Illustrated Physics
physics, drawn
Concept

Group velocity — where it appears

The speed of a wave packet's envelope, being the slope of the dispersion relation rather than the speed of any individual crest. It is the speed at which energy and information travel, and it can differ from the phase velocity by any factor including sign.

Named by 33 essays across 7 fields — each of them below, with the objects they name alongside it.

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.

waves · Wave packets 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.

fluids · Wave packets 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.

waves · Wave packets 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.

waves · Wave packets The lowest note a pipe will carry. The dispersion relation of a guided wave for three cutoffs, in units where the free wave speed is one. Each curve leaves the vertical axis at its own cutoff and bends toward the diagonal, which is the free wave. Above the cutoff the phase velocity is the slope of the line from the origin and always exceeds one, while the group velocity is the slope of the curve and never does: at k = 2 their product is 1.0000, 1.0000, 1.0000, which is one to four decimal places in every case and is an identity rather than a coincidence. Below the cutoff there is no curve, because there is no travelling wave to draw.

The pipe that will not carry a low note

A wave squeezed sideways acquires a lowest frequency. Below it nothing travels — the field is there, it is large, and it goes nowhere. Above it the guide is dispersive whether or not anything in it is, and the pattern inside runs faster than light while the signal does not.

waves · Guided waves Water's permittivity across six decades of frequency. The real and imaginary parts of water's permittivity against frequency, on logarithmic axes, from 10 to 16 in powers of ten hertz. ε′ begins at 80.1 — the number a textbook prints — falls through the Debye relaxation near twenty gigahertz, is dragged down again by the librational, bending and stretching bands of the molecule, and settles at 1.777 in the visible, whose square root is 1.3330: water's refractive index. The identity n = √ε_r is exact and is an identity between two numbers taken at the same frequency; using the static permittivity in it predicts an index of 8.9 and is the most instructive wrong answer in the subject. The curve is a Debye term and four Lorentz oscillators, with two parameters solved so that the two plateaus are the measured ones rather than fitted by eye.

The constant that depends on how fast it is asked

Water's relative permittivity is 80.1. Its refractive index is 1.333, and the square of 1.333 is 1.777. The identity n = √ε_r is exact, the two numbers differ by a factor of forty-five, and nothing is wrong with either — because a permittivity is a function of frequency and the two measurements were made eight orders of magnitude apart.

electromagnetism · Dielectrics A band 3.60 eV wide, and the curvature is the whole story. Energy against wavenumber for one tight-binding band, α + 2β·cos(ka), with α = -4 eV, β = -0.9 eV and a repeat of 300 pm. The band is 3.60 eV from floor to ceiling and periodic in k, so the zone boundary at ka = π is not an edge of anything: it is where the curve turns over. Near the bottom it is a parabola, which is the free-electron dispersion with a different coefficient — a mass of 0.470 m_e rather than one — and near the top it is a parabola the other way up, a mass of -0.470. The inflection between them is at ka = π/2, and it is the point at which a constant force stops producing any acceleration.

The mass a curve decides

An electron in a solid answers a force with a mass that is nothing to do with the mass of an electron. It is set by how sharply the band bends, it is smaller than the free value in a wide band and larger in a narrow one, and near the top of any band it is negative — which is why aluminium's Hall voltage has the sign of a positive carrier and no adjustment to an electron count can repair it.

quantum · Bands One line, 21 decades of density, and every mirror on it. Plasma frequency against electron density, both logarithmic, with the horizontal rules at frequencies a reader already has a feel for. A wave is reflected by everything to the right of where its own rule meets the line and passes through everything to the left. a fluorescent tube at 1.0e+17 m⁻³ cuts off at 2.84 GHz; ionosphere, D layer at night at 1.0e+8 m⁻³ cuts off at 90 kHz; ionosphere, E layer at 1.0e+11 m⁻³ cuts off at 2.84 MHz; ionosphere, F2 layer at 1.0e+12 m⁻³ cuts off at 8.98 MHz; aluminium's conduction electrons at 1.8e+29 m⁻³ cuts off at 3819.9 THz. So the F2 layer turns back a 1 MHz broadcast and lets a 100 MHz one straight out, which is why one of them is heard across an ocean at night and the other stops at the horizon; and aluminium's cutoff sits in the far ultraviolet, which is why it is a mirror for everything visible and a window above 78 nm. The dependence is a square root, so the line has slope one half: a hundredfold denser plasma reflects only ten times the frequency. Arriving at an angle helps by a factor of sec θ — 1.00 at 0°, 1.15 at 30°, 2.00 at 60° — because only the component of the motion along the density gradient has to be turned round.

