Question

What is the relation between the group velocity and phase velocity in a non-dispersive medium?

• A. \(v_g>v_p\)
• B. \(v_g<v_p\)
• C. \(v_g = v_p\)
• D. \(v_g = 2v_p\)

Hint:

In a {non-dispersive medium}: \[ v_g = v_p \] In a {dispersive medium}: \[ v_g \ne v_p \]

Answer 1

The Correct Option is C

Concept: Phase velocity \(v_p\) is the velocity with which a single wave phase propagates through space. \[ v_p = \frac{\omega}{k} \] where

• \(\omega\) = angular frequency
• \(k\) = wave number

Step 1: Define group velocity.} Group velocity \(v_g\) is the velocity at which the envelope of a wave packet (or energy) travels. \[ v_g = \frac{d\omega}{dk} \]
Step 2: Condition for a non-dispersive medium.} In a non-dispersive medium, the wave speed does not depend on frequency. This means \[ \omega \propto k \]
Step 3: Resulting relationship.} Under this condition, \[ v_g = v_p \] Thus, in a non-dispersive medium, the group velocity equals the phase velocity.