The treatment of forces in Bloch analysis

In periodic lattice structures, wave propagation on the infinite domain can be greatly simplified by invoking the Floquet–Bloch theorem. The theorem allows a system's degrees of freedom to be reduced to a small subset contained in a repeating unit cell. The equations of motion governing this subset contain internal force terms, which must be eliminated before establishing the eigenvalue problem for the dispersion relationships. There are subtle issues with regard to the elimination of these forces, which we address in this paper. We demonstrate that for any two- or three-dimensional periodic lattice, the internal forces vanish when acted upon by the linear transformation engendered by the degree of freedom reduction.