= Waveguide phase and group velocities
{title2=$v_{\mathrm{ph}}v_g=c^2$}
For a propagating branch of an ideal <vacuum>-filled <rectangular electromagnetic waveguide>, the <phase velocity> tracks constant <wave phase> along the axis, whereas the <group velocity> describes a sufficiently narrow, weakly distorted <wave packet>. Using $\omega^2=c^2\beta^2+\omega_c^2$ gives
$$
v_{\rm ph}=\frac{\omega}{\beta}=\frac{c}{\sqrt{1-(\omega_c/\omega)^2}},\qquad
v_g=\frac{d\omega}{d\beta}=\frac{c^2\beta}{\omega}=c\sqrt{1-(\omega_c/\omega)^2}.
$$
Thus $v_{\rm ph}>c$, $0<v_g<c$, and $v_{\rm ph}v_g=c^2$. Differentiating the <dispersion relation> proves the second formula; dividing it by $\beta^2$ proves the first. This is the same mathematical structure as <Klein-Gordon packet velocities>. In the stipulated $f=1.25f_c$ example,
$$
v_{\rm ph}=\frac53c,\qquad v_g=\frac35c.
$$
Over $0.30\,\mathrm m$, the corresponding narrow-envelope delay is $(0.50\,\mathrm m)/c$, approximately $1.668\,\mathrm{ns}$. Timing a carrier <wave crest> would give $0.6004\,\mathrm{ns}$; it does not time the arrival of a newly introduced message.
There is an independent <energy> check. For the forward $\mathrm{TE}_{10}$ fields, the average axial <Poynting vector> is $\langle S_z\rangle=\beta|E_y|^2/(2\mu_0\omega)$. Integrating across the guide gives average <power> $P$ and average <electromagnetic energy> per unit length $U'$:
$$
P=\frac{ab\beta E_0^2}{4\mu_0\omega},\qquad U'=\frac{\epsilon_0abE_0^2}{4},\qquad \frac{P}{U'}=\frac{c^2\beta}{\omega}=v_g.
$$
The transverse <sine> and <cosine> squared each integrate to $a/2$; $\beta^2+(\pi/a)^2=\omega^2/c^2$ then makes the integrated electric and magnetic contributions to $U'$ equal. The axial <energy> transport therefore agrees with the <group velocity> in this ideal model.
As $\omega\downarrow\omega_c$, $v_g\to0$ and $v_{\rm ph}\to\infty$; exactly at cutoff $\omega/\beta$ is undefined. Strong <wave dispersion> close to cutoff also limits the narrow-packet approximation. Neither a large <phase velocity> nor a pulse peak establishes superluminal <information velocity>; <front velocity> asks a separate causal question. In a uniform filling with constant positive <permittivity> $\varepsilon$ and <magnetic permeability> $\mu$, replace $c$ by $u=1/\sqrt{\varepsilon\mu}$, giving $v_{\rm ph}v_g=u^2$. Frequency-dependent material response or significant loss requires differentiating the actual <dispersion relation>, so this product is not universal.
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