The frequency below which nothing gets in

Free charges give a medium a permittivity that is negative, and a negative permittivity is not an absorbing medium — it is one in which no wave exists at all. Below that frequency the reflection is total, exactly rather than nearly, because there is no transmitted wave and nothing to absorb. The same expression puts the number at 9 MHz for the ionosphere and 3.8 PHz for aluminium.

astrophysics · Plasma oscillation The cross a shaken cylinder leaves in a stratified fluid. The beams radiated by a small body oscillating in a fluid of buoyancy frequency 1.053e-2 per second — a period of 9.9 minutes — at 0.3, 0.6, 0.9 times that frequency. The disturbance does not spread in circles. It leaves along four rays, and the angle of those rays to the horizontal is fixed entirely by the ratio of the driving frequency to the buoyancy frequency: 17.5° at 0.3N, 36.9° at 0.6N, 64.2° at 0.9N. Nothing about the size of the body, the amplitude of the shaking or the wavelength enters. Drive it faster and the beams stand up; drive it slower and they lie down; drive it above N and there are no beams at all, because the dispersion relation ω = N cos φ has no solution. The short arrows across each beam are the wavevector, which is perpendicular to the beam — the dot products drawn here are 6e-17 — so the crests travel sideways across the ray while the energy travels along it, and a fluid doing this looks, in a photograph, as though its waves are moving at right angles to where they are going.

The wave that picks an angle

Shake a rod slowly in a tank of salty water layered by density and the disturbance leaves along four straight beams, at an angle fixed entirely by how fast the shaking is. Change the wavelength and the angle does not move. Change the frequency and it does. Above the buoyancy frequency there is no wave at all.

fluids · Stratification The wavelength at which a fibre stops smearing a pulse. Group-delay dispersion against wavelength for fused silica, computed by differencing Malitson's Sellmeier fit twice rather than read off a table, in picoseconds of spread per nanometre of source width per kilometre of fibre. The material curve crosses zero at 1273 nm — a property of the glass, fixed by where the ultraviolet and infrared absorptions balance, and not adjustable. Below it a fibre is normally dispersive and above it anomalously so, and at 1550 nm, where silica is most transparent, it is 21.9. The second curve adds a waveguide term, which is negative because a mode confined by a core spreads into the cladding differently at different wavelengths, and which depends on the fibre's geometry rather than on the glass; it is drawn here as a constant offset chosen to put the total zero at 1310 nm, which is what standard single-mode fibre is built to do. The consequence in a system is the thing worth carrying away: a source one nanometre wide sent 100 km through fibre at 1550 nm arrives 1833 picoseconds broader than it left, against 0.0 at the zero. That is the whole reason a network runs at one wavelength rather than another, and the reason the two quantities an engineer wants — lowest loss and zero dispersion — sit at different wavelengths and have to be reconciled by design rather than chosen.

The wavelength a fibre does not smear

Fused silica's index has a second derivative that passes through zero at 1,273 nanometres, and that is not a design choice — it is where the ultraviolet and infrared absorptions balance. A pulse sent at that wavelength arrives the shape it left. A pulse at the wavelength of least loss arrives 1,800 picoseconds wider after a hundred kilometres.

