 Monatomic-chain dispersion relation

For equal masses $m$ at spacing $d$ joined by nearest-neighbour springs of spring constant $K$, Newton's second law gives $m{\ddot{u}}_j=K(u_{j+1}+u_{j-1}-2u_j)$. A normal mode $u_j=u_0{e}^{i(kjd-\omega t)}$ givesThis single acoustic phonon branch is linear at small wavevector and has zero group velocity at the first Brillouin zone edge.