Dispersion, Phase Velocity and Group Velocity

Derivation

Dispersion, Phase Velocity and Group Velocity

D-071 Home PU-103 Threads waves Depends on The Wave Equation as a Continuum Limit, Standing Waves and Boundary Quantization

Statement

For a linear wave medium in which a plane wave e^i(kx−ωt) obeys a dispersion relation ω = ω(k), the surface of constant phase moves at the phase velocity v_p = ω/k, whereas a narrow-band wave packet built from a range of k moves — and carries its energy and information — at the group velocity v_g = dω/dk. The two coincide only when ω/k is independent of k (a non-dispersive medium).

Why it matters

Almost every real wave medium is dispersive: light in glass, water waves, electrons in a crystal, matter waves, waves on a stiff or loaded string. Once ω depends non-linearly on k, "the speed of the wave" is ambiguous, and one must distinguish the speed of the ripples inside a packet from the speed of the packet as a whole. Only the latter transports energy and signals.

This distinction is the gateway to pulse spreading in optical fibres, the meaning of refractive index versus signal delay, the resolution limit of wave packets, and the identification of v_g with a particle velocity in quantum mechanics. It also resolves the apparent paradox of superluminal phase velocities: v_p may exceed c without violating causality because it carries no information.

Assumptions

Linearity of the medium.If the wave equation is non-linear, distinct Fourier components are not independent, the superposition below is invalid, and the packet does not simply translate — solitons and shocks appear instead. A well-defined dispersion relation ω = ω(k).If plane waves are not eigen-solutions (e.g. an inhomogeneous or time-varying medium), no single ω(k) exists and neither velocity is defined globally. Narrow bandwidth: the spectrum A(k) is sharply peaked about k₀.If the band is broad, the linear Taylor expansion of ω(k) fails; second-order dispersion (group-velocity dispersion) reshapes and broadens the envelope, and a single v_g no longer describes the whole packet. Real k and negligible absorption over the packet.If ω(k) has an imaginary part (loss or gain), or one works in an anomalous-dispersion absorption band, v_g = dω/dk can exceed c or turn negative and ceases to be the energy-transport speed; a strict signal-front velocity must be used.

Derivation

ψ(x,t) = ∫ A(k) e^{i(kx − ω(k)t)} dk

By linearity, any solution is a superposition of plane-wave eigenmodes; A(k) is the (complex) spectral amplitude fixed by the initial profile. A

ω(k) ≈ ω₀ + (k − k₀) ω′(k₀)

Taylor-expand about the spectral peak k₀ and keep first order, legitimate because A(k) is narrow. Write ω₀ = ω(k₀), ω′ = dω/dk|_k₀, and κ = k − k₀. B

kx − ωt = (k₀ + κ)x − (ω₀ + κω′)t

Substitute the expansion into the phase; purely algebraic. A

kx − ωt = (k₀x − ω₀t) + κ(x − ω′t)

Group the carrier terms (subscript 0) separately from the envelope terms (proportional to κ). A

ψ = e^{i(k₀x − ω₀t)} ∫ A(k₀+κ) e^{iκ(x − ω′t)} dκ

The carrier phase is independent of κ and factors out of the integral. B

ψ(x,t) = e^{i(k₀x − ω₀t)} · E(x − ω′t)

The remaining integral defines an envelope E that depends on x and t only through the combination x − ω′t — a rigid shape translating at speed ω′. B

carrier surfaces: k₀x − ω₀t = const ⇒ x = (ω₀/k₀)t

A surface of constant carrier phase advances at v_p = ω₀/k₀, while step 6 shows the envelope advances at v_g = ω′ = dω/dk. The two speeds are logically independent. C

v_g = dω/dk = d(v_pk)/dk = v_p + k dv_p/dk

Write ω = v_pk and differentiate by the product rule. This exposes the exact condition for the two velocities to agree. B

v_g = v_p ⇔ k dv_p/dk = 0 ⇔ dv_p/dk = 0

For non-zero k, group and phase speeds coincide precisely when the phase speed is independent of wavenumber — i.e. the medium is non-dispersive. A

Result

v_p = ω/k v_g = dω/dk = v_p + k dv_p/dk

Reading. The phase velocity is the speed of an individual crest of the carrier; the group velocity is the speed of the packet's envelope, which is what actually moves energy and information. In a non-dispersive medium (v_p constant) they are equal; in normal dispersion (v_p decreasing with k) the group lags the phase, so crests appear to be born at the back of the packet and die at the front.

