= Group velocity
{title2=$v_g=\nabla_{\mathbf k}\omega$}
{wiki}
The <group velocity> of a differentiable, real <dispersion relation> branch is
$$
\mathbf c_g=\nabla_{\mathbf k}\omega(\mathbf k_0),\qquad c_g=\frac{d\omega}{dk}(k_0)\ \text{in one dimension}.
$$
It describes the leading translation of a narrow-band <wave packet> centred on <wavevector> $\mathbf k_0$, assuming a linear model, a locally uniform medium and negligible loss. Narrow-band means that the appreciably excited <wavevectors> occupy a small neighbourhood on which one smooth branch can be linearized. Write its complex wavefield as
$$
\psi(\mathbf x,t)=\int a(\mathbf q)e^{i[(\mathbf k_0+\mathbf q)\cdot\mathbf x-\omega(\mathbf k_0+\mathbf q)t]}\,d\mathbf q.
$$
The <Taylor expansion> $\omega(\mathbf k_0+\mathbf q)\simeq\omega_0+\mathbf c_g\cdot\mathbf q$ gives
$$
\psi(\mathbf x,t)\simeq e^{i(\mathbf k_0\cdot\mathbf x-\omega_0t)}A(\mathbf x-\mathbf c_gt),\qquad A(\mathbf y)=\int a(\mathbf q)e^{i\mathbf q\cdot\mathbf y}\,d\mathbf q.
$$
Thus the rapid <wave phase> follows the <phase velocity>, while the slowly varying <wave envelope> follows the <group velocity>. This <Fourier analysis> argument is an outline of the approximation, rather than a statement that every <wave packet> translates rigidly forever. In one dimension a useful short-time condition is $|\omega''(k_0)|t(\Delta k)^2\ll1$, where $\Delta k$ is the characteristic spectral width; neglected higher derivatives must also remain small. Curvature of the <dispersion relation> eventually changes the <wave envelope>.<Pulse Propagation by Richard Fitzpatrick>[equations 9.9–9.16]{ref}
Two nearby <plane waves> make the derivative visible without an integral. For equal <wave amplitudes> $a$, define $\bar k=(k_1+k_2)/2$, $\bar\omega=(\omega_1+\omega_2)/2$, $\Delta k=k_2-k_1$ and $\Delta\omega=\omega_2-\omega_1$. The <superposition principle> gives the exact identity
$$
a\cos(k_1x-\omega_1t)+a\cos(k_2x-\omega_2t)=2a\cos\!\left(\frac{\Delta k\,x-\Delta\omega\,t}{2}\right)\cos(\bar kx-\bar\omega t).
$$
The periodic <wave envelope> has speed $\Delta\omega/\Delta k$, approaching <group velocity> as the two <wavenumbers> coalesce, while the carrier <wave phase> has speed $\bar\omega/\bar k$. Two tones alone form an infinite beat train, not a localized <wave packet>. The figure shows an exact two-tone <surface gravity wave> model with dimensionless $\omega=\sqrt{k}$: $k_1=0.9$, $k_2=1.1$. Its carrier <phase velocity> is approximately $0.99875$, its finite-difference envelope velocity approximately $0.50063$, and $\omega'(1)=0.5$ is the limiting <group velocity>. The marked carrier phase line need not coincide exactly with a maximum of the full modulated wave.
\Image[/phase-group-dispersion-beats.png]
{title=Exact analytical two-tone gravity-wave model: carrier phase advances through a slower periodic envelope}
The deep-water <surface gravity wave> therefore has $c_g=c_p/2$, whereas the deep-water <capillary wave> limit has $c_g=3c_p/2$: crests can drift either forward or backward relative to a <wave envelope>. For finite-depth <surface gravity waves>, differentiating the <dispersion relation> gives
$$
c_g=\frac{c_p}{2}\left(1+\frac{2kh}{\sinh(2kh)}\right),\qquad k>0.
