The group velocity in physics is the velocity at which the overall shape or envelope of a wave packet propagates. This is in contrast to the phase velocity, which is the velocity at which the individual wave crests or troughs propagate.
The group velocity can be calculated by taking the derivative of the dispersion relation, which relates the frequency and wavevector of a wave. In most cases, the group velocity will be equal to the phase velocity, but in cases of dispersion, they may differ.
One example of a real-world application of group velocity is in fiber optics, where the group velocity is used to determine the speed at which information can be transmitted through a fiber optic cable. Group velocity is also used in radar and sonar systems to determine the speed of moving objects.
No, according to the theory of relativity, the group velocity, like the phase velocity, cannot exceed the speed of light. However, in certain materials with negative refractive index, the group velocity can appear to be greater than the speed of light, but this is a result of the way the wave is propagating through the material and does not violate the laws of relativity.
The group velocity can affect the behavior of waves in a number of ways. For example, in dispersive media, the group velocity can cause different frequencies of a wave to travel at different speeds, resulting in the separation of colors in a prism. In quantum mechanics, the group velocity plays a role in the behavior of particles, such as in the phenomenon of wave-particle duality.