Wave Packet

From infinite sine to localized pulse

A single sine wave, A·cos(kx − ωt), stretches from −∞ to +∞. It has one perfectly defined wavelength λ = 2π/k but no location at all — it is everywhere. A real disturbance — a tap on a string, a flash from a laser, an electron in a beam — is localized: it exists here and not there. To build something localized out of sine waves you must add many of them together. That sum is a wave packet.

The idea is pure interference. Take a band of waves whose wavenumbers cluster around a central k₀. At one point in space they all happen to be in phase and add constructively, producing a tall bump. Everywhere else their phases scatter and they cancel by superposition. The more wavelengths you include, the more completely they cancel away from the bump, and the narrower the packet becomes.

Mathematically, a one-dimensional packet is the Fourier synthesis

ψ(x, t) = ∫ A(k) · exp[ i(kx − ω(k)t) ] dk

where A(k) is the amplitude of each component (the spectrum) and ω(k) is the dispersion relation that tells you how fast each wavelength travels. Choose A(k) to be a narrow band around k₀ and ψ is a compact pulse; choose A(k) to be a single spike and ψ is back to an infinite sine wave.

The simplest case: two waves and a beat envelope

You can see the mechanism with just two waves of nearly equal wavenumber, k₀ ± Δk, and frequency ω₀ ± Δω:

cos[(k₀+Δk)x − (ω₀+Δω)t] + cos[(k₀−Δk)x − (ω₀−Δω)t]
   = 2 · cos(Δk·x − Δω·t) · cos(k₀·x − ω₀·t)

The result is a fast carrier wave, cos(k₀x − ω₀t), multiplied by a slow envelope, cos(Δk·x − Δω·t). The carrier crests move at the phase velocity v_p = ω₀/k₀. The envelope — the slowly varying amplitude that bunches the carrier into groups — moves at Δω/Δk. In the limit of many closely spaced waves this becomes the derivative dω/dk, the group velocity. This two-wave picture is exactly the phenomenon of beats, lifted into a traveling pattern.

Group velocity vs phase velocity

These two speeds are the heart of the subject, and they are usually different.

Quantity	Definition	What moves at this speed
Phase velocity v_p	ω / k	The individual crests / ripples inside the packet
Group velocity v_g	dω / dk	The envelope — the pulse that carries energy and information
Relationship	v_g = v_p + k · dv_p/dk	Equal only when v_p is independent of k

When the phase velocity does not depend on wavelength — light in vacuum, waves on an ideal string — every component travels at the same speed, v_g = v_p, and the packet glides along without changing shape. This is a non-dispersive medium. When v_p does depend on k, the medium is dispersive, the two speeds split, and the packet begins to deform.

A famous visual is deep-water gravity waves, where ω = √(gk). Then v_p = √(g/k) and v_g = ½√(g/k) = v_p/2. The crests travel twice as fast as the group. Watch a wave group on the ocean and you see crests appear at the rear, sweep through the group toward the front, and disappear — the carrier outrunning its own envelope.

Why packets spread: dispersion

Expand the dispersion relation around the central wavenumber k₀:

ω(k) ≈ ω₀ + v_g·(k − k₀) + ½ β·(k − k₀)²,   where β = d²ω/dk²

The constant term sets the carrier oscillation. The linear term moves the whole envelope at the group velocity v_g. The quadratic term β — the group-velocity dispersion — is what spreads the packet. If β ≠ 0, the components with different k accumulate different phase errors, the envelope broadens, and its peak shrinks because the total energy is conserved while being smeared over a wider region.

For a Gaussian packet of initial width Δx₀, the width at time t grows as

Δx(t) = Δx₀ · √( 1 + (β t / Δx₀²)² )

so for large t the width grows linearly: Δx ≈ |β| t / Δx₀. Counterintuitively, a packet that starts narrower (small Δx₀) spreads faster, because squeezing it in space requires a broader band of wavelengths Δk, and a broad band disperses more.

The quantum wave packet

In quantum mechanics a free particle of mass m has the dispersion relation ω = ħk²/(2m), straight from E = p²/2m with E = ħω and p = ħk. Differentiating:

v_p = ω/k = ħk/(2m) = p/(2m)
v_g = dω/dk = ħk/m   = p/m   = v_classical

The group velocity equals the ordinary particle velocity p/m — this is why the packet, not the individual ripples, represents the particle. The phase velocity is exactly half of it and has no direct physical meaning. Because d²ω/dk² = ħ/m ≠ 0, free-space quantum mechanics is intrinsically dispersive: every free wave packet spreads. An electron localized to Δx₀ = 1 nm spreads to roughly double that width in about 10⁻¹⁵ s; a 1-kg object localized to a micron would take longer than the age of the universe to spread measurably, which is why the macroscopic world looks classical.

