Past exam of the physics A course of the University of Cambridge / 2019 / Paper 2 / Section B / B9 / e

For positive angular frequency, the first three branches areEach branch is even in $k$, has minimum ${\omega }_n(0)=n\pi c_m/a$, and starts with zero slope. At large $|k|$, all approach the straight asymptote $\omega =c_m|k|$. Their group velocities are $v_{g,n}=c_m^{2}k/{\omega }_n$, whereas their longitudinal phase velocities are $v_{p,n}={\omega }_n/k$ for $k\ne 0$, giving $v_{p,n}{v}_{g,n}=c_m^2$.