Group speed

The green dots move at group speed,
the red at phase speed .

The group speed is the speed at which the envelope (i.e. the amplitude curve) of a wave packet moves ${\ displaystyle v _ {\ mathrm {g}}}$

${\ displaystyle v _ {\ mathrm {g}} = {\ frac {\ partial \ omega} {\ partial k}}}$,

thus the partial derivative of the angular frequency of the wave with respect to the circular wave number . ${\ displaystyle \ omega}$ ${\ displaystyle k}$

Connections

With the phase velocity

Using a Fourier series , one can imagine a wave packet as a superposition of individual waves of different frequencies . The individual waves each propagate with a certain phase velocity , which indicates the velocity with which points of constant phase move: ${\ displaystyle v _ {\ mathrm {p}}}$

${\ displaystyle v _ {\ mathrm {p}} = {\ frac {\ omega} {k}} = \ lambda \, f}$

With

• the wavelength ${\ displaystyle \ lambda}$
• the frequency .${\ displaystyle f}$

By inserting into the definition of the group speed, the Rayleigh relationship results after applying the product rule : ${\ displaystyle \ omega = v _ {\ rm {p}} \ cdot k}$

${\ displaystyle v _ {\ mathrm {g}} = v _ {\ mathrm {p}} + k {\ frac {\ mathrm {\ partial} v _ {\ mathrm {p}}} {\ mathrm {\ partial} k} }}$

With the wavelength it can also be written as: ${\ displaystyle \ lambda = 2 \ pi / k}$

${\ displaystyle v _ {\ mathrm {g}} = v _ {\ mathrm {p}} - \ lambda {\ frac {\ mathrm {\ partial} v _ {\ mathrm {p}}} {\ mathrm {\ partial} \ lambda}}}$

With the dispersion

The dispersion relation describes how from depends: ${\ displaystyle \ omega (k)}$${\ displaystyle \ omega}$${\ displaystyle k}$

• is proportional to :${\ displaystyle \ omega}$ ${\ displaystyle k}$
${\ displaystyle {\ frac {\ omega} {k}} = v _ {\ mathrm {p}} = {\ text {const.}}}$
${\ displaystyle \ Rightarrow {\ frac {\ partial v _ {\ mathrm {p}}} {\ partial k}} = {\ frac {\ partial v _ {\ mathrm {p}}} {\ partial \ lambda}} = 0}$
so the group velocity is identical to the phase velocity:
${\ displaystyle \ Rightarrow v _ {\ mathrm {p}} = v _ {\ mathrm {g}}}$
and the shape of the envelope is retained.
• If is not proportional to :${\ displaystyle \ omega}$ ${\ displaystyle k}$
${\ displaystyle {\ frac {\ omega} {k}} = v _ {\ mathrm {p}} = {\ text {f}} (f) \ neq {\ text {const.}} \ Rightarrow v _ {\ mathrm {p}} \ neq v _ {\ mathrm {g}}}$
there is dispersion . In this case the envelope of the wave packet widens as it propagates, e.g. B. for signals in fiber optics .

With the signal speed

In practically lossless media

One often thinks of group speed as the signal speed at which the wave packet transports energy or information through space: ${\ displaystyle v_ {s}}$

${\ displaystyle v _ {\ mathrm {s}} = v _ {\ mathrm {g}}}$

This is true in most cases, whenever losses can be neglected:

In lossy media

In lossy media, the signal speed is not identical to the group speed:

${\ displaystyle v _ {\ mathrm {s}} \ neq v _ {\ mathrm {g}}}$

In the case of light pulses in heavily lossy media, the phase velocity can be considerably greater than the group velocity and even greater than the velocity of light in a vacuum. However, it is not possible to transmit information at faster than light speed , since the front speed is decisive, as it can never reach faster than light speed: ${\ displaystyle c_ {0}}$

${\ displaystyle v _ {\ mathrm {s}} = v _ {\ mathrm {f}} \ leq c_ {0}}$

The front speed is the speed at which the wave fronts (i.e. areas of equal amplitude) and discontinuities of the wave are moving. It is defined as the limit value of the phase velocity for an infinitely large circular wave number:

${\ displaystyle v _ {\ mathrm {f}} = \ lim _ {k \ to \ infty} v _ {\ mathrm {p}}}$