Group velocity
The group velocity of a wave is the velocity with which the overall shape of the waves' 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. The expanding ring of waves is the wave group, within which one can discern individual wavelets of differing wavelengths traveling at different speeds. The longer waves travel faster than the group as a whole, but their amplitudes diminish as they approach the leading edge. The shorter waves travel more slowly, and their amplitudes diminish as they emerge from the trailing boundary of the group.
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.
One derivation of the formula for group velocity is as follows.
Consider a wave packet as a function of position x and time t: α(x,t).
Let A(k) be its Fourier transform at time t=0,
By the superposition principle, the wavepacket at any time t is
where ω is implicitly a function of k.
Assume that the wave packet α is almost monochromatic, so that A(k) is sharply peaked around a central wavenumber k0.
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