Group Velocity — Physica

Equivalent forms

v_g = \frac{\Delta\omega}{\Delta k}

v_g = v_p - \lambda\frac{dv_p}{d\lambda}

Energy and information travel at the slope of the dispersion curve, not at the speed of the crests — a distinction that only a derivative can capture.

Symbol	Name	SI unit	Dimension	Range
$domega$	Angular frequency span Change in angular frequency across the packet's band	rad/s	T^-1	0.1 – 10
$dk$	Wavenumber span Change in wavenumber across the packet's band	rad/m	L^-1	0.1 – 10
$v_g$	Group velocityoutput Speed of the wave-packet envelope (energy/information)	m/s	L T^-1	0 – 100

Unit systems

Symbol	SI	CGS	Imperial
$domega$	rad/s	rad/s	rad/s
$dk$	rad/m	rad/cm	rad/m
$v_g$	m/s	cm/s	ft/s

Where it holds

Valid for a narrow-band wave packet in a medium with smooth dispersion omega(k). Near sharp resonances the group velocity loses its meaning as a signal speed; the front velocity is then the relevant bound and never exceeds c.

Discovery

W. R. Hamilton and Lord Rayleigh · 1877

Hamilton introduced the idea of group velocity, and Rayleigh, in his Theory of Sound, gave the clear distinction between the velocity of individual wavelets and the velocity of the group, explaining why energy travels at the latter.

Try this

Why can the ripples inside a patch of water waves race ahead of the patch itself and vanish at its front edge?

A wave packet is built from components near omega = 2*10^15 rad/s and k = 6.7*10^6 rad/m. Over a small band, omega rises by 1.0*10^14 rad/s as k rises by 4.0*10^5 rad/m. What is the group velocity?

Research status: stable

Real-world applications

Fiber-optic communications budget pulse spreading using group-velocity dispersion.
Slow-light and fast-light experiments tailor v_g near atomic resonances.
Ultrashort-pulse lasers compensate group delay to keep pulses femtosecond-short.
Ocean and seismic wave forecasting tracks energy at the group velocity of swells.

Common misconceptions

Group velocity always equals phase velocity — only in non-dispersive media.
A group velocity above c violates relativity — it does not, because no information rides faster than the wavefront (front velocity < $= c)$ .
Group velocity is the speed of any single wave — it is the speed of the envelope built from a band of waves.

Experimental verification

Time-of-flight of light pulses through dispersive fibers, and the lag of ultrasonic pulse envelopes versus their carrier, both measure

v_g = d(\omega )/dk

distinct from the phase velocity, confirming that energy arrives at the group velocity.

Derivation

Superpose two waves of nearly equal (omega, k):

\cos (k_{1}*x - \omega 1*t) + \cos (k_{2}*x - \omega 2*t) = 2*\cos ((dk*x

- domega*t)/2)*cos((k*x - omega*t)).

The fast factor (the carrier) moves at the phase velocity omega/k; the slow envelope factor moves where dk*x - domega*t is constant, i.e. at

x/t =

domega/dk.

In the continuum limit this is

v_g = d(\omega )/dk

.

Limiting cases

Non-dispersive medium⟶

v_g = v_p

When

\omega = v*k

exactly, the slope equals the ratio and the packet keeps pace with its crests — light in vacuum, sound in air.

Anomalous dispersion⟶ v_g > c or even negativeNear a strong absorption line the envelope can appear to move faster than light or backward, without ever transmitting information faster than c.

What if…

What if the dispersion curve is flat (domega -> 0 over a band)?

The group velocity approaches zero — 'stopped light.' The envelope freezes while the carrier crests keep oscillating, a hallmark of slow-light media.

What if the medium is non-dispersive?

$\omega = v*k$ makes $d(\omega )/dk = \omega /k = v$ , so group and phase velocities coincide and the packet propagates without spreading.

1

Group velocity from a finite-difference slope

Given ·

domega:: 1
dk:: 4

Find · v_g

Steps

Approximate the derivative by the finite slope: $v_g =$ domega/dk.
Substitute the band values domega $= 1.0*10^14\,\mathrm{rad/s}$ , $dk = 4.0*10^5\,\mathrm{rad/m}$ .
$v_g = 2.5*10^8\,\mathrm{m/s}$ (about 0.83 c).

Result · Using the band values,

v_g =

domega/

dk = (1.0*10^14)/(4.0*10^5) = 2.5*10^8\,\mathrm{m/s}

. With the slider scale (domega

= 1.0

,

dk = 4.0)

the ratio is 0.25 in the displayed units.

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