The group velocity is the velocity at which the energy or information of a wave packet (a group of waves) travels, while the phase velocity is the velocity at which the individual waves in the packet travel.
Group and phase velocity are related through the dispersion relation, which is the mathematical relationship between the frequency and wavelength of a wave. The group velocity is equal to the phase velocity times the group index, which takes into account the dispersion of the medium the wave is traveling through.
It is important to distinguish between group and phase velocity because they have different physical meanings and can have different values. The group velocity is related to the propagation of energy and information, while the phase velocity is related to the propagation of the wave itself. This distinction is particularly important in the study of wave phenomena, such as in optics and acoustics.
For different types of waves, the group and phase velocities can have different values and behaviors. For example, in dispersive media, the group velocity may be different for waves of different frequencies, leading to phenomena such as dispersion and pulse spreading. In non-dispersive media, the group and phase velocities are equal and there is no dispersion.
In some cases, the group velocity can appear to exceed the speed of light, but this does not violate the fundamental principle of special relativity. This is because the group velocity is not the speed at which any single wave travels, but rather the speed at which the energy or information of the wave packet travels. Therefore, the group velocity is not limited by the speed of light, but individual waves within the group cannot travel faster than the speed of light.