Wave packets — the series · Illustrated Physics

A packet on deep water, ω = √(gk), 1.6 s apart. A wave packet built from a Gaussian spread of wavenumbers about 1.57 per metre, drawn at two times 1.6 seconds apart, with its computed envelope ghosted around it. Between the two frames the envelope's peak moves 2.01 metres and the marked crest moves 3.95 metres, so the packet travels at 1.26 metres per second and the crests at 2.47 — a ratio of 0.51.

The packet that moves at another speed than its own crests

Watch a group of water waves and the individual crests run forward through it, rise in the middle, and vanish off the front. The group travels at half the speed of the crests, and both numbers are real.

part 1 · waves

Speed against wavelength. Phase and group speed for waves on water 4 m deep. Short waves are dispersive — the speed rises as the square root of the wavelength and the group travels at half the phase speed — and long ones all travel at √(gh) = 6.26 m/s together, which is why a tsunami keeps its shape across an ocean while a wind sea spreads out into swell.

The speed that depends on the length

Long waves on water travel faster than short ones, which is why a distant storm arrives as a slow swell and a tsunami crosses an ocean without spreading. One relation covers both, and the two familiar rules taught separately are its two limits.

part 2 · fluids

The narrow packet is the one that spreads. Three packets on deep water, ω = √(gk), all built on the same 4 m carrier and differing only in bandwidth — 10%, 18%, 28% of the carrier wavenumber. Each curve is the width of the emitted envelope, measured as the second moment of its intensity about its own centroid in a frame moving at the group velocity, divided by that width at the start. The starting widths are 4.50 m, 2.50 m, 1.61 m, and the order of the curves is the reverse of the order of the widths: the shortest packet, which is the one with the widest spectrum, is the one that comes apart first. The dashed curves are √(1 + (t/τ)²) with τ = σ₀²/|d²ω/dk²| — computed from the dispersion relation, not fitted. They agree with the measured widths to 0.3% at 10% bandwidth, 3.3% at 18% bandwidth, 8.5% at 28% bandwidth, and that ordering is the second thing the figure says: the closed form keeps only the curvature of ω(k), so it is exact for a narrow spectrum and starts to fail for a wide one, by about as much as the cubic term is worth. A packet with no bandwidth would never spread at all, and would also never begin or end.

The packet that will not keep its shape

A group velocity is only the first thing a dispersion relation says. The second is that the packet spreads — at a rate fixed by its own bandwidth and by the curvature of ω(k) — so a short pulse comes apart quickly and a long one hardly at all. It is a trade quantum mechanics is usually given credit for, and classical waves make it too.

part 3 · waves

A peak that leaves before it should have arrived. Two pulses at the far face of a cell, both normalised to the peak the vacuum one reaches. One has crossed empty space; the other has crossed a medium with two gain lines either side of its carrier, whose group index there is -3.85 — negative, so the envelope's peak should emerge early, and it does. Measured off the two curves the advance is 616 in units where the carrier period is 2π, against 582 predicted from the group index alone; the difference is the higher-order dispersion the group index leaves out. The advance is 0.21 of the pulse's own duration, and the peak leaves the far face before the input peak has entered the near one. Nothing has outrun anything. The emergent pulse is a reshaped version of the input's leading edge, which arrived in plenty of time and already contained — for a smooth pulse — everything needed to reconstruct the rest; the medium amplifies it by 1.14× and delivers it early. Give the pulse a genuine front, a moment before which it is exactly zero, and that front travels at the speed of light in every medium there is.

The speed that carries no signal

In the right medium a pulse's peak emerges from the far side before it entered the near one. The measurement is real, it has been made, and nothing has outrun light — because the peak of a smooth pulse was never carrying any information in the first place.

part 4 · waves

The pulse two failures keep alive

Dispersion spreads a pulse until it is nothing. Nonlinearity steepens it until it breaks. Each on its own destroys a disturbance, and there is exactly one height for each width at which the two cancel completely — leaving a shape that travels for ever and survives being run into by another one.

part 5 · waves

A hump that is not a soliton comes apart into solitons. A single smooth hump of height 6, shaped as the square of a hyperbolic secant, released into the Korteweg–de Vries equation and followed by a pseudo-spectral integration, drawn at times 0.00, 0.15, 0.35, 0.60, each snapshot raised above the last. The hump is too tall for its width to be a soliton, and it separates: by the last time there are 2 crests, of heights 8.00 and 2.00, running apart at different speeds, with a small ripple left behind. Read as a potential well, the same hump holds 2 bound states, at κ = 2.000 and 1.000, and a soliton of height 2κ² belongs to each: 8.00 and 2.00. The integration conserved the hump's area to 3.1·10⁻¹⁵.

The solitons a hump already contains

A soliton is one height for one width. Release a hump of any other shape and it does not keep that shape or simply spread — it comes apart into a fixed number of solitons of fixed heights, running off in order of size, with a ripple left behind. The number and the heights can be read off before anything moves, by treating the hump upside down as a well and counting the levels it holds.

part 6 · waves