\]  
This second-degree polynomial eigen-equation is equivalent to the first-degree system of equations  
\[\mathbf{v}=\lambda\mathbf{u}, \tag{3}\] \[\mathbf{Ku}+\lambda\mathbf{V}\mathbf{u}+\lambda\mathbf{M}\mathbf{ v}=0, \tag{4}\]  
or finally equivalent to the double-size eigenvalue problem \[\begin{pmatrix}\mathbf{K}&0\\ 0&1\end{pmatrix}\begin{pmatrix}\mathbf{u}\\ \mathbf{v}\end{pmatrix}=\lambda\begin{pmatrix}-\mathbf{V}&-\mathbf{M}\\ 1&0\end{pmatrix}\begin{pmatrix}\mathbf{u}\\ \mathbf{v}\end{pmatrix}. \tag{5}\]  
This asymmetric eigenvalue problem yields complex eigenvalues, and hence the complex-eigenfrequency band structure. The wavevector \(\mathbf{k}\) enters via the dependence of matrices \(\mathbf{K}\) and \(\mathbf{V}\). Note that the eigensystem can be written in an Hermitian symmetric form, providing the mass matrix can be inverted and is symmetric, and both \(K\) and \(V\) are Hermitian symmetric, e.g.,  
\[\begin{pmatrix}\mathbf{K}&0\\ 0&-\mathbf{M}^{-1}\end{pmatrix}\begin{pmatrix}\mathbf{u}\\ -\mathbf{M}\mathbf{v}\end{pmatrix}=\lambda\begin{pmatrix}-\mathbf{V}&1\\ 1&0\end{pmatrix}\begin{pmatrix}\mathbf{u}\\ -\mathbf{M}\mathbf{v}\end{pmatrix} \tag{6}\]  
As a result, eigenvalues come in complex conjugate pairs \((\lambda,\lambda^{*})\). Each pair of eigenfrequencies have the same real part (and hence propagation constant) but the opposite imaginary part. The eigenfrequency with positive imaginary part is a mode of vibration that attenuates in time, whereas the eigenfrequency with the negative imaginary part is a mode of vibration that amplifies in time. Note that their respective excitation is dictated by initial boundary conditions. For instance, specifying that the energy in the mode cannot grow to infinity with increasing time disqualifies the amplifying mode of vibration.