$$
Here $\sinh$ is the <hyperbolic sine>. The shallow-water limit has $c_g\simeq c_p\simeq\sqrt{gh}$; the deep-water limit has $c_g\simeq c_p/2$.<Free Surface Water Waves (13.012 Hydrodynamics for Ocean Engineers, Reading 5) by A. H. Techet>[equations 6.8–6.11]{ref}
In the wave tank, produce a finite train with a smooth turn-on and turn-off and time the same <wave envelope> feature at two probes separated by distance $L$; its transit estimate is $L/\Delta t$. Independently follow <wave crests> through that train. Techet's 2003 lab asks students to observe a deep-water train's foremost crest disappearing as the group moves forward, and to compare deep- and shallow-water cases. The interpretation is that a <wave crest> overtakes the slower <wave envelope>, dwindling at its front while other crests become visible at its rear. This is a qualitative expected observation in the teaching protocol, not a published set of measured curves.<Waves, phase speed, and group speed (13.012 Hydrodynamics for Ocean Engineers, Laboratory 1) by A. H. Techet>{ref}
A useful controlled version keeps the paddle <wave amplitude> small, records $h$ and $\lambda$, and checks both $ka\ll1$ and $a/h\ll1$. Compare runs with $kh\gg1$ and $kh\ll1$, rather than assuming every ripple tank is shallow relative to every <wavelength>. Increase train duration at fixed central <frequency> to narrow its spectrum, and check that the envelope transit estimate stabilizes. Calibrate both probes, sample fast enough to resolve the carrier, and compare records before reflected waves return; an absorbing beach reduces this contamination. A current changes laboratory velocities, large <wave amplitudes> introduce nonlinear effects, and very short <wavelengths> require <surface tension>. Strong <damping> or noticeable reshaping makes the chosen threshold or peak an unreliable marker of simple <group velocity>. These are proposed controls and model limits, not claims that the cited lab measured every correction.
Three velocities should consequently be kept separate. The <group velocity> predicts narrow-band <wave envelope> motion. An energy-transport velocity is <energy flux> divided by <energy density>, using the appropriate time average; it agrees with <group velocity> in standard lossless linear wave models when all stored <energy> is included.<Mathematical Methods in Nanophotonics, MIT 18.369, Spring 2016 — lectures 18–19 by Steven G. Johnson>{ref} A signal-front velocity concerns the earliest response to a new disturbance and requires a causal model over its full spectrum. A strictly band-limited <wave packet> has no sharply bounded onset, and a mathematical two-tone envelope cannot test front arrival. In strongly absorbing or broadband propagation, a single <group velocity> need not describe the observed peak, energy transport, or first detectable response. A complex $\omega$ or $k$ also prevents interpreting its derivative as an ordinary real transport velocity without further analysis.
In particular, an unusually large or negative <group velocity> from pulse reshaping does not establish superluminal information transmission. For the causal dielectric model analysed by Fitzpatrick, the field ahead of the front vanishes for $t<x/c$, even when its <phase velocity> exceeds the vacuum <speed of light> $c$. That result concerns the electromagnetic model's <causality>; the ideal water-wave diagrams above contain no relativistic signal-front claim.<Wave-Front Propagation by Richard Fitzpatrick>[equations 871–873]{ref}
Finally, zero <wavenumber> requires care: the positive-frequency <wave equation> branch $c|k|$ is not differentiable at $k=0$, so it has no unique <group velocity> there, although its one-sided values are $\pm c$. A smooth gapped branch such as the <Klein-Gordon equation> instead has $\omega'(0)=0$ despite undefined <phase velocity> at $k=0$. The same signed branch must be used in both derivatives and ratios; a negative <group velocity> means an envelope moving in the negative coordinate direction, not a negative elapsed time. The practical value of <group velocity> is precisely this translation between a local slope on a <dispersion diagram> and a measurable packet transit, with its assumptions made explicit.
The <Gaussian envelope in a Klein-Gordon medium> supplies a quantitative next step: the first <derivative> translates the <wave envelope>, the second <derivative> spreads it, and higher <derivatives> limit that approximation. At finite spectral width, a peak's measured speed need not equal the <group velocity> at the central <wavenumber>. Neither a peak nor a fixed fraction of its height generally identifies <front velocity>.
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