The characteristic vibrations of a rectangular lattice

Summary

For a rectangular lattice with one particle in each unit cell, it is shown that the group velocity of the waves vanishes for the six characteristic frequencies and that the state of movements of the particles arising out of an initial disturbance tends to a superposition of these six characteristic vibrations of the lattice. These six frequencies would reduce to three for a square lattice on account of its symmetry; in all these two cases however, the amplitudes of vibrations of the particles vary inversely as the time elapsed from the instant of the initial disturbance. The physical interpretation of these results and their applicability to the case of a finite lattice are discussed.