= Solution
For a real propagating positive-frequency branch, require $\omega^2>a^2$ and $c\ne0$. If $a$ is the usual nonnegative cutoff, the source condition $\omega>a$ gives this branch. With an arbitrary real negative $a$, that inequality alone is insufficient: $a=-2$ and $\omega=1$ satisfy it but make the displayed square root imaginary.
On the positive-frequency real branch, the <phase velocity> $v_P=\omega/k$ gives
$$
v_P=\frac{c\omega}{\sqrt{\omega^2-a^2}},\qquad k=\frac{\sqrt{\omega^2-a^2}}c.
$$
Differentiate the <dispersion relation> to obtain the <group velocity>:
$$
\frac{dk}{d\omega}=\frac{\omega}{c\sqrt{\omega^2-a^2}},\qquad v_G=\frac{d\omega}{dk}=\frac{c\sqrt{\omega^2-a^2}}\omega.
$$
Therefore
$$
\boxed{v_G=\frac{c^2}{v_P},\qquad v_Pv_G=c^2}.
$$
For $c>0$ and a nonzero cutoff, $v_P>c$ while $v_G<c$; when $a=0$, both equal $c$. The constant $c$ is the <velocity> scale in this assumed medium, and need not independently be identified with the vacuum speed of light. A <group velocity> describes a narrow-band wave packet on the chosen differentiable branch, rather than automatically fixing a signal front <velocity> in every dispersive model.
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