Wave Physics
Group Velocity vs Phase Velocity
v_phase = ω/k (carrier wave) vs v_group = dω/dk (envelope, energy, info)
A monochromatic wave cos(kx − ωt) has phase velocity v_p = ω/k. A wave packet (sum of nearby frequencies) propagates with group velocity v_g = dω/dk — the speed of the envelope, which carries energy and information. In a non-dispersive medium (ω/k constant), v_p = v_g. In a dispersive medium (refractive index n(λ) frequency-dependent), they differ. Examples: deep-water surface waves: v_g = v_p/2 (group lags), making ship wakes form their characteristic V pattern; light in glass: v_g < v_p slightly (normal dispersion); near absorption resonances: anomalous dispersion can give v_p > c (no information violation; v_g still ≤ c); in plasmas above plasma frequency, v_p > c but v_g < c — radio waves in ionosphere. The Schrödinger wave packet group velocity = ℏk/m = particle velocity.
- Phase velocityv_p = ω/k
- Group velocityv_g = dω/dk
- Non-dispersivev_p = v_g
- Deep waterv_g = v_p/2
- Plasmav_p > c, v_g < c
- Information speedv_g (≤ c)
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Why group/phase velocity matters
- Fiber optics dispersion. Every long-haul Internet pulse spreads in time as different wavelength components travel at different v_g. Standard fiber dispersion (~17 ps/nm/km at 1550 nm) limits bit-rate × distance. Dispersion-compensating fiber, dispersion-shifted fiber, and electronic equalization keep v_g spread within budget for 100G and 400G coherent links.
- Plasma physics. The ionosphere reflects radio waves below the plasma frequency (a wall) and transmits above it (with v_p > c, v_g < c). Long-distance shortwave radio bounces off the F-layer ionosphere; satellite links must use frequencies above f_p ≈ 10 MHz to penetrate. Pulsars detected via dispersion measure (frequency-dependent v_g delay through interstellar plasma) calibrate galactic electron content.
- Medium dispersion engineering. Photonic crystal fibers, microstructured waveguides, and metamaterials are designed to give exotic dispersion: zero v_g (slow light), v_g > v_p, even negative v_g. Slow-light buffers can store optical pulses for nanoseconds; supercontinuum sources exploit anomalous dispersion to broaden a pulse into white light spanning octaves.
- Quantum mechanics. The Schrödinger wave packet for a free particle has ω = ℏk²/(2m), giving v_p = ℏk/(2m) and v_g = ℏk/m = p/m — the classical particle velocity. Group velocity is what experimentalists detect when they "see" the particle move; the half-as-fast phase velocity has no observational consequence.
- Tsunami warning. Tsunamis are shallow-water waves with v_p = v_g = √(gh) — non-dispersive at oceanic depth (~h = 4 km gives v ≈ 200 m/s). Predictable arrival times let warning systems forecast arrivals to the second from a seismic origin.
- Radar and communications. Pulse compression in chirped radar relies on dispersive delay lines that delay different frequencies by different amounts (engineered v_g profile), then re-compress in the receiver. Satellite GPS signals correct for ionospheric v_g delay using dual-frequency measurements.
- Acoustics in stratified media. Ocean SOFAR channels guide sound for thousands of kilometers via depth-dependent v_g. Whale song, submarine detection, and seismic-wave propagation all depend on the ω(k) relation in their respective media.
Common misconceptions
- "Phase velocity is the signal speed." No — v_g is. v_p describes how individual peaks of an idealized infinite sine wave move, but real signals are packets, and packets move at v_g. Confusing the two leads people to think faster-than-light signaling is possible from v_p > c examples.
- "v_p > c violates relativity." Only group velocity (or strictly, the front velocity of a causal step) is bounded by c. Phase velocities exceed c routinely in plasmas, waveguides above cutoff, and X-ray reflection — without any causality issue because nothing real moves at v_p.
- "v_p and v_g are always equal in vacuum." True for electromagnetic waves in free vacuum (ω = ck, perfectly linear). But quantum-vacuum effects, gravitational fields, and modified-dispersion-relation theories break this slightly. For all engineering purposes, vacuum is non-dispersive.
