Solution
= Solution
The <vacuum> <wave equation> gives $\omega^2=c^2[k_g^2+(\pi/a)^2]$. The <cutoff frequency> is $f_c=c/(2a)=1.499\,\mathrm{GHz}$. In a lossless empty guide, <energy> travels at <group velocity>,
$$
\boxed{v_E=\frac{d\omega}{dk_g}=c\sqrt{1-\frac{f_c^2}{f^2}}
=0.8003c\simeq2.40\times10^8\,\mathrm{m\,s^{-1}}.}
$$
The larger <phase velocity> $\omega/k_g=c^2/v_E$ is not the <speed> of <energy> transport.