Dispersion & Group Velocity
Wave packet spreading. Builds on Traveling EM Waves.
You send a short laser pulse (a wave packet) through a block of glass. The glass is dispersive — different frequencies travel at different speeds inside it.
Is the group velocity (speed of the pulse envelope) the same as the phase velocity (speed of the individual wave crests)?
They’re generally different! In a dispersive medium, wave crests can slide through the pulse envelope, moving at a different speed. The group velocity v_g = dω/dk can be greater than, less than, or even opposite to the phase velocity v_p = ω/k. In normal dispersion (like glass for visible light), v_group < v_phase. In anomalous dispersion, v_group > v_phase. The pulse also spreads out over time as its frequency components separate.
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Watch the wave packet move through the dispersive medium. Notice how the crests (colored lines) move at a different speed than the envelope (the pulse shape). Toggle between normal and anomalous dispersion to see both regimes.
vₑ = dω/dk = vₚ + k(dvₚ/dk)
When dvₚ/dk = 0 (no dispersion), group velocity equals phase velocity. When dvₚ/dk ≠ 0, they differ — and the pulse spreads. The faster the medium’s ‘speed vs. frequency’ curve changes, the faster pulses distort.
Glass is dispersive: blue light travels slower than red inside the prism. This frequency-dependent speed is exactly what makes a prism spread white light into a rainbow spectrum.
In long fiber-optic cables, dispersion causes light pulses to spread. If they spread too much, adjacent bits overlap and data is lost. Engineers use dispersion-compensating fibers and soliton pulses to fight this.
Deep-ocean waves are dispersive: longer wavelengths travel faster. After a distant storm, long-period swell arrives first, followed by shorter waves. Surfers know — the ‘good waves’ (long period) arrive before the ‘chop.’
Radio pulses from pulsars are dispersed by interstellar plasma: lower frequencies arrive later. By measuring this delay, astronomers calculate the electron density along the line of sight across the galaxy!
“In a dispersive medium, crests and pulses part ways. The envelope carries the energy; the crests are just the ripples riding along.”
Dispersion Lab
Send wave packets through media with different dispersion relations. Watch group and phase velocity diverge. See how pulses spread and distort in real time.
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Start with zero dispersion — the pulse travels without changing shape. Now increase the dispersion parameter and send another pulse. Watch it broaden as it travels! Compare group and phase velocity arrows. Switch to anomalous dispersion and watch the crests slide backward through the pulse. Try an extremely short pulse (many frequencies) vs. a long one (nearly monochromatic) — the short pulse spreads much faster.
Dispersion means different frequencies travel at different speeds. Short pulses (which contain many frequencies) spread faster than long pulses. This fundamental trade-off between pulse duration and bandwidth governs all of telecommunications.