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Group velocity: speed of a wave packet and its physical significance

Group velocity is the rate at which a wave packet's envelope—and often energy or information—propagates. It differs from phase velocity and is central to dispersion, pulse spreading, and signal transport.

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Overview

Group velocity is the velocity at which the overall shape (envelope) of a modulated wave packet propagates through space. In many contexts it corresponds to the speed at which energy or a modulation travels, and it is closely related to but distinct from the phase velocity of individual sinusoidal components. For a simple wave described by angular frequency ω and wavenumber k, the group velocity is commonly written as v_g = dω/dk.

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Definition and basic properties

When a wave packet is formed by superposing waves of nearby frequencies, its envelope moves with the group velocity. Mathematically v_g = dω/dk follows from linearizing the dispersion relation ω(k). If ω is proportional to k (nondispersive medium), phase and group velocities are equal and the packet preserves its shape. In dispersive media they differ and the packet typically spreads.

Dispersion and observable effects

Dispersion occurs when different frequency components travel at different phase velocities, so d^2ω/dk^2 ≠ 0. This causes pulse broadening and can alter signal fidelity. Optical fibers, water waves, plasmas and waveguides all show dispersion governed by their specific dispersion relation. Engineers commonly quantify group velocity dispersion (GVD) to predict pulse evolution.

Uses and examples

  • In optics, group velocity determines pulse arrival times in fibers and impacts telecommunications.
  • In mechanics and seismology, it predicts how earthquake wave packets convey energy across Earth.
  • In quantum mechanics the group velocity of a particle's wave packet corresponds to its classical velocity in many cases.

Important distinctions and cautions

Group velocity is often treated as the speed at which information or energy is conveyed, but exceptions exist. In regions of strong dispersion or absorption one can observe superluminal or negative group velocities without violating causality—information speed (signal velocity) remains constrained by fundamental limits. Also note the difference from phase velocity, which applies to individual monochromatic waves and can exceed group velocity.

Historical note: The concept has its roots in classical wave theory and was developed to explain observations of wave packets and pulse propagation. Today it is a standard tool in physics and engineering for analyzing wave behavior across many systems.

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

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

with

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

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

With wavelength λ \lambda =2\pi /kit can also be written as:

{\displaystyle 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 kdepends on

  • ω is \omega proportional to k:

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

{\displaystyle \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:

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

and the shape of the envelope is maintained.

  • If ω is \omega not proportional to k

{\displaystyle {\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:

{\displaystyle 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:

{\displaystyle 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:

{\displaystyle 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:

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

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