The width relation Δx·Δk ≥ ½ is just a fact about Fourier pairs: a function and its transform cannot both be narrow. Multiply by ħ and substitute p = ħk and you recover Heisenberg's uncertainty principle Δx·Δp ≥ ħ/2. The Gaussian packet is the unique minimum-uncertainty state that achieves equality.

Concrete numbers

System	Group velocity / spreading behavior
Light pulse in vacuum	v_g = v_p = c; no spreading (β = 0)
1550 nm pulse in standard single-mode fiber	Spreads ~17 ps per nm of bandwidth per km — limits long-haul bit rate
Deep-water ocean swell	v_g = ½ v_p; crests race through the group
Free electron localized to 1 nm	Doubles in width in ~1 fs (β = ħ/m)
Femtosecond laser pulse, 800 nm, 10 fs	Δλ ≈ 100 nm of bandwidth needed (large Δk for small Δx)
Sound pulse in air	Nearly non-dispersive; a clap stays a clap over short ranges

JavaScript — building and propagating a packet

// Sum many sinusoids with a Gaussian spectrum to make a packet
function packet(x, t, { k0 = 8, sigmaK = 1.0, omega }) {
  // omega(k) is the dispersion relation; default = non-dispersive (omega = k)
  omega = omega || (k => k);
  let re = 0, im = 0;
  const N = 200, kMin = k0 - 4 * sigmaK, kMax = k0 + 4 * sigmaK;
  const dk = (kMax - kMin) / N;
  for (let i = 0; i < N; i++) {
    const k = kMin + i * dk;
    const A = Math.exp(-((k - k0) ** 2) / (2 * sigmaK ** 2)); // spectrum
    const phase = k * x - omega(k) * t;
    re += A * Math.cos(phase) * dk;
    im += A * Math.sin(phase) * dk;
  }
  return Math.hypot(re, im); // envelope magnitude
}

// Phase and group velocity from a dispersion relation
function velocities(omega, k0, h = 1e-4) {
  const vp = omega(k0) / k0;
  const vg = (omega(k0 + h) - omega(k0 - h)) / (2 * h); // numerical dω/dk
  return { vp, vg };
}

// Non-dispersive: vg === vp, packet keeps its shape
console.log(velocities(k => k, 8));            // { vp: 1, vg: 1 }

// Deep water: omega = sqrt(g*k)  →  vg = vp / 2
const g = 9.81;
console.log(velocities(k => Math.sqrt(g * k), 8)); // vg ≈ vp/2

// Free quantum particle: omega = hbar*k^2/(2m)  →  vg = 2*vp
const hbar = 1, m = 1;
console.log(velocities(k => hbar * k * k / (2 * m), 8)); // vg = 2*vp

// Spreading of a Gaussian packet: width grows with time
function gaussianWidth(dx0, beta, t) {
  return dx0 * Math.sqrt(1 + (beta * t / (dx0 * dx0)) ** 2);
}
console.log(gaussianWidth(1, 1, 0));  // 1   (start)
console.log(gaussianWidth(1, 1, 5));  // ~5.1 (spread)
console.log(gaussianWidth(0.5, 1, 5)); // ~20  (narrower start spreads faster)

Where wave packets matter

• Quantum mechanics. The localized "particle" is a packet; its group velocity is the classical particle speed, and its inevitable spreading sets the scale of quantum uncertainty.
• Fiber-optic communication. Each bit is a light pulse. Chromatic dispersion broadens it over kilometers; engineers add dispersion-compensating fiber and chirped pulses to keep bits from overlapping.
• Ultrafast lasers. Mode-locking sums thousands of phase-locked modes into femtosecond packets used for eye surgery, micromachining, and optical frequency combs.
• Radar, sonar, ultrasound. A finite pulse is a packet; its bandwidth sets range resolution — shorter packet, finer resolution, broader spectrum.
• Oceanography and seismology. Wave groups travel at the group velocity, which is how energy and arrival times of swells and seismic surface waves are predicted.
• Slow and fast light. Engineered dispersion can drop group velocity to meters per second or push it past c, the basis of optical buffering experiments.

Common mistakes

• Thinking the crests carry the energy. Energy and information travel with the envelope at the group velocity, not with the phase velocity of the ripples.
• Assuming v_g > c violates relativity. In anomalous dispersion the group velocity can exceed c or go negative, but the front velocity — the speed of the leading edge — never exceeds c, so no signal does.
• Forgetting that narrower packets spread faster. Tight localization demands a wide band of wavelengths, and a wider band disperses more strongly.
• Confusing dispersion with attenuation. Dispersion reshapes the packet (spreads it) without absorbing energy; attenuation removes energy. They are independent.
• Using v_g = v_p. True only in non-dispersive media. In water, glass, plasmas, and free-particle quantum mechanics they differ, sometimes by a factor of two.
• Treating ω(k) as linear. The linear term moves the packet; the curvature d²ω/dk² is what spreads it. Drop the curvature and you lose the spreading entirely.