Constant phase-group velocity product for cutoff dispersion ($v_P{v}_G=c^2$)

On a real positive-frequency branch of ${\omega }^2=a^2+c^2{k}^2$, phase velocity is $v_P=\omega /k$ and group velocity is $v_G=d\omega /dk=c^{2}k/\omega$. Their product is $c^2$. Real propagation requires ${\omega }^2>a^2$ when solving for a nonzero real $k$; the inequality $\omega >a$ alone assumes $a$ is nonnegative. The phase-group product does not, without further assumptions on the medium, prove a general statement about signal-front propagation.