= Solution
A <dispersive> medium has frequency-dependent <phase velocity>, so different Fourier components generally propagate with different phases. A <dispersion relation> specifies $\omega$ as a function of the <wave number> $k$, or an equivalent relation between them. For a monochromatic factor $e^{i(kx-\omega t)}$, a point of fixed <phase> moves at
$$
\boxed{v_p=\frac\omega k.}
$$
For a narrow <wave packet> centered at $k_*$, expand $\omega(k)\simeq\omega(k_*)+(k-k_*)\omega'(k_*)$. The envelope then translates at the <group velocity>
$$
\boxed{v_g=\frac{d\omega}{dk}.}
$$
The <phase velocity> tracks crests of a monochromatic wave. The <group velocity> tracks a slowly varying envelope when the narrow-band approximation is valid; in the ideal lossless guide here it also equals the longitudinal energy-transport speed. It is not a universal formula for signal-front speed in every possible dispersive medium.
For the propagating branch of the supplied <electromagnetic waveguide> relation, take $\omega_0>0$, $k>0$ and $\omega=\sqrt{\omega_0^2+c^2k^2}$. Differentiation gives
$$
\boxed{v_p=\frac{\sqrt{\omega_0^2+c^2k^2}}k>c,\qquad
v_g=\frac{c^2k}{\sqrt{\omega_0^2+c^2k^2}}<c,\qquad v_pv_g=c^2.}
$$
The product alone would also allow $v_p=v_g=c$; the strictly positive cutoff is what makes the inequalities strict. At $k=0$ the phase-speed expression is undefined, and these formulas are interpreted through the propagating limit.
A moving monochromatic crest does not carry a new message. An information-bearing modulation is made from a range of frequencies; here its packet and energy move below $c$, and the high-frequency limit approaches $c$. Causal propagation of a newly switched-on disturbance is governed by the electromagnetic equations, so the superluminal <phase velocity> does not permit faster-than-light signalling. These are the <waveguide phase and group velocities> for a mode with nonzero cutoff, rather than a zero-cutoff mode with both velocities equal to $c$.
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