Show that the dispersion relation for the lattice vibrations of a chain of identical masses M, in which each is connected to its first and second nearest neighbors by springs of spring constants K and K2 respectively, is given by:
ω^2 = 2K[1 - cos(ka)] + 2K2[1 - cos(2ka)]
where a is the equilibrium spacing. Show that this dispersion relation reduces to that for sound waves in the long-wavelength limit (ensuring that the velocity corresponds to that predicted by the elastic modulus of the crystal). Also, show that the group velocity vanishes at k = π/a and that ω is periodic in k with a period of 2π/a.