optics · Dispersion A click that arrives sorted by pitch. Arrival time against frequency for a broadband pulse travelling 40 megametres along a magnetic field line through a plasma of 1e+8 electrons per cubic metre in a field of 300 nanotesla — the magnetosphere, roughly. The gyrofrequency is 8.4 kilohertz and the plasma frequency 90 kilohertz, so this is the regime where a right-hand circularly polarised wave travels happily below both of them. It does not travel at one speed. The group velocity is 2c√ω(ω_c − ω)^(3/2)/(ω_p ω_c), which is zero at zero frequency and zero again at the gyrofrequency and peaks in between — at 2.10 kilohertz here, which is a quarter of the gyrofrequency, found by searching the drawn function rather than quoted. Everything either side of that peak is slower, so the curve has a nose: 3.21 s at 500 Hz, 2.20 s at 2000 Hz, 3.59 s at 5000 Hz. The branch below the nose is the classic whistler — a lightning stroke in one hemisphere reaching a receiver in the other as a note gliding downward over about a second, which is what gave the phenomenon its name — and the branch above it arrives as a rising tone at the same time. Both are observed, and a recording that shows the two joined at the nose is how the gyrofrequency along the path is read off directly. Read the other way it is an instrument: the product of the delay and the square root of the frequency is 79 s·Hz^½ here, nearly constant across the low end of the band, and it measures the electron content of a path through the magnetosphere that nothing else could reach.

The whistle that arrives sorted

A lightning stroke in one hemisphere reaches a receiver in the other as a note gliding downward over about a second. Nothing dispersed the sound; there was no sound. A radio pulse travelled forty megametres along a magnetic field line through a plasma whose group velocity depends on frequency, and arrived with its frequencies separated by up to three seconds.

astrophysics · Plasma oscillation 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.

waves · Wave packets The chain's dispersion, and the frequency it stops at. Frequency against wavenumber for a chain of equal masses joined by equal springs, in units where the spacing, the mass and the spring are one. At long wavelength the curve is a straight line through the origin — the chain behaves as a continuous string with a sound speed, and the departure from the line is second order in the wavenumber, which is why a lattice is invisible until the wavelength approaches the spacing. At the zone edge, where neighbouring masses move in exact opposition, the curve flattens: the frequency stops rising, the group velocity falls to zero, and the mode is a standing wave that carries nothing. Above that frequency there is no travelling solution at all.

The frequency a lattice cannot carry

A continuous string carries every note. A row of masses joined by springs does not: there is a highest frequency, set by nothing but the time one mass takes to be pushed back by its neighbours, and above it a disturbance does not travel at all.

waves · Periodic media The delay that stops caring how thick the wall is. The Wigner phase time — how late the transmitted packet's peak arrives, compared with a free particle covering the same distance — for an electron under a 3 eV barrier, at 0.2 nm, 0.5 nm, 1 nm thick. 0.2 nm: 0.3728 fs at half the barrier height; 0.5 nm: 0.4372 fs at half the barrier height; 1 nm: 0.4388 fs at half the barrier height. The widths span a factor of 5.0 and the times span 1.18. A thicker barrier is exponentially harder to cross and the delay in crossing it barely moves — which is either a remarkable fact about tunnelling or a sign that the phase time is not a traversal time, and the rest of the essay is about which.

How long the crossing takes

Tunnelling has a probability, and asking how long it takes turns out to be a different kind of question. The delay a transmitted packet shows stops growing once the barrier is opaque, so a thicker wall is crossed in the same time — and several defensible clocks give several different answers.

quantum · Tunnelling Three speeds, drawn against direction. The phase speed of the three magnetohydrodynamic waves against the angle between the wavevector and the field, drawn as a polar diagram with the field horizontal and the sound speed 0.6 times the Alfvén speed. Along the field the two magnetosonic branches take the values of the sound speed and the Alfvén speed themselves and swap which is which as the ratio crosses one; across the field the slow branch vanishes entirely and the fast branch runs at the quadrature sum. The shear branch is the figure-of-eight, v_A cos θ, and it is the only one of the three whose speed contains no thermodynamic quantity — no pressure, no temperature, no sound speed. It does not know what the gas is made of.

