Editing Group velocity (section)

== Definition and interpretation ==
[[File:Wave packet.svg|thumb|{{legend-line|dodgerblue solid|A [[wave packet]].}}{{legend-line|red dashed|The ''envelope'' of the wave packet. The envelope moves at the group velocity.}}]]
The group velocity {{math|''v''<sub>g</sub>}} is defined by the equation:<ref>{{Citation | publisher = Dover | isbn = 978-0-486-49556-9 | last = Brillouin | first = Léon | author-link = Léon Brillouin | title = Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices | year = 2003 | orig-year = 1946 | page=75 }}</ref><ref>{{Citation | publisher = Cambridge University Press | isbn = 978-0-521-01045-0 | last = Lighthill | first = James | author-link=James Lighthill | title = Waves in fluids | year = 2001 | orig-year=1978 | page=242 }}</ref><ref>{{harvtxt|Lighthill|1965}}</ref><ref>{{harvtxt|Hayes|1973}}</ref>
:<math>v_{\rm g} \ \equiv\  \frac{\partial \omega}{\partial k}\,</math>

where {{math|''ω''}} is the wave's [[angular frequency]] (usually expressed in [[radians per second]]), and {{math|''k''}} is the [[angular wavenumber]] (usually expressed in radians per meter). The [[phase velocity]] is: {{math|''v''<sub>p</sub> {{=}} ''ω''/''k''}}.

The [[function (mathematics)|function]] {{math|''ω''(''k'')}}, which gives {{math|''ω''}} as a function of {{math|''k''}}, is known as the [[dispersion relation]].

* If  {{math|''ω''}} is [[proportionality (mathematics)|directly proportional]] to  {{math|''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 {{math|(''ω'' {{=}} ''ak'' + ''b'', ''b'' ≠ 0)}}, then the group velocity and phase velocity are different. The envelope of a [[wave packet]] (see figure on right) will travel at the group velocity, while the individual peaks and troughs within the envelope will move at the phase velocity.
* If {{math|''ω''}} is not a linear function of {{math|''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 {{math|''k''}}), the group velocity {{math|''∂ω/∂k''}} will be different for different values of {{math|''k''}}. Therefore, the envelope does not move at a single velocity, but its wavenumber components ({{math|''k''}}) move at different velocities, distorting the envelope. If the wavepacket has a narrow range of frequencies, and {{math|''ω''(''k'')}} is approximately linear over that narrow range, the pulse distortion will be small, in relation to the small nonlinearity. See further discussion [[#Higher-order terms in dispersion|below]]. For example, for [[Gravity wave#Deep water|deep water]] [[gravity waves]], <math display="inline">\omega = \sqrt{gk}</math>, and hence {{math|''v''<sub>g</sub> {{=}} ''v''<sub>p</sub> /2}}.{{paragraph}} 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.<ref>G.B. Whitham (1974). ''Linear and Nonlinear Waves'' (John Wiley & Sons Inc., 1974) pp 409–410 [https://archive.org/details/LinearAndNonlinearWaves Online scan]</ref>            
{{Further|Airy wave theory#Table of wave quantities}}

===Derivation===
One derivation of the formula for group velocity is as follows.<ref name=Griffiths>
{{cite book
 | author=Griffiths, David J.
 | title=Introduction to Quantum Mechanics
 | url=https://archive.org/details/introductiontoqu00grif_200
 | url-access=limited
 | publisher=[[Prentice Hall]]
 | year=1995
 | page=[https://archive.org/details/introductiontoqu00grif_200/page/n61 48]
| isbn=9780131244054
 }}</ref><ref>
{{cite book
 | title = Quantum Mechanics: An Introduction for Device Physicists and Electrical Engineers
 | edition = 2nd
 | author = David K. Ferry
 | publisher = CRC Press
 | year = 2001
 | isbn = 978-0-7503-0725-3
 | pages = 18–19
 | url = https://books.google.com/books?id=imvYBULWPMQC&pg=PA18
| bibcode = 2001qmid.book.....F
 }}</ref>

Consider a [[wave packet]] as a function of position {{math|''x''}} and time {{math|''t'': ''α''(''x'',''t'')}}.

Let {{math|''A''(''k'')}} be its [[Fourier transform]] at time {{nowrap|{{math|''t'' {{=}} 0}}}},
:<math> \alpha(x, 0) = \int_{-\infty}^\infty dk \, A(k) e^{ikx}.</math>

By the [[superposition principle]], the wavepacket at any time  {{math|''t''}} is
:<math> \alpha(x, t) = \int_{-\infty}^\infty dk \, A(k) e^{i(kx - \omega t)},</math>
where  {{math|''ω''}} is implicitly a function of  {{math|''k''}}.

Assume that the wave packet  {{math|''α''}} is almost [[monochromatic]], so that  {{math|''A''(''k'')}} is sharply peaked around a central [[wavenumber]]  {{math|''k''<sub>0</sub>}}.

Then, [[linearization]] gives
:<math>\omega(k) \approx \omega_0 + \left(k - k_0\right)\omega'_0</math>
where 
:<math>\omega_0 = \omega(k_0)</math> and <math>\omega'_0 = \left.\frac{\partial \omega(k)}{\partial k}\right|_{k=k_0}</math> 
(see next section for discussion of this step). Then, after some algebra,
:<math> \alpha(x,t) = e^{i\left(k_0 x - \omega_0 t\right)}\int_{-\infty}^\infty dk \, A(k) e^{i(k - k_0)\left(x - \omega'_0 t\right)}.</math>

There are two factors in this expression. The first factor, <math>e^{i\left(k_0 x - \omega_0 t\right)}</math>, describes a perfect monochromatic wave with wavevector  {{math|''k''<sub>0</sub>}}, with peaks and troughs moving at the [[phase velocity]] <math>\omega_0/k_0</math>  within the envelope of the wavepacket.

The other factor, 
:<math>\int_{-\infty}^\infty dk \, A(k) e^{i(k - k_0)\left(x - \omega'_0 t\right)}</math>, 
gives the envelope of the wavepacket. This envelope function depends on position and time ''only'' through the combination <math>(x - \omega'_0 t)</math>.

Therefore, the envelope of the wavepacket travels at velocity 
:<math>\omega'_0 = \left.\frac{d\omega}{dk}\right|_{k=k_0}~,</math> 
which explains the group velocity formula.

=== Other expressions ===
For light, the refractive index {{math|''n''}}, vacuum wavelength  {{math|''λ<sub>0</sub>''}}, and wavelength in the medium  {{math|''λ''}}, are related by
:<math>\lambda_0 = \frac{2\pi c}{\omega}, \;\; \lambda = \frac{2\pi}{k} = \frac{2\pi v_{\rm p}}{\omega}, \;\; n = \frac{c}{v_{\rm p}} = \frac{\lambda_0}{\lambda},</math>
with  {{math|''v''<sub>p</sub>&nbsp;{{=}}&nbsp;''ω''/''k''}} the [[phase velocity]].

The group velocity, therefore, can be calculated by any of the following formulas,
:<math> \begin{align}
  v_{\rm g} &= \frac{c}{n + \omega \frac{\partial n}{\partial \omega}}
       = \frac{c}{n - \lambda_0 \frac{\partial n}{\partial \lambda_0}}\\
      &= v_{\rm p} \left(1 + \frac{\lambda}{n} \frac{\partial n}{\partial \lambda}\right)
       = v_{\rm p} - \lambda \frac{\partial v_{\rm p}}{\partial \lambda} = v_{\rm p} + k \frac{\partial v_{\rm p}}{\partial k}.
\end{align}</math>