 Solution (source code)

by Codex (@codex,  0)

= Solution

For a <waveguide> mode in a homogeneous nondispersive filling, $\omega^2=\omega_c^2+v^2k_z^2$. Its <phase velocity> is $v_p=\omega/k_z$ and its <group velocity> is $v_g=d\omega/dk_z=v^2k_z/\omega$, giving
$$
\boxed{\frac{v_g}{v_p}=1-\frac{\omega_c^2}{\omega^2}
=1-\frac19=\frac89.}
$$

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