The wave that does not know what the gas is made of

Give the magnetic tension of a bent field line an inertia and it becomes a string. The wave that runs along it goes at the same speed at every wavelength, compresses nothing anywhere, and has a speed containing no temperature, no pressure and no sound speed — it is the only wave in classical physics that is indifferent to what the medium is. Its energy travels along the field whatever direction the wave was sent in.

astrophysics · Flux freezing The pattern that stands still while the air goes through it. Streamlines of a steady 20 metre-per-second wind over a bell-shaped ridge 800 metres high and 6.0 kilometres wide, in air whose buoyancy frequency is 1.05e-2 per second. Nothing in the picture is moving: the air crosses it from left to right at 20 metres a second and the waves stay where they are, because standing still is what selects them. The vertical wavelength is 2πU/N = 11.9 kilometres, and the crests lean upstream as they rise — checked here by finding the maximum displacement a quarter of a wavelength up, which sits well upstream of the ridge. That tilt is the signature of energy travelling upward, and it is the feature every hand-drawn version of this picture gets backwards. Cloud forms at the crest of each wave where the air is highest and coldest, which is why the lens-shaped clouds sit stationary in a moving airstream.

The wave that is required to stand still

A stratified fluid supports internal waves at every wavelength there is. Put a steady wind over a ridge and the requirement that the pattern stay put picks exactly one of them — 2πU/N, and nothing about the mountain appears in it. The clouds that mark the crests sit still while the air goes through them at twenty metres a second, and the momentum the wave carries away is delivered thirty kilometres up.

fluids · Stratification A reflection that keeps the angle to gravity and not to the wall. An internal wave beam of frequency 0.5N reflecting from a slope of 12°, with the wavelength ratio for slopes of 12°, 20°, 28° computed beside it. The frequency fixes the angle the energy makes with the horizontal — 30.0° here — because the restoring force is gravity and gravity is vertical, so the reflected beam must leave at that same angle whatever the wall is doing. Incident and reflected rays are therefore not mirror images, and the wavelength changes on reflection by sin(θ+α)/sin(θ−α). A flat floor gives one, checked exactly. A slope approaching the ray's own angle gives infinity, checked as the limit, and that is where a basin's internal tide is compressed until it breaks.

The reflection that changes the wavelength

An internal wave's frequency fixes the angle its energy makes with gravity, so a sloping wall cannot send it back the way a mirror would. The reflected beam leaves at the same angle to the vertical rather than the same angle to the wall, its wavelength changes by a factor that diverges when the slope matches the ray, and in a closed basin the changes accumulate until every ray in the fluid lies on one line.

fluids · Stratification Two intersections, and the rule that picks one. The phase-matching construction for light arriving at 40° from a medium of index 1 into one of index -1. The circles are each medium's own relation between wavevector and frequency; the vertical line is the tangential wavenumber, which the boundary conserves. The line crosses the second circle twice, and the construction alone does not say which point is the answer — the rule that does is that energy must travel away from the interface. For a positive index the energy runs along the wavevector and the upper point is taken. For a negative one the energy runs against it, so the lower point is taken, the wavevector points back toward the boundary, and the ray leaves at -40.0° — on the same side of the normal as it arrived. Nothing in the drawing has changed except which intersection is circled.

The ray on the wrong side of the normal

The phase-matching construction draws a circle and a line, and the line crosses the circle twice. Every earlier construction silently took the upper intersection. Which one is physical is decided by where the energy goes rather than by where the wavevector points, and in a medium whose group velocity opposes its phase velocity the answer is the other one — so the refracted ray leaves on the same side of the normal it arrived on, a flat slab focuses, and a lens can beat the diffraction limit until loss stops it.

optics · Refraction The latitude past which the tide cannot shed its energy by halves. Frequency in cycles per day against latitude. The curve is the inertial frequency, 2Ω sin(latitude), below which no internal wave can oscillate; the shaded region above it is where internal waves exist. The horizontal lines are the semidiurnal and diurnal tides and the frequencies half of each, where a parametric instability would put the waves the tide decays into. Each line ends where it meets the curve, which is its critical latitude: M2, semidiurnal at 1.932 per day, 74.5°; M2 ÷ 2 at 0.966 per day, 28.8°; K1, diurnal at 1.003 per day, 30.0°; K1 ÷ 2 at 0.501 per day, 14.5°. Equatorward of 28.8° the semidiurnal tide can feed waves at half its frequency; poleward of it those waves cannot exist and that route is closed. The diurnal tide's subharmonic is confined within 14.5° of the equator, and the diurnal tide itself cannot propagate as a free internal wave poleward of 30°.

