Calculate phase velocity and group velocity of waves. Explore dispersion in ocean waves, plasma, optical fiber, and custom media with frequency sweeps.
The wave velocity calculator computes both phase velocity and group velocity for waves in different media, revealing the critical phenomenon of dispersion. While the phase velocity describes how fast individual wave crests move, the group velocity tells you how fast energy and information actually travel — a distinction that becomes crucial in dispersive media.
In non-dispersive media like sound in air, both velocities are equal. But in dispersive media — optical fibers, ocean surfaces, plasma — the two speeds diverge. Deep-ocean swells have a group velocity exactly half their phase velocity, which is why surfers see waves approaching faster than the wave groups that carry them. In plasma, the phase velocity can exceed the speed of light (without violating relativity, since no information travels faster than vg < c).
This calculator models real dispersion relations for deep-water gravity waves, ionospheric plasma, and optical media. Enter a frequency, choose your medium, and see how phase and group velocities differ across a range of frequencies.
Understanding the difference between phase and group velocity is essential whenever waves travel through a real medium. Optical engineers must manage dispersion to maintain signal integrity over thousands of kilometers of fiber. Naval engineers predict wave energy arrival times for coastal structures. Atmospheric scientists track radio wave propagation through the ionosphere.
This calculator goes beyond simple v = fλ to reveal the rich physics of dispersion, showing how phase and group velocities diverge in real-world media and how that divergence varies with frequency.
Phase velocity: vₚ = ω/k = fλ Group velocity: vg = dω/dk Specific dispersion relations: • Deep-water gravity waves: ω² = gk → vₚ = √(g/k), vg = vₚ/2 • Plasma waves: ω² = ωₚ² + c²k² → vₚ = c/√(1-(fₚ/f)²), vg = c·√(1-(fₚ/f)²) • Non-dispersive: vₚ = vg = constant Where: • ω = angular frequency (rad/s), k = wave number (rad/m) • g = 9.81 m/s² (gravitational acceleration) • ωₚ = plasma frequency, c = speed of light
Result: Phase velocity ≈ 15.6 m/s, Group velocity ≈ 7.8 m/s
A 10-second ocean swell (f = 0.1 Hz) has phase velocity vₚ = √(g/k) ≈ 15.6 m/s and group velocity vg = vₚ/2 ≈ 7.8 m/s. Wave crests move twice as fast as the wave packet (energy), which is why individual crests appear to emerge from the back of a swell group and vanish at the front.
The distinction between phase and group velocity is one of the most profound concepts in wave physics. Phase velocity vₚ = ω/k describes the speed of individual wave crests — the rate at which a surface of constant phase moves through space. Group velocity vg = dω/dk describes the speed of the wave envelope — the rate at which energy, information, and wave packets propagate.
In a non-dispersive medium where ω = vk (linear dispersion), phase and group velocities are identical. This is the familiar case for sound in air or light in vacuum. But most real media are dispersive: glass slows blue light more than red (normal dispersion), while the ionosphere slows low-frequency radio waves more than high-frequency ones (also normal dispersion by a different mechanism).
Ocean surface waves provide one of the most intuitive examples of dispersion. For deep-water gravity waves (depth > half the wavelength), the dispersion relation is ω² = gk, giving vₚ = √(g/k) = gT/(2π). Longer-period swells travel faster.
The group velocity for these waves is exactly vₚ/2 — a beautiful result that means wave energy propagates at half the speed of visible wave crests. Watch ocean swells carefully and you'll see individual crests form at the back of a group, propagate forward through the group, and disappear at the front. This has practical importance: when a storm generates ocean swell, the energy arrives at a distant coast at the group speed, not the (faster) phase speed.
Modern telecommunications depend on managing dispersion. In an optical fiber carrying 100 Gbps of data, each data pulse occupies roughly 10 picoseconds. Standard single-mode fiber has chromatic dispersion of about 17 ps/(nm·km) at 1550 nm. Over 1000 km, a pulse with 0.1 nm spectral width spreads by 1.7 ns — far wider than the original 10 ps — destroying the data stream unless compensated.
Engineers use dispersion-compensating fiber, dispersion-shifted fiber, and digital signal processing to combat this effect. The entire field of ultrafast optics—generating and manipulating femtosecond laser pulses—requires precise control of dispersion through prisms, gratings, and chirped mirrors.
Yes. In plasma and other anomalous-dispersion media, phase velocity can exceed c. This doesn't violate relativity because phase velocity doesn't carry information or energy — the group velocity (always ≤ c in these cases) is what matters for causality. The superluminal vₚ simply means wavefronts don't represent signal propagation.
Group velocity is the speed at which a wave packet's envelope moves, which corresponds to the speed of energy or information transfer. If you modulate a carrier wave with a signal, the modulation pattern moves at the group velocity. For a laser pulse in fiber, vg determines signal timing.
For deep-water gravity waves, the dispersion relation ω² = gk gives vg = dω/dk = vₚ/2. Physically, individual crests continually form at the back of a wave group, pass through it, and disappear at the front. The energy moves at vg while crests move at 2vg.
A medium where wave speed depends on frequency. In dispersive media, different frequency components of a pulse travel at different speeds, causing the pulse to spread out (disperse) over time and distance. Glass, water surfaces, and plasma are dispersive; air for sound and vacuum for light are non-dispersive.
In optical fiber, chromatic dispersion spreads light pulses over long distances, limiting data rates. A pulse containing wavelengths from 1549-1551 nm will spread because each wavelength has a slightly different group velocity. Engineers compensate with dispersion-shifted fiber and dispersion compensation modules.
Normal dispersion: higher frequencies travel slower (vg < vₚ), like light in glass at visible wavelengths. Anomalous dispersion: higher frequencies travel faster (vg > vₚ), like light in glass at certain UV wavelengths or surface waves in shallow water. The type of dispersion depends on the medium and frequency range.