Theory of damped Bloch waves in elastic media

We present a theory for Bloch wave propagation in damped elastic media. We expand the eigenvalue problem governing the dispersion relation using a set of Bloch mode eigenvectors at each wave-vector point. With the assumption of Rayleigh damping, this decomposition allows us to derive the band structure in the Brillouin zone. The damping ratio corresponding to each Bloch mode is also generated. We show that damping qualitatively alters the shape of the dispersion curves. Damping also results in a branch-overtaking phenomenon that has a significant effect on band gaps. As the damping is increased, a band-gap size can drop abruptly.