Wave Packet

3 min read#waves

A localized burst of oscillation — a carrier wave wrapped in a traveling envelope — whose envelope and crests move at different speeds in a dispersive medium.

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Wave Packet

A wave packet is a localized lump of wave: a fast oscillating carrier wrapped inside a smooth, slowly varying envelope. Unlike an endless sine, a packet is concentrated in a region of space, which is what makes it physical — real signals, pulses of light, and quantum particles are all wave packets. It is the bridge between a wave (spread everywhere) and a particle (somewhere).

You build one by adding many sinusoids of nearby wavenumbers via the Superposition Principle. Away from the center they interfere destructively and cancel; near the center they reinforce. A Gaussian envelope is the cleanest example:

u(x,t)=\underbrace{e^{-\frac{(x-v_g t)^2}{2\sigma^2}}}_{\text{envelope, moves at }v_g}\;\underbrace{\cos\!\big(k_0 x-\omega_0 t\big)}_{\text{carrier, crests move at }v_p}.

Two speeds at once

The defining feature of a packet in a dispersive medium is that the envelope and the carrier move at different speeds. The envelope — the bundle of energy and information — travels at the group velocity v_g=d\omega/dk. The individual crests inside travel at the phase velocity v_p=\omega/k. Watch closely and you can see crests born at the trailing edge of the packet, sweeping forward through the envelope, and dissolving at the leading edge.

A Gaussian wave packet. The faint outline is the envelope, gliding at the group velocity; the orange oscillation inside is the carrier, whose crests slide through the envelope at a different (here faster) phase velocity. Watch a crest appear at the back, march forward, and vanish at the front.

Localization costs bandwidth

A packet narrow in space needs a wide spread of wavenumbers to build it, and a packet narrow in wavenumber must be broad in space. This reciprocal trade — \Delta x\,\Delta k \gtrsim \tfrac12 — is a pure property of the Fourier Transform, and in quantum mechanics, where p=\hbar k, it becomes the Uncertainty Principle. You cannot have a wave that is both perfectly located and perfectly monochromatic.

In a non-dispersive medium the packet would glide along rigidly; with dispersion the constituent frequencies drift apart and the packet spreads as it travels — the same broadening that limits how fast pulses can be sent down an optical fiber.

See also

Related topics

  • DispersionWhen wave speed depends on frequency, a pulse made of many frequencies spreads and reshapes as it travels, and phase and group velocities part ways.
  • Fourier TransformThe generalization of the Fourier series to non-periodic signals — an integral that resolves any signal into a continuous band of frequencies.
  • Superposition PrincipleWhen two waves meet, the medium's displacement is simply the sum of what each wave would do alone — the linearity that underlies all of wave physics.
  • Uncertainty PrincipleA signal cannot be sharply localized in both time and frequency at once — the tighter one view, the broader the other.
  • Wave EquationThe partial differential equation whose solutions are anything that travels left or right at a fixed speed without changing shape.
  • Fourier AnalysisAny signal is a sum of pure sinusoids — and reading it as such turns the tangled language of time into the clean language of frequency.

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