= 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 $k>0$,
$$
\boxed{v_p=\sqrt{\omega/m}.}
$$
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 $m\ne0$ 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>.
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