Fig. 4. Schematic illustration of how the Bloch theorem defined in eqn (5) generates four of the phonon modes of a monoatomic 2D lattice. The displacement of the atom in the primitive cell defines the cell-periodic function u(r) (blue), which is replicated across the other unit cells in the lattice with a phase pattern defined by plane-wave terms exp(iq·r) (red). In (a) and (b) the wavevector q is oriented along the real-space x direction with two different wavelengths corresponding to different modulation periods. In (c) and (d), the wavevector is aligned along the y and xy directions, respectively, with the same wavelength as in (b).