The latitude past which a tide cannot split

The ocean's internal tide carries about a terawatt, and somewhere it has to be broken into waves small enough to mix the water. One of the ways it breaks is by pumping waves at half its own frequency, the way a child on a swing pumps at twice the swing's. Those half-frequency waves cannot exist where the planet's rotation forbids oscillations that slow — poleward of 28.8° for the semidiurnal tide — so the route has an edge on the map, fixed by the Moon's period and the Earth's spin.

fluids · Stratification The block has to move, and which way decides the momentum. A pulse of light crosses a free block of glass of index 1.5 and length L, with time in units of L/c. The pulse falls behind where it would have been in vacuum by 0.50 L while it is inside. The block's displacement, in units of EL/Mc², is obtained from momentum conservation alone: if the light inside carries Abraham's E/nc the block takes up the rest of E/c, moves forward, and ends 0.500 units along — Balazs's (n − 1). The sum of the pulse's lag and the block's shift, which is the centre of energy's departure from uniform motion, is then zero to within 0.0004 units at every moment. If instead the light carried Minkowski's nE/c, the block would have to recoil backwards and the centre of energy would end 1.25 units behind a straight line, which no isolated system can do. For a one-joule pulse through a kilogram of glass ten centimetres long the real shift is 5.6 × 10⁻¹⁹ m.

The momentum light carries into glass

A pulse of light in glass has two momenta, nE/c and E/nc, and for a century the argument was about which one is right. Both are measured. A free block that the light crosses must move forward, which only Abraham's value explains; a Cherenkov cone exists at all, which only Minkowski's allows. They answer different questions, and the difference between them is momentum carried by the glass.

electromagnetism · Field energy The orbits of an electron in a travelling wave. The possible motions of an electron in a plasma wave whose phase moves at γₚ = 10 and whose field is 0.2 of the cold wave-breaking field, drawn as its energy against its position in the wave over two wavelengths — the level curves of H = γ − βₚ u − ε cos ψ, which the electron's motion conserves. Inside the separatrix (thick) an electron is trapped: it slides back and forth through the wave, gaining energy while it outruns the wave and losing it while the wave outruns it. The separatrix reaches γ = 99 at its top, near 4γₚ²ε = 80, starting from electrons moving at the wave's own speed. Outside it, electrons that are too slow are overtaken by the wave and those that are fast enough outrun it; neither gains energy on average.

The wave an electron rides to a gigaelectronvolt

A laser pulse fired into a thin plasma pushes its electrons aside and leaves them oscillating behind it, a wave of charge travelling at nearly the speed of light. The field in that wave is a thousand times stronger than any metal accelerator can hold, and an electron caught in it at the right moment is carried along like a surfer, gaining a gigaelectronvolt in a few centimetres. Then it outruns the wave. The distance it can ride, and the energy it can gain, are both set by how nearly the wave keeps pace with it — which is to say, by relativity.

relativity · Relativistic dynamics Colours that leave a grating on different roads. A pair of parallel diffraction gratings with 1,200 lines per millimetre and light arriving at 30°, drawn to scale for three wavelengths, 760, 800 and 840 nm. The first grating sends each colour off at its own angle — 24.3° for 760 nm, 27.4° for 800 nm, 30.5° for 840 nm — and the second turns them all back parallel to the input, side by side. Between the gratings the red travels the longest road: for gratings 10 mm apart the slant paths are 10.97 mm, 11.26 mm, 11.61 mm. Measured to a common wavefront after the second grating, the longer road still wins, so red arrives last: the geometry has given the pulse the opposite ordering to the one glass gives it.

The dispersion made of angles

Glass delays blue light more than red, so a short pulse that passes through glass comes out stretched with its red end first. At the wavelengths where most short-pulse lasers work, no transparent glass does the opposite. Two diffraction gratings do it with no material at all. The first spreads the colours in angle, the second turns them back parallel, and in between the red has travelled the longer road. A spread in angle, however it is made, always adds this kind of dispersion, of the sign glass cannot give. It is how every high-power ultrafast laser squeezes its pulse back after amplifying it.

optics · Dispersion A delay the Planck scale would write into a burst. The arrival delay, relative to a low-energy photon emitted at the same moment, of photons of each energy from a gamma-ray burst at redshift 0.903, if light slowed in proportion to its energy over an energy scale, on logarithmic axes, for that scale at the Planck energy and at ten times and a tenth of it. At the Planck energy a 31 GeV photon would arrive 1.16 s late. The 31 GeV photon from GRB 090510 arrived 0.83 s after the burst began (marked), so a delay of more than that is excluded, and with it any scale below 1.4 times the Planck energy on this simple reading. Fermi's own analysis, allowing for when in the burst the photon could have been emitted, set the limit at 1.2 times the Planck energy: a speed that depends on energy at first order has been excluded at the scale where it was expected.

