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We study wave propagation in periodic and frequency dependent materials. Theapproach in this paper leads to spectral analysis of a quadratic operatorpencil where the spectral parameter relates to the quasimomentum and thefrequency is a parameter. We show that the underlying operator has a discretespectrum, where the eigenvalues are symmetrically placed with respect to thereal and imaginary axis. Moreover, we discretize the operator pencil withfinite elements and use a Krylov space method to compute eigenvalues of theresulting large sparse matrix pencil.

On the Spectrum of an Operator Pencil with Applications to Wave Propagation in Periodic and Frequency Dependent Materials