Group velocity

Author: Leandro Alegsa

07-09-2021

The group velocity $v_{{\mathrm {g}}}$ is the velocity at which the envelope (i.e. the amplitude response) of a wave packet travels

$v_{\mathrm {g} }={\frac {\partial \omega }{\partial k}}$ ,

that is, the partial derivative of the angular frequency ω of $\omega$ the wave with respect to the angular wavenumber .

The green points move with group velocity, the red with phase velocity.

Interrelationships

With the phase velocity

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

$v_{\mathrm {p} }={\frac {\omega }{k}}=\lambda \,f$

with

the wavelength λ $\lambda$
of the frequency .

Substituting ω $\omega =v_{\rm {p}}\cdot k$ into the definition of group velocity, after applying the product rule, yields Rayleigh's relation:

$v_{\mathrm {g} }=v_{\mathrm {p} }+k{\frac {\mathrm {\partial } v_{\mathrm {p} }}{\mathrm {\partial } k}}$

With wavelength λ $\lambda =2\pi /k$ it can also be written as:

$v_{\mathrm {g} }=v_{\mathrm {p} }-\lambda {\frac {\mathrm {\partial } v_{\mathrm {p} }}{\mathrm {\partial } \lambda }}$

With the dispersion

The dispersion relation ω $\omega (k)$ describes how ω $\omega$ depends on

ω is $\omega$ proportional to :

${\frac {\omega }{k}}=v_{\mathrm {p} }={\text{konst.}}$

$\Rightarrow {\frac {\partial v_{\mathrm {p} }}{\partial k}}={\frac {\partial v_{\mathrm {p} }}{\partial \lambda }}=0$

then the group velocity is identical to the phase velocity:

$\Rightarrow v_{\mathrm {p} }=v_{\mathrm {g} }$

and the shape of the envelope is maintained.

If ω is $\omega$ not proportional to

${\frac {\omega }{k}}=v_{\mathrm {p} }={\text{f}}(f)\neq {\text{konst.}}\Rightarrow v_{\mathrm {p} }\neq v_{\mathrm {g} }$

dispersion is present. In this case, the envelope of the wave packet widens as it propagates, e.g. in the case of signals in optical waveguides.

With the signal speed

In virtually lossless media

Often the group velocity is thought of as the signal velocity $v_{s}$ at which the wave packet transports energy or information through space:

$v_{\mathrm {s} }=v_{\mathrm {g} }$

This is true in most cases, and always when losses are negligible:

In lossy media

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

$v_{\mathrm {s} }\neq v_{\mathrm {g} }$

For light pulses in strongly lossy media, the phase velocity can be much larger than the group velocity and even larger than the speed of light $c_{0}$ in vacuum. However, information transfer with faster than light speed is not possible, because for this the front speed is crucial, which can never reach faster than light speed:

$v_{\mathrm {s} }=v_{\mathrm {f} }\leq c_{0}$

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

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

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