The photons that raced for seven billion years

Nothing on a laboratory bench can reach the Planck scale, but a gamma-ray burst halfway across the universe can amplify a Planck-scale effect until it is a second long. If spacetime has structure at the Planck length, the simplest guess is that light's speed depends slightly on its energy. Then a burst's high-energy photons would arrive measurably later than its low-energy ones, after a race lasting billions of years. In 2009 a 31 GeV photon from a burst 7.5 thousand million light-years away arrived within 0.83 s of the burst's start. That excludes the simplest version of the idea at the Planck energy itself.

astrophysics · Planck scale Light from one guide, spreading through a row of them. The power in each of a row of identical coupled waveguides after light launched into the central one has travelled 1.5, 3, 5 and 7 coupling lengths (1/C) along the row, drawn as stacked bars with the distance increasing upward: |Jₙ(2Cz)|², where n counts guides from the centre. The light does not spread into a single widening hump as it would in a uniform slab. It forms two beams that run outward at a constant rate, carrying most of the power in their leading edges near n = ±2Cz, and leaves the centre nearly dark: after 7 coupling lengths the central guide holds 2.9 per cent and the brightest guides, at ±12, hold 8.1 per cent each. This is discrete diffraction: the spreading of light that can only step between neighbouring sites.

The light that spreads in two beams

Shine light into one of a row of identical optical fibres laid side by side and close enough to leak into each other, and it does not spread out into a widening glow as it would in a slab of glass. It splits into two beams that run outward at a fixed speed, bright at their leading edges, leaving the middle almost dark. The reason is that light hopping between neighbours has a fastest possible sideways speed, and piles up at it. The same periodicity gives the row two tricks no uniform medium has: a direction in which a beam travels without spreading at all, and a direction in which diffraction runs backwards, so that a spread beam can be gathered up again.

waves · Periodic media Rays at a frequency the ionosphere cannot stop head-on. Ray paths at 14 MHz launched at angles from 10° to 75° from the vertical into a Chapman layer peaking at 300 km with a critical frequency of 8 MHz, over a flat Earth. A wave sent straight up at 14 MHz passes through: it is above the layer's plasma frequency everywhere. A ray at φ₀ from the vertical turns back where the plasma frequency equals f cos φ₀ — the secant law — so only rays more oblique than arccos(8/14) = 55.2° return. Those that do land beyond about 1133 km; nothing comes down closer than that: the skip zone, a ring of silence round a transmitter that is audible much further away. Vertical scale exaggerated about six times.

The frequency the sky returns only at a slant

The ionosphere turns back any radio wave below its plasma frequency, about eight megahertz on a summer afternoon, and lets anything above it out into space — if the wave is sent straight up. Send a fourteen-megahertz wave up at a slant and it comes back down, a thousand kilometres away. Only the part of the wave's motion across the layer has to be stopped, and that part is the frequency times the cosine of the angle. The rule turns one frequency into a range of them, sets the highest frequency that can reach any given distance, and leaves a ring of silence round every short-wave transmitter, inside which it cannot be heard at all.

astrophysics · Plasma oscillation 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.

waves · Wave packets What a station hears from one thunderstorm region. The power spectrum of the vertical field from a single region of thunderstorms, received 20° and 90° of arc away, from a sum of the first seven resonant patterns, each with the measured peak frequency and a quality factor of 4 to 6, both taken as inputs. Measured spectra fall off with frequency rather more than this one, because real strokes and the real ionosphere are not as simple. The resonances are so broad that neighbouring peaks run into each other; a quality factor of five means a peak about a fifth as wide as its frequency, and the ringing dies in a few cycles. At 20° every tone is present, the higher ones merging. At 90° the first, third and fifth have vanished, because those patterns have a node a quarter of the world away, and the spectrum is made of the even tones alone. Real stations receive the sum of the world's storms, which move with the time of day, and the heights of the peaks drift with them.

