Group velocity

Lore & Background

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. For surface gravity waves on deep water, the phase velocity is twice the group velocity, which underlies the Kelvin wake pattern for the bow wave of all ships and swimming objects. Regardless of how fast they are moving, as long as their velocity is constant, on each side the wake forms an angle of 19.47° with the line of travel.

In a wave packet, individual waves 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. New waves seem to emerge at the back of a wave group, grow in amplitude until they are at the center of the group, and vanish at the wave group front. For example, if a stone is thrown into a very still pond, a circular pattern of waves with a quiescent center appears; the expanding ring of waves is the wave group, within which individual waves travel faster than the group.

If the angular frequency ω is directly proportional to the wavenumber k, then the group velocity equals the phase velocity and a wave of any shape travels undistorted. If ω is a linear function of k but not directly proportional, the group and phase velocities differ. If ω is not a linear function of k, the envelope of a wave packet becomes distorted as it travels because different wavenumber components move at different velocities. For deep water gravity waves, ω is proportional to the square root of k, and the group velocity is half the phase velocity.

Reader's Guide

Group velocity is significant because it represents the speed at which the envelope of a wave packet—and often the energy or information it carries—propagates. In most cases, the group velocity can be thought of as the signal velocity of the waveform. However, in lossy or gainful media, the group velocity may not be a well-defined or meaningful quantity; Brillouin argued that in a lossy medium the group velocity ceases to have a clear physical meaning. An example concerning the transmission of electromagnetic waves through an atomic gas is given by Loudon. Another example is mechanical waves in the solar photosphere, where the energy velocity is often substantially lower than the waves' group velocity due to damping.

Despite this ambiguity, a common way to extend the concept to complex media is to consider spatially damped plane wave solutions and apply the usual formula to the real part of the wavevector. This generalization can behave strangely, and the example of anomalous dispersion serves as a good illustration. The legacy of group velocity is its central role in understanding dispersion effects, such as the distortion of wave packets in optical fibers and the design of high-power, short-pulse lasers. The concept also explains why the wake of a ship always forms a specific angle, independent of the ship's speed.

Did You Know?

• For deep water gravity waves, the phase velocity is twice the group velocity.
• The Kelvin wake pattern for ships forms an angle of 19.47° on each side of the line of travel.