What is the Difference Between Phase Velocity and Group Velocity?

Phase velocity and group velocity are two important concepts in wave mechanics, and they differ in the way they describe the propagation of waves. Here are the main differences between them:

• Phase Velocity: This is the velocity at which the phase of a wave travels. In other words, it is the speed at which the individual wave crests move. Phase velocity is given by the equation:

$$V_p = \frac{\omega}{k}$$

where $$\omega$$ is the angular velocity and $$k$$ is the angular wave number.

• Group Velocity: This is the velocity with which a wave packet travels. The group velocity is related to the phase velocity through the following equation:

$$Vg = Vp + k \frac{dV_p}{dk}$$

The group velocity is directly proportional to the phase velocity, meaning that when the group velocity increases, the phase velocity also increases, and vice versa.

In summary, phase velocity describes the speed at which the individual wave crests move, while group velocity describes the speed at which the wave packet, or the envelope containing the wave information, travels. These two velocities are related through the equation mentioned above.

Comparative Table: Phase Velocity vs Group Velocity

The difference between phase velocity and group velocity can be summarized in the following table:

In general, phase velocity is greater than the group velocity, and it is greater than the velocity of light, while group velocity is equal to the particle velocity. The relationship between phase velocity and group velocity can be expressed as $$Vg = Vp + k\left(\frac{dVp}{dk}\right)$$. For dispersive and non-dispersive waves, the relation between phase velocity and group velocity is given by $$\left(\frac{dVp}{dk}\right) \ne 0$$ and $$Vp \ne Vg$$ for dispersive waves, and $$\left(\frac{dVp}{dk}\right) = 0$$ and $$Vp = V_g$$ for non-dispersive waves.