Step 1: Define group delay and its role in distortion.
Group delay is $\tau_g = -d\phi/d\omega$. For a signal to pass without distortion, all its frequency components must arrive together. This requires $\tau_g$ to be constant for all $\omega$.
Step 2: Relate constant group delay to the phase constant.
For phase velocity $v_p = \omega/\beta$ to be the same at every frequency (no dispersion), $\beta/\omega$ must be constant, meaning $\beta = k\omega$ (a linear relationship with frequency).
Step 3: Verify consistency with distortionless condition.
If $\beta = k\omega$, then $\tau_g = d\beta/d\omega = k = $ constant. Constant group delay ensures all spectral components arrive simultaneously. \[ \boxed{\text{Varies directly with frequency}} \]