Units check. ω has units rad·s⁻¹ and k has rad·m⁻¹, so ω/k and dω/dk both carry (rad·s⁻¹)/(rad·m⁻¹) = m·s⁻¹, a speed. The correction term k dv_p/dk has (rad·m⁻¹)·(m·s⁻¹)/(rad·m⁻¹) = m·s⁻¹, consistent.

Limiting cases

Non-dispersive line (ω = ck, ideal string or vacuum light): v_p = v_g = c; packets travel undistorted.
Normal dispersion (dv_p/dk < 0, e.g. glass in the visible): v_g < v_p.
Anomalous dispersion (dv_p/dk > 0): v_g > v_p, and near a resonance v_g can exceed c or go negative.
Deep-water gravity waves (ω = √(gk)): v_g = ½ v_p exactly — the classic factor of two.
Free non-relativistic particle (ω = ℏk²/2m): v_g = ℏk/m = p/m, the particle speed, while v_p = v_g/2.

Breaks when

Broad spectrum / strong second-order dispersion. When the band is not narrow, the neglected term ½ω″(k₀)κ² matters. The envelope is no longer rigid: it broadens as it propagates, its peak still moves at v_g but the shape disperses on a scale set by ω″ = d²ω/dk². A single group velocity then describes only the centroid, not the packet.
Strong absorption / anomalous-dispersion band. Where ω(k) is complex or dω/dk becomes superluminal or negative, v_g stops being the energy-transport speed. The physically meaningful causal limit is then Sommerfeld's signal-front velocity, which never exceeds c.
Non-linear media. If the medium is non-linear (high intensity, amplitude-dependent ω), Fourier superposition fails outright; the packet may form a soliton whose speed is neither v_p nor the small-amplitude v_g.

Failure modes

Using v = fλ = ω/k for signal delay. Students compute phase velocity and then time a pulse's arrival with it; the pulse actually arrives after L/v_g, not L/v_p.
Differentiating ω/k as if it were v_g. Writing v_g = d(ω/k)/dk instead of dω/dk. The correct group velocity differentiates ω, not the ratio.
Confusing "superluminal v_p" with faster-than-light signalling. Concluding relativity is violated because v_p = c/n > c for n < 1 (X-rays); no energy or information moves at v_p.
Evaluating v_g at the wrong k. Both velocities are functions of wavenumber; quoting them without stating k₀ (or λ) is meaningless in a dispersive medium.
Dropping the factor of two for water waves. Assuming v_g = v_p and predicting wave groups that keep pace with their crests, when in fact crests overtake the group.

Discussion

The deepest content of this derivation is that "the velocity of a wave" is not a property of a single sinusoid at all — a pure plane wave is infinite in extent and carries no marker whose arrival one could time. Velocity in the operational sense (energy, signal, particle) belongs to a modulation, and any modulation necessarily involves a spread of wavenumbers. The group velocity is the leading-order answer to "how fast does that spread's centre of mass move", and it emerges the moment one keeps the first derivative of ω(k).

Physically, v_g = dω/dk is also the energy-transport velocity in a lossless medium: for a mechanical wave the time-averaged energy flux divided by the energy density equals dω/dk, and for light it equals the ray velocity. This is why v_g, not v_p, sets pulse-arrival times, fibre-optic timing budgets, and the recoil of a wave-driven object.

The identity v_pv_g = c², which holds for a relativistic matter wave (ω = √(c²k² + (mc²/ℏ)²)), ties the two speeds together with striking economy: since a massive particle moves at v < c and this is v_g, the phase velocity v_p = c²/v > c is forced to be superluminal — harmlessly, because it transports nothing. De Broglie's whole scheme requires the particle velocity to be the group velocity, which is the historical origin of taking v_g seriously.

At next order the packet is governed by the term ½ω″κ² discarded in step 2. This gives a Schrödinger-type equation for the envelope in the frame moving at v_g, with ω″ = d²ω/dk² playing the role of an inverse mass. A Gaussian packet of initial width σ then spreads on the dispersion length L_D ∼ σ²/|ω″|; the same ω″ (as the fibre parameter β₂) sets chromatic pulse broadening and, with non-linearity, the balance that forms optical solitons.

Common misconceptions. That phase velocity is "the real" wave speed and group velocity a correction — it is the reverse for anything observable. That v_g can never exceed c — it can, in absorbing bands, without violating causality, because in those regimes it is no longer the signal velocity. And that the two velocities are simply related by a fixed factor — the ratio depends entirely on the dispersion relation and on k.

