Illustrated Physics
physics, drawn
Series

Wave packets — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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
  2. 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
  3. 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
  4. 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
  5. Two ways of destroying a pulse, and their cancellation. The same starting pulse, carried forward three times by three equations that differ only in which terms are present, all shown at t = 0.097. Dotted: where it began. With the dispersive term alone the pulse spreads and sheds an oscillating tail, because its Fourier components travel at different speeds and drift out of step — the profile departs from what it was by 27 per cent of its own height. With the nonlinear term alone the tall part overtakes the shallow part and the front leans forward: the steepest gradient is 9.3 times what it started as and is on its way to vertical. With both, neither happens — the profile has moved to the right and is otherwise identical to what it was, to 2.7e-12 of its own height. The balanced run is a pseudo-spectral integration and the other two are exact, and the integration conserves the two quantities the equation conserves — mass to 2.2e-16 and the squared integral to 4.7e-15 — which is the check that the answer belongs to the equation rather than to the integrator. What the picture cannot show is why the cancellation is stable: a pulse of the wrong height for its width does not persist in the wrong shape, it sheds the excess as a dispersive tail and settles on the shape that works, which is why these objects turn up in canals and optical fibres rather than only in equations.

    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
  6. 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
  7. A tiny ripple that grows, peaks, and gives the energy back. The highest point of a wave train's envelope against time, starting from an even train of amplitude 1 with a ripple of 1 per cent at the fastest-growing wavenumber, integrated from the nonlinear Schrödinger equation. For a while nothing happens that the eye could see; the ripple grows exponentially from a per cent. Then, near t = 5.6, the energy of a whole modulation period gathers into one crest 2.41 times the original amplitude — 1 + √2, the height of the Akhmediev breather this wavenumber produces. Then it spreads out again, and by t ≈ 11 the train is almost even once more, its highest point 1.01: the energy returns to where it came from, and the cycle repeats, the crest rising again near t = 17. The wave's total norm is conserved throughout to a part in a hundred million.

    The even wave train that cannot stay even

    In the mid-1960s a team at the National Physical Laboratory tried to make a long, perfectly regular train of waves in a towing tank, and could not: however carefully they set the paddle going, a few hundred metres down the tank the waves had bunched into groups. Their paddle was not at fault. A uniform train of steep waves on deep water is unstable — small modulations of its height grow, the energy of many waves gathers into a few, and then, astonishingly, gives itself back. Out of that instability comes a wave three times the height of the sea around it, with no history before and none after.

    part 7 · waves
  8. A wave that falls with nothing pulling it. The intensity of Berry and Balazs's Airy packet, |Ai(x − t²/4)|², drawn at six moments t = 0 to 5 in units where ħ = m = 1, each profile raised by its time. It is an exact solution of the free Schrödinger equation — no potential, no force — and equally of the equation that governs a light beam in the paraxial approximation, with t then the distance travelled. Its shape never changes and never spreads, and its main lobe, which starts at x = −1.02, moves along the parabola x = t²/4 (dashed): −0.77 at t = 1, −0.02 at t = 2, 1.23 at t = 3, 2.98 at t = 4, 5.23 at t = 5. A uniform acceleration of a half, with nothing accelerating it. The packet carries infinite energy, which is the price of the trick: its oscillating tail stretches to minus infinity, and the tail is what pushes the main lobe along.

    The packet that bends with nothing pushing it

    Every wave packet left to itself spreads, and every free particle moves in a straight line. In 1979 Berry and Balazs found a packet that does neither: it keeps its shape for ever, and its peak accelerates sideways along a parabola, in empty space, with no force acting. Nothing in it breaks Newton's laws — the packet's centre of mass stays exactly where it was, and only its bright head runs off, fed by a long dim tail — and the trick turns out to be the rainbow's: the bright lobe is the envelope of a family of straight rays. Thirty years later it was made with light, and then with electrons.

    part 8 · waves
  9. The long pulse that arrives as a spike. The output of the matched filter for the chirp, in decibels against delay in units of 1/B, beside the same filter's output for an unmodulated pulse of the same duration (dashed). The plain pulse's echo is a broad triangle 100 units wide; the chirp's is a spike 0.878/B wide at its half-power points — compressed by the time-bandwidth product, 50. The compression is not free of shape: the spike has sidelobes, the first at −13.6 dB, from the abrupt edges of the band, and a weak echo sitting in a strong echo's sidelobe can be hidden by it. The energy is all there — the filter has concentrated the whole pulse's echo into one instant of output.

    The long pulse that arrives as a spike

    A radar that wants to see far must send a lot of energy, and a transmitter limited in peak power can only do that with a long pulse; a radar that wants to tell close targets apart needs a short one. A chirp — a pulse whose pitch rises steadily from start to finish — is both. Received through a filter that delays its low notes more than its high ones, every part of the echo arrives at the same instant, and fifty microseconds of pulse leave the receiver as a spike one microsecond wide. Resolution turns out to belong to bandwidth, not to duration, and the price of the trick is paid in sidelobes and in a target's speed being mistaken for its range.

    part 9 · waves

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