Introduction:
In the study of waves, two important velocities are often discussed - the phase velocity and the group velocity. These velocities describe different aspects of wave propagation in a medium. Whether these velocities are equal or different depends on the nature of the medium.

Explanation:
1. Non-dispersive medium:
In a non-dispersive medium, the velocity of propagation of waves does not depend on their frequency. This means that all the waves of different frequencies travel at the same speed. In such a medium, the phase velocity and the group velocity are equal.

2. Dispersive medium:
In a dispersive medium, the velocity of propagation of waves depends on their frequency. This means that waves of different frequencies travel at different speeds. In such a medium, the phase velocity and the group velocity are not the same.

3. Phase velocity:
The phase velocity of a wave is the velocity at which the phase of the wave propagates through space. It can be calculated by dividing the wavelength of the wave by its period (or frequency). Mathematically, the phase velocity (v_phase) is given by the equation:
v_phase = λ / T

4. Group velocity:
The group velocity of a wave is the velocity at which the shape or envelope of the wave propagates through space. It can be calculated by taking the derivative of the wave's angular frequency (ω) with respect to its wave vector (k). Mathematically, the group velocity (v_group) is given by the equation:
v_group = dω / dk

5. Relationship between phase velocity and group velocity:
In a non-dispersive medium, the phase velocity and the group velocity are the same. This means that the shape or envelope of the wave propagates at the same speed as the phase of the wave. In such a case, the wave does not get distorted as it propagates.

In a dispersive medium, the phase velocity and the group velocity are different. This means that the shape or envelope of the wave propagates at a different speed compared to the phase of the wave. In such a case, the wave gets distorted as it propagates, leading to dispersion.

Conclusion:
In conclusion, in a non-dispersive medium, the group velocity is equal to the phase velocity, while in a dispersive medium, the group velocity and the phase velocity are not the same. The distinction between these velocities is important in understanding wave propagation and the behavior of waves in different media.