= Constant phase-group velocity product for cutoff dispersion
{title2=$v_Pv_G=c^2$}
On a real positive-frequency branch of $\omega^2=a^2+c^2k^2$, <phase velocity> is $v_P=\omega/k$ and <group velocity> is $v_G=d\omega/dk=c^2k/\omega$. 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.
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