Final answer:
The relationship between group velocity and phase velocity involves the definitions of these velocities in terms of angular frequency and wavenumber. In non-dispersive media, they are equal, while in dispersive media, they differ. Group velocity is important in the context of energy transfer.
Step-by-step explanation:
The relationship between the group velocity and phase velocity of a wave is a concept studied in physics, specifically within the field of wave mechanics. To show the relationship between them, consider a wave with angular frequency ω and wavenumber k, where phase velocity Vp is defined as Vp = ω/k and group velocity Vg as the derivative of ω with respect to k.
In cases where the wave equation allows for the wavelengths involved in a wave packet to disperse over time, the group velocity can be understood as the velocity at which the overall shape of the waves' amplitudes—known as the modulation or envelope—propagates through space.
For a non-dispersive medium, where all wavelengths travel at the same speed, the group velocity is equal to the phase velocity. However, in a dispersive medium, these velocities can differ.
The group velocity is particularly important for understanding the transfer of energy in wave packets.