- "v_g is always less than v_p." Often, but not always. In anomalous dispersion regions (near absorption lines), v_g can exceed v_p — even exceed c. In Bose-Einstein condensates, v_g has been slowed to walking pace; in left-handed metamaterials, v_g and v_p point in opposite directions.
- "Energy travels at v_p." Energy travels at v_g (more precisely, at the energy-transport velocity, which equals v_g in most ordinary media). Phase velocity has zero energy density associated with the carrier alone — energy needs envelope variations to be located somewhere.
- "Group velocity is well-defined for any pulse." Only when the pulse bandwidth is narrow enough that ω(k) is roughly linear across it. For ultrashort pulses, the higher-order terms (group-velocity dispersion, third-order dispersion) become important — pulses chirp, broaden, and develop substructure not captured by a single v_g.
Frequently asked questions
Why are group and phase velocity different?
A pure monochromatic wave has a single frequency ω and wavenumber k, and travels at v_p = ω/k — the speed at which crests move. A pulse, in contrast, is built from a band of frequencies via Fourier synthesis. If ω is exactly proportional to k (non-dispersive medium), all components move at the same speed and the pulse rigidly translates at v_p = v_g. If ω(k) is curved (dispersive), each component moves at a slightly different v_p, and the envelope of their superposition moves at v_g = dω/dk — generally different from any individual v_p.
Can phase velocity exceed c?
Yes — and it does, in many real systems. In a plasma above the plasma frequency, ω² = ω_p² + c²k² gives v_p = c·√(1 + (ω_p/(ck))²) > c. In an X-ray waveguide, near-absorption-line gas, or any medium with refractive index n < 1 at some wavelength, v_p = c/n > c. This doesn't violate relativity because no energy or information rides on a pure phase wave — it's an idealization. Information requires a modulated packet, which moves at v_g, and v_g remains ≤ c in all causal media.
Why does v_g carry information?
Information must be encoded as some change — a pulse turn-on, a bit transition, a click. Any such change has finite duration and therefore finite Fourier bandwidth, making it a wave packet. The packet's envelope is what your detector senses; the carrier underneath is irrelevant to the message. Since the envelope moves at v_g, that's how fast the bit arrives. Special relativity demands signals satisfy v_signal ≤ c, and a more careful analysis (Sommerfeld and Brillouin, 1914) shows the leading edge of a strictly causal pulse travels at exactly c regardless of medium — anomalous v_g > c reflects pulse reshaping, not faster-than-light signaling.
What is normal vs anomalous dispersion?
Normal dispersion: dn/dλ < 0 (n decreases with wavelength), which is the typical case for transparent media in regions far from absorption. Blue light bends more than red — Newton's prism. Group velocity is less than phase velocity. Anomalous dispersion: dn/dλ > 0, occurring near absorption lines where the index curve hooks upward briefly. Group velocity can exceed phase velocity; in extreme cases v_g > c (no causality violation, since pulse shape distorts and the leading edge stays subluminal). Most lasers operate in normal-dispersion regimes; ultrafast pulse compression exploits anomalous regions deliberately.
How does ship wake show v_g < v_p?
Deep-water gravity waves obey ω = √(gk), so v_p = √(g/k) and v_g = (1/2)√(g/k) = v_p/2. Watch a ship's wake: individual wave crests appear at the back of the wake, race forward through the wake pattern, and disappear at the front. The wake itself (the envelope of crests) only travels half as fast. This v_g = v_p/2 ratio is also why ship wakes form Kelvin's characteristic 19.5° half-angle V — group propagation directions interfere on a fixed cone.
What is the group delay in fiber optics?
An optical pulse in a fiber has a spread of wavelengths around the carrier. Different wavelengths have slightly different v_g = c/n_g, where n_g = n − λ dn/dλ is the group index. Over a long fiber, this dispersion of group velocities stretches a sharp pulse into a broader one — chromatic dispersion, measured in ps/(nm·km). Standard single-mode fiber has zero-dispersion wavelength near 1310 nm; at 1550 nm (where attenuation is lowest) dispersion is ~17 ps/(nm·km), forcing dispersion compensation. Fiber dispersion limits the bandwidth-distance product of every fiber link.