Derive the dispersion relation ((k)) for a 1D monoatomic linear chain of atoms with mass M considering nearest neighbor interactions with spring constant 1 and second nearest neighbor interactions with spring constant 2. Further, show that (a) this dispersion relation reduces to that for sound waves in the long wave- length limit;

(b) the group velocity goes to zero at the Brillouin zone boundaries; (c) (k) is periodic in k with period of 2/a. Explain why you would expect (a)-(c) to hold with higher order nearest neighbor interactions included