Group velocity and dispersion

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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, within which one can discern individual wavelets of
differing wavelengths traveling at different speeds. The shorter waves travel faster than the group as a
whole, but their amplitudes diminish as they approach the leading edge of the group. The longer waves
travel more slowly, and their amplitudes diminish as they emerge from the trailing boundary of the group.

group velocity vg is defined by the equation
where,
ω =2πf (angular frequency)
k =2π radians/ λ (wave number)
And, vp = ω/k (phase velocity)
∂ω/∂k
vg =

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 will travel at the group velocity,
while the individual peaks and troughs within the envelope will
move at the phase velocity.

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. If the wave packet has a narrow range of frequencies,
and ω(k) is approximately linear over that narrow range, the pulse distortion will
be small, in relation to the small nonlinearity. For example, for deep water
gravity waves, ω=√ gk , and hence vg = vp/2.
 This 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° =
arcsin(1/3) with the line of travel.