The tone the whole Earth rings at

The ground conducts and so does the ionosphere, eighty kilometres up, and between them lies a waveguide the size of the planet. Two walls, unlike a pipe, carry waves of any length, so the guide takes radio waves forty thousand kilometres long — longer than the world is round — and, closed on itself, it rings. Lightning strikes fifty times a second and keeps it ringing at about 7.8, 14.3 and 20.8 hertz. Perfect walls would put the tones at 10.6, 18.4 and 25.9; the difference is the ionosphere, a ceiling that leaks.

waves · Guided waves Where the spot is, and when, for three sweep speeds. A spacetime diagram of the spot a turning beam makes on a flat screen at distance D: position along the screen across, the time at which light arrives there up, both in units of D and D/c, for beams sweeping the screen's nearest point at ωD = 0.5c, 1.0c, 3.0c. Each curve is the arrival time t(x) = atan(x/D)/ω + √(D² + x²)/c. Every curve has a lowest point on the approaching side — at x = −1.25, −0.79, −0.32 D respectively — because far out the beam strikes the screen obliquely and its sweep along the screen outruns light whatever ω is. Light reaches that point first: the spot is born there, as a pair, one running back up the screen and one forward. The faster the sweep, the nearer the centre the birth. The faint dashed lines are light signals, slope one. Wherever a spot's track is flatter than they are, it is a sequence of events no signal could connect.

The spot that outruns light and carries nothing

Flick a laser pointer across the face of the Moon and the spot it makes crosses the Moon faster than light. Nothing is wrong with relativity: the spot is not a thing, and no part of it travels from one side to the other. Each lit point is lit by its own light, sent from the hand. What makes the spot worth studying is how strange such a non-thing is allowed to be. It can be born in the middle of the screen as a pair running apart; its speed can be made infinite, or reversed, merely by watching it from a moving frame; and the composition of velocities that caps everything real has nothing to say about it.

relativity · Velocity-addition Which waves a moving object can make. The phase speed of waves on deep water against wavelength (red), with horizontal lines at the speeds of an object moving steadily across the surface: 15.0 cm/s, 23.1 cm/s, 30.0 cm/s, 50.0 cm/s. A steady object can only feed waves that keep pace with it, whose phase speed equals its own, and those are where its line meets the curve. At 15 cm/s the line misses the curve entirely: no wave can keep up, and the object leaves no wake. At 23.1 cm/s it touches the curve once. Above that it meets it twice: at 30 cm/s a ripple of 5.6 mm and a gravity wave of 5.2 cm; at 50 cm/s a ripple of 1.9 mm and a gravity wave of 15.8 cm.

The speed below which nothing on water leaves a wake

Every wave on clean water travels at 23 centimetres a second or faster: long waves are fast because gravity drives them, short ones because surface tension does, and in between there is a slowest wave. A boat, a duck or a fishing line can only make waves that keep pace with it, so anything crossing the water more slowly than 23 centimetres a second makes no waves at all. Faster, it makes two kinds at once — short ripples that run ahead of it and long waves that trail behind — and the threshold between nothing and both is set by gravity, surface tension and density alone.

waves · Wave motion Every wave a plate can carry, at every frequency. The phase velocities of the guided waves in an aluminium plate (longitudinal speed 6.32 km/s, shear 3.13), against the product of frequency and thickness in MHz·mm, from the Rayleigh–Lamb equations: symmetric modes, in which the two faces bulge and thin together (light), and antisymmetric ones, in which they bend together (dark). At low frequency only two exist. The symmetric S₀ is a stretching wave at the plate velocity, 5.44 km/s; the antisymmetric A₀ is a bending wave whose speed falls towards zero, 0.96 km/s at 0.1 MHz·mm. At high frequency both approach the 2.92 km/s of a Rayleigh wave on each face. Higher modes enter at cut-off frequencies where the plate's thickness holds a whole number of half-wavelengths of one bulk wave — the first at 1.56 MHz·mm — arriving from infinite phase velocity and slowing as the frequency rises.

