Group velocity

The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space. For example, if a stone is thrown into the middle of a very still pond, a circular pattern of waves with a quiescent center appears in the water, also known as a capillary wave. The expanding ring of waves is the wave group or wave packet, within which one can discern individual waves that travel faster than the group as a whole. The amplitudes of the individual waves grow as they emerge from the trailing edge of the group and diminish as they approach the leading edge of the group. The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877. The group velocity vg is defined by the equation: where ω is the wave's angular frequency (usually expressed in radians per second), and k is the angular wavenumber (usually expressed in radians per meter). The phase velocity is: vp = ω/k. The function ω(k), which gives ω as a function of k, is known as the dispersion relation. If ω is directly proportional to k, then the group velocity is exactly equal to the phase velocity. A wave of any shape will travel undistorted at this velocity. If ω is a linear function of k, but not directly proportional (ω = ak + b), then the group velocity and phase velocity are different. The envelope of a wave packet (see figure on right) will travel at the group velocity, while the individual peaks and troughs within the envelope will move at the phase velocity. If ω is not a linear function of k, the envelope of a wave packet will become distorted as it travels. Since a wave packet contains a range of different frequencies (and hence different values of k), the group velocity ∂ω/∂k will be different for different values of k. Therefore, the envelope does not move at a single velocity, but its wavenumber components (k) move at different velocities, distorting the envelope.