Step 1: Understanding the Concept:
Phase velocity ($v_p$) represents the speed at which a single specific phase of a wave travels. Group velocity ($v_g$) represents the speed at which the overall envelope (or energy) of a wave packet travels.
Step 2: Key Formula or Approach:
The formulas defining these velocities are:
Phase velocity: $v_p = \frac{\omega}{k}$
Group velocity: $v_g = \frac{d\omega}{dk}$
where $\omega$ is the angular frequency and $k$ is the wave number.
Step 3: Detailed Explanation:
From the phase velocity formula, we can express the angular frequency as:
\[ \omega = v_p k \]
To find the group velocity, we differentiate $\omega$ with respect to $k$:
\[ v_g = \frac{d\omega}{dk} = \frac{d}{dk} (v_p k) \]
By definition, a non-dispersive medium is one where the wave speed is completely independent of the frequency or wave number. This means $v_p$ is a constant.
Since $v_p$ is constant, we can pull it out of the derivative:
\[ v_g = v_p \frac{dk}{dk} = v_p (1) \]
Therefore, in a non-dispersive medium, the group velocity is exactly equal to the phase velocity ($v_g = v_p$).
Step 4: Final Answer:
The correct option is (C).