# Group velocity The **group velocity** of a wave is the speed at which the envelope of a wave packet, the slowly varying outline traced by a group of superposed waves of nearby frequency, propagates through space, as distinct from the phase velocity at which any one crest inside that envelope moves. Adding two [[Wave|waves]] of slightly different [[Angular_frequency|angular frequency]] and wavenumber produces a beat pattern whose fast oscillation travels at the phase velocity while its slower amplitude envelope travels at the group velocity; a wave packet built from a continuous spread of frequencies carries the same idea to its limit. The two velocities coincide only when every frequency component travels at the same speed. In a dispersive medium, one where wave speed depends on frequency, they differ, so a signal's envelope, and with it the energy and information the signal carries, moves at a different rate from any individual crest inside it; group velocity is accordingly the more physically meaningful of the two speeds for most transmission problems, and the one a receiver's timing actually tracks. The primary microsim on this page superposes two sine waves of slightly different wavenumber and animates their sum. Adjusting the carrier wavenumber, the spread between the two components, and a dispersion parameter changes how fast the resulting envelope travels relative to a tracked phase crest, with both speeds read out live. ## History The distinction between the speed of a wave's individual crests and the speed of the slowly varying envelope that carries its energy was worked out over the nineteenth century by mathematicians studying [[Wave|water waves]] and optical dispersion, most notably William Rowan Hamilton and, later, Lord Rayleigh, who is credited with the now-standard formula relating group velocity to how phase velocity itself changes with wavelength.[^cite-hist1] The term "group velocity" and its formal identification with the speed at which energy is transported through a dispersive medium followed from that same body of nineteenth-century wave theory, well before the concept found its later and more surprising applications in quantum mechanics and in the study of anomalous dispersion.[^cite-hist2] ## Definition and interpretation A [[Plane_wave|plane wave]] with a single, sharply defined frequency ω and wavenumber k travels at the phase [[Velocity|velocity]] `v_p = w/k`, the speed of any one of its crests. A real signal is never perfectly monochromatic; a [[Fourier_transform|Fourier decomposition]] of it shows a narrow band of frequencies clustered around some centre, and the envelope formed by superposing that whole band travels at the group velocity `v_g = dw/dk`, the rate of change of frequency with wavenumber evaluated at the band's centre. Where the two speeds are equal the distinction is academic, but wherever they differ, it is the group velocity that describes how quickly a recognisable feature of the signal, a pulse peak or a modulation pattern, actually crosses space. ### Derivation Adding two equal-amplitude waves at wavenumbers k0 ± Δk and frequencies ω0 ± Δω, related by a dispersion relation ω(k), gives a carrier oscillating at (k0, ω0) multiplied by a slower envelope oscillating at (Δk, Δω); the carrier's crests move at ω0/k0 while the envelope's peaks move at Δω/Δk, which becomes the derivative dω/dk in the limit of a narrow spread between the two components. The sketch's top panel draws the two component waves separately and its bottom panel their sum, marking one phase crest and the envelope peak so the two speeds can be read and compared directly as the spread control is widened or narrowed. ### Other expressions Group velocity can equally be written in terms of wavelength, `v_g = v_p - lambda*dv_p/dlambda`, a form that makes explicit that the two velocities coincide exactly when phase velocity does not depend on wavelength. In quantum mechanics, a free particle's wavefunction obeys the [[Schrödinger_equation|Schrödinger equation]]'s dispersion relation `w = hbar*k^2/(2*m)`; its group velocity, `hbar*k/m`, equals the particle's ordinary classical velocity, while the underlying phase velocity is exactly half that and carries no direct classical meaning of its own. ## Dispersion A dispersion relation ω(k) fixes both velocities at once: phase velocity is the slope of the line from the origin to the point (k, ω(k)) on a plot of frequency against wavenumber, and group velocity is the local slope of the curve itself at that point. When ω(k) is a straight line through the origin, as it is for [[Sound|sound]] in air or light in a vacuum, the two slopes are identical everywhere and every frequency component travels together, so a pulse built from many frequencies keeps its shape as it travels. The sketch models dispersion with a simple quadratic relation, `w(k) = c*k + alpha*k^2` in the sketch, and dragging its dispersion slider away from zero curves the line, separating the phase and group velocities and visibly detaching the tracked crest marker from the envelope marker as the two speeds pull apart. A medium is called normally dispersive when group velocity falls as frequency rises, the ordinary case for visible light passing through window glass, and anomalously dispersive in the narrow bands, usually near an absorption line, where it does the opposite. ## Relation to phase velocity, refractive index and transmission speed Because phase velocity is what most directly defines a medium's refractive index, `v_p = c/n`, and group velocity is what a real signal's envelope actually travels at, the two enter differently into how a medium is characterised. A group index, `n_g = c/v_g`, can be defined the same way, and the two indices coincide only where the ordinary refractive index is flat with frequency; wherever n itself changes with frequency, the group index departs from it by a term proportional to that slope, `n_g = n + w*dn/dw`. The transmission speed of a modulated signal, the rate at which useful information travels down a real channel, is the group velocity of the carrier and its sidebands taken together, not the phase velocity of the carrier alone. This distinction matters