Past exam of the mathematical methods II course of the University of Oxford / 2021 / 11 / iii / Solution

A fixed wave phase satisfies $kx-\omega t=\text{constant}$, so the signed phase velocity is $v_p=\omega /k$. The group velocity $v_g=d\omega /dk$ gives the leading translation of the wave envelope of a narrow-band wave packet on a differentiable dispersion relation branch.

Here $v_p=k/m$. For the conventional positive branch $m>0$, $\omega >0$ and $k>0$,On the opposite propagation branch $k<0$ it is $-\sqrt{\omega /m}$, so $\sqrt{\omega /m}$ is the phase speed for either sign of $k$. More generally, for real $m\ne 0$ use $v_p=k/m$ with $k^2=m\omega$, allowing the sign of $\omega$ to follow that of $m$. The $k=0$ constant solution has no distinguished phase velocity.