The plate that carries two kinds of sound

Tap the edge of a sheet of aluminium and the sound that runs along it is not one wave but two, and neither is the sound of the metal. One stretches the sheet and travels at 5.44 km/s, faster than a shear wave and slower than a compression wave in the same metal. The other bends it, travels at a speed that depends on how high the note is and how thick the sheet is, and at low frequency crawls at a few hundred metres a second. Both are made of the same two bulk waves bouncing between the faces, and the two faces together decide everything.

waves · Guided waves The sideways shaking that arrives longest waves first. The transverse ground motion 3000 km from a sudden shear source, synthesised from the computed dispersion of the fundamental Love wave in a 35 km crust, with periods from about 8 to 120 seconds. Nothing arrives before 11.0 minutes, the travel time at the mantle's shear speed. Then the longest periods come in first — 80 s waves at their group velocity of 4.23 km/s, after 11.8 min — and the period shortens steadily through the train, 30 s waves arriving at 14.3 min. The strongest shaking comes last, near 14.8 min, where the group velocity passes through its minimum and waves over a band of periods arrive together. Read off a real seismogram, the arrival time of each period is the crust's dispersion curve, and the crust's thickness with it.

The shaking a uniform Earth could not carry

Early seismographs recorded a large sideways shaking that arrived after the body waves and before the Rayleigh wave, and no theory of a uniform Earth allowed it: a horizontal shear wave at a free surface reflects as itself and nothing holds it there. Augustus Love showed in 1911 that a layer does. A crust slower than the rock beneath traps the shaking between a perfect mirror at the top and total reflection at the bottom, and because the long waves reach deeper and run faster, the order in which the periods arrive is a measurement of how thick the crust is.

waves · Wave motion The readings of the moving clocks along one instant: a wave. The cosine of each moving clock's phase, read along a single laboratory instant, against position in light-periods of the clock, for clocks moving at 0.3c, 0.6c, 0.85c. The readings make a wave whose wavelength, measured from the drawing, is 3.18, 1.33, 0.62 light-periods: c²/(γvν₀), shrinking as the clocks move faster. For a clock that ticks at its rest energy over Planck's constant, ν₀ = mc²/h, that is h/γmv — de Broglie's wavelength, h over the momentum. Nothing about waves was assumed: only clocks, synchronised where they rest, read from a frame in which they move.

The wave a row of moving clocks leaves on an instant

Line up identical clocks, synchronise them where they stand, and send the whole row past a laboratory at a steady speed. On any one laboratory instant the clocks do not agree: the ones further ahead read less, by an amount that grows steadily along the row. The readings form a wave laid out in space, its crests are the moving row's instants, and they sweep along faster than light. Give each clock the frequency of its rest energy and the wavelength is exactly h/p. De Broglie's matter wave began, in 1923, as an argument about the relativity of simultaneity.

relativity · Simultaneity Waves that turn to face the beach. The paths of crests' motion for a 10 s swell arriving at 50° on a plane beach of slope 1 in 50, seen from above, with depth contours dashed. Snell's law, sin α over speed is constant, holds along the ray because the depth changes only in the offshore direction, and the integrated paths were checked against it. The end of each ray is nearly normal to the shore: 5° from square-on by the point where it is about to break. In this sense all surf comes in nearly straight whatever direction the storm was in. The bend is complete because the speed falls almost to zero at the shore, whereas a prism's angles are bounded by its index. What remains of the angle drives the current along a beach.

The waves that always arrive square to the beach

A storm a thousand kilometres off sends out swell in whatever direction it pleases, and that swell arrives on the beach running almost straight in. It does so whichever way the coast faces. Waves slow down in shallow water, so the part of a crest nearest the shore lags and the whole line swings round, as light does in glass. It also explains why waves shrink before they grow, and why they gather on a headland and leave the bay alone.

waves · Wave motion

Named alongside it

The objects these essays reach for when they reach for this one.

DispersionDispersion relationPhase velocityWave packetCutoffRefractive indexEvanescent waveAnisotropyCausalityGuided wavesInternal wavesStratification

All concepts