Worked examples

Deep-water gravity waves: ω(k) = √(gk), λ = 100 m

Ocean swell; find phase and group speed and the ratio. A

k = 2π/λ = 2π/100 = 0.0628 rad·m⁻¹

Wavenumber from wavelength. A

v_p = ω/k = √(g/k) = √(9.81/0.0628) = 12.5 m·s⁻¹

Insert ω = √(gk); symbols first, then numbers. B

v_g = dω/dk = ½√(g/k) = ½ v_p = 6.25 m·s⁻¹

Differentiate √(gk); the half is exact for this dispersion law. B

v_p ≈ 12.5 m·s⁻¹, v_g ≈ 6.25 m·s⁻¹, v_g/v_p = ½

Reading. A wave group crosses the ocean at half the speed of the individual crests within it; crests visibly march forward through the group and vanish at its leading edge.

Light in glass: n(λ) = 1.500 at λ = 500 nm, dn/dλ = −5.0×10⁴ m⁻¹

Normal dispersion (index falls with wavelength); find v_p and v_g. B

v_p = c/n = (3.00×10⁸)/1.500 = 2.00×10⁸ m·s⁻¹

Definition of refractive index. A

v_g = c / (n − λ dn/dλ)

Standard group-velocity form in terms of wavelength dispersion, obtained from v_g = dω/dk with ω = ck/n. Symbols before numbers. C

n − λ dn/dλ = 1.500 − (500×10⁻⁹)(−5.0×10⁴) = 1.500 + 0.025 = 1.525

Group index n_g; the negative slope raises it above n. B

v_g = c/n_g = (3.00×10⁸)/1.525 = 1.97×10⁸ m·s⁻¹

Divide. A

v_p = 2.00×10⁸ m·s⁻¹, v_g = 1.97×10⁸ m·s⁻¹

Reading. In normal dispersion the pulse (group) travels slightly slower than the phase, so a timing budget based on n alone underestimates the delay by about 1.7%.

Problems

A stretched string has the ideal dispersion ω = ck with c = 30 m·s⁻¹. Find v_p and v_g at λ = 0.50 m, and state whether a pulse spreads.
Solution
k = 2π/0.50 = 12.6 rad·m⁻¹. Since ω = ck, v_p = ω/k = c = 30 m·s⁻¹ and v_g = dω/dk = c = 30 m·s⁻¹. Both are 30 m/s, independent of k: the line is non-dispersive, ω″ = 0, and a pulse propagates without spreading.
A stiff (bending) rod has ω = αk² with α = 4.0 m²·s⁻¹. Compute v_p and v_g at k = 5.0 rad·m⁻¹ and give the ratio.
Solution
v_p = ω/k = αk = 4.0×5.0 = 20 m·s⁻¹. v_g = dω/dk = 2αk = 2×4.0×5.0 = 40 m·s⁻¹. Ratio v_g/v_p = 2: the group outruns the phase, and since ω″ = 2α = 8.0 ≠ 0 pulses disperse.
Deep-water waves obey ω = √(gk). Show generally that v_g = ½v_p, then evaluate both for λ = 2.0 m (g = 9.81 m·s⁻²).
Solution
v_p = ω/k = √(gk)/k = √(g/k). v_g = dω/dk = ½g/√(gk) = ½√(g/k) = ½v_p. Numerically k = 2π/2.0 = 3.14 rad·m⁻¹, v_p = √(9.81/3.14) = 1.77 m·s⁻¹, v_g = 0.88 m·s⁻¹.
A free non-relativistic electron has ω = ℏk²/2m. Show v_g equals the classical velocity p/m, and find v_g for λ = 1.0×10⁻¹⁰ m (ℏ = 1.055×10⁻³⁴ J·s, m = 9.11×10⁻³¹ kg).
Solution
v_g = dω/dk = ℏk/m. With de Broglie p = ℏk this is p/m, the particle velocity. (Note v_p = ω/k = ℏk/2m = ½v_g.) Numerically k = 2π/1.0×10⁻¹⁰ = 6.28×10¹⁰ rad·m⁻¹, so v_g = (1.055×10⁻³⁴×6.28×10¹⁰)/9.11×10⁻³¹ = 7.3×10⁶ m·s⁻¹.
In a plasma, ω² = ω_p² + c²k² with plasma frequency ω_p. Prove the exact relation v_pv_g = c² and comment on causality when v_p > c.
Solution
v_p = ω/k. Differentiate the dispersion relation implicitly: 2ω dω = 2c²k dk, so v_g = dω/dk = c²k/ω. Then v_pv_g = (ω/k)(c²k/ω) = c². Since ω > ck (propagation requires ω > ω_p), v_p = ω/k > c, hence v_g = c²/v_p < c. Causality is safe: energy and signals travel at v_g < c, while the superluminal phase velocity carries no information.