Waves and Oscillations · Dispersion, group velocity, and energy transport

Superluminal phase without superluminal transport

Problem

A lossless guided mode has longitudinal dispersion relation \[ \omega^2=\omega_c^2+c^2k^2, \] where $\omega_c>0$ is the cutoff angular frequency and $c$ is the underlying bulk wave speed. For $\omega>\omega_c$, take $k>0$. Derive the phase and group velocities and prove their product is $c^2$. Show which is greater than $c$, which is less, and find both limits as $\omega\downarrow\omega_c$ and $\omega\to\infty$. Evaluate $k c/\omega_c$, $v_p/c$, and $v_g/c$ at $\omega=1.25\omega_c$. For $0<\omega<\omega_c$, derive the positive attenuation constant $\kappa$ when $k=i\kappa$, and explain why this is an evanescent rather than a propagating mode in an infinitely long ideal guide. Explain why $v_p>c$ above cutoff does not imply superluminal energy or information transport and state the narrowband assumption behind identifying $v_g$ with envelope transport.

Hint

Use $v_p=\omega/k$ and $2\omega\,d\omega/dk=2c^2k$.

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