Past exam of the mathematical methods II course of the University of Oxford / 2013 / 10 / d

Differentiate the positive-frequency dispersion relation with respect to $k$:Where $k\ne 0$, $c_p{c}_g=c^2$. For $a>0$ and $c\ne 0$, the magnitude of the group velocity is below $|c|$ even though the positive-$k$ phase velocity exceeds $|c|$. At $k=0$, the group velocity is zero. The two velocities describe envelope and phase transport, respectively.