whenever the two velocities differ by more than a rounding error: a radio signal's timing, and the delay a receiver actually measures, tracks the envelope, so group velocity, not phase velocity, is the number that belongs in a link budget or a propagation-delay calculation for anything but a pure, unmodulated carrier. Phase velocity itself is free to exceed the speed of light in vacuum without contradiction, since no energy or information rides on a single crest in isolation; it is only ever the group, or more precisely the leading edge of a causally complete signal, whose speed [[Special_relativity|special relativity]] constrains. ## In three dimensions In more than one dimension, wavenumber becomes a vector k and frequency a function ω(k) of its components; group velocity generalises to the gradient `v_g = grad_k(w)`, a vector that points in the direction the wave's [[Energy|energy]] actually flows. In an isotropic medium, where ω depends only on the magnitude of k, that gradient is parallel to k itself and the three-dimensional case adds nothing beyond bookkeeping. In an anisotropic medium, such as a crystal whose own [[Wave_equation|wave equation]] treats one direction differently from another, or a metal waveguide whose walls pick out a preferred direction, the energy-carrying group velocity vector can point in a genuinely different direction from the wavevector it is derived from, so a wave's crests and the energy the wave carries do not, in general, travel the same way at all. ## In lossy or gainful media The clean picture of a single group velocity breaks down close to a medium's resonance, where absorption or amplification is strong and changes rapidly with frequency. There, the dispersion relation itself becomes [[Complex_analysis|complex]]-valued, and the simple derivative dω/dk no longer describes a single, well-defined envelope speed for a pulse whose spectrum spans the resonance; the pulse reshapes as it propagates rather than merely sliding along at one fixed rate, and no single number captures its motion honestly once absorption or gain varies quickly enough across the pulse's own bandwidth. ### Superluminal group velocities In the narrow anomalous-dispersion band very close to an absorption or gain line, the formula for group velocity can formally return a value greater than the speed of light in vacuum, or even a negative value, and experiments with tailored absorbing or amplifying media, including some built around [[Laser|laser]] gain lines, have reported pulse peaks emerging from such a medium sooner than an equal length of vacuum would allow.[^cite-superluminal] This does not let information or energy outrun light: the effect reshapes a pulse whose leading edge already carries everything needed to reconstruct its peak, rather than advancing any new information into the medium ahead of when it arrived, and it remains consistent with special relativity's prohibition on faster-than-light signalling once the whole, causally connected pulse is analysed rather than just its reshaped peak.[^cite-superluminal2] ## Microsims The primary sketch superposes two travelling sine waves of nearly equal wavenumber and plots them individually in a top panel and summed in a bottom panel. A carrier-wavenumber slider sets k0, a spread slider sets the wavenumber difference between the two components, a dispersion slider sets the quadratic coefficient α in the sketch's dispersion relation, and a fourth slider scales the animation's playback speed; the space bar pauses time and the 'r' key resets it. Two vertical markers track a single phase crest and the envelope's peak as they drift apart, and a live readout gives the phase [[Velocity|velocity]] `v_p = w/k` and the group velocity `v_g = dw/dk` in the sketch's own units. A companion three.js sketch, in preparation, renders the same wave packet in a fuller three-dimensional view, with crests and envelope moving at their own, separately tracked speeds. *Try:* Set the dispersion slider to zero and watch the two markers stay locked together, then move it away from zero and watch the phase-crest marker slide out ahead of, or fall behind, the envelope marker. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Group_velocity) : [Wikitube](https://en.wikitube.io/wiki/Group_velocity) Skeleton mirrored at revision 1368878350. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Transmission_line]] - [[Wave]] - [[Special_relativity]] - [[Fourier_transform]] ## References [^cite-hist1]: Citation needed: a primary citation (paper or treatise title, venue, year) for Hamilton's and Rayleigh's nineteenth-century work on group velocity has not been pinned to a Portal Book page in this pass. [^cite-hist2]: Citation needed: a primary or secondary source for the early history of the term "group velocity" itself has not been pinned to a Portal Book page in this pass. [^cite-superluminal]: Citation needed: a specific primary paper (authors, journal, year) reporting a measured superluminal or negative group velocity has not been pinned to a Portal Book page in this pass. [^cite-superluminal2]: Citation needed: a specific secondary source explaining why superluminal group velocity does not permit faster-than-light signalling has not been pinned to a Portal Book page in this pass. ### Notes The dispersion relation `w(k) = c*k + alpha*k^2` used above is the sketch's own simplified model, chosen to make phase and group velocity separate cleanly under one slider; a real dispersive medium's ω(k) is set by its own physics and is rarely an exact quadratic (style guide §5.2, ILLUSTRATIVE). ### Further reading - Christian Tiberius; Max Mulder. *Engineering Signal Analysis: From Fourier to filtering — Theory* (2026). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory - Don Johnson. *Fundamentals of Electrical Engineering I* (2014). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1 ## External links - Group velocity, live sketch: https://editor.p5js.org/sciencenibber/full/WbKkd5qOY - Group velocity, editor source: https://editor.p5js.org/sciencenibber/sketches/WbKkd5qOY <!-- Hubs: Signal_processing. Portals: PORTAL_Signal_Processing. Signal Processing portal wave 1 · 2026-09-17 · drafted. -->