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⭐ Core Definition: 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.

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👉 Group velocity in the context of Wave shoaling

In fluid dynamics, wave shoaling is the effect by which surface waves, entering shallower water, increase in wave height. It is caused by the fact that the group velocity, which is also the wave-energy transport velocity, decreases with water depth. Under stationary conditions, a decrease in transport speed must be compensated by an increase in energy density in order to maintain a constant energy flux. Shoaling waves will also exhibit a reduction in wavelength while the frequency remains constant.

In other words, as the waves approach the shore and the water gets shallower, the waves get taller, slow down, and get closer together.

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⭐ Core Definition: Group velocity
👉 Group velocity in the context of Wave shoaling
Group velocity in the context of Phase velocity
Group velocity in the context of Signal velocity
Group velocity in the context of Dispersion relation

Group velocity in the context of Phase velocity

The phase velocity of a wave is the speed of any wavefront, a surface of constant phase. This is the velocity at which the phase of any constant-frequency component of the wave travels. For such a spectral component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase velocity of light waves is not a physically meaningful quantity and is not related to information transfer.

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Group velocity in the context of Signal velocity

The signal velocity is the speed at which a wave carries information. It describes how quickly a message can be communicated (using any particular method) between two separated parties. No signal velocity can exceed the speed of a light pulse in a vacuum (by special relativity).

Signal velocity is usually equal to group velocity (the speed of a short "pulse" or of a wave-packet's middle or "envelope"). However, in a few special cases (e.g., media designed to amplify the front-most parts of a pulse and then attenuate the back section of the pulse), group velocity can exceed the speed of light in vacuum, while the signal velocity will still be less than or equal to the speed of light in vacuum.

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Group velocity in the context of Dispersion relation

In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the dispersion relation, one can calculate the frequency-dependent phase velocity and group velocity of each sinusoidal component of a wave in the medium, as a function of frequency. In addition to the geometry-dependent and material-dependent dispersion relations, the overarching Kramers–Kronig relations describe the frequency-dependence of wave propagation and attenuation.

Dispersion may be caused either by geometric boundary conditions (waveguides, shallow water) or by interaction of the waves with the transmitting medium. Elementary particles, considered as matter waves, have a nontrivial dispersion relation, even in the absence of geometric constraints and other media.

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Group velocity in the context of "Wave shoaling"

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⭐ Core Definition: Group velocity

👉 Group velocity in the context of Wave shoaling

Group velocity in the context of Phase velocity

Group velocity in the context of Signal velocity

Group velocity in the context of Dispersion relation