Waves

A wave is a shape that travels, and nothing else does

In a wave on water, no water goes anywhere. What moves is the shape — and separating the two motions is the whole of wave physics.

When two waves meet, they simply add

Waves pass through each other unchanged and their displacements add point by point. From that one impoverished-sounding rule comes interference, beats, and the evidence that light is a wave at all.

Only some notes fit, and that is where discreteness comes from

A string clamped at both ends can vibrate at some frequencies and not others. A continuous object producing a whole-number list is the oldest quantisation in physics.

The note that changes on approach, and the two ways of getting it

A moving source and a moving listener produce different formulas for the same shift, because the medium is watching. The difference is small, real, and the reason light had to be treated differently.

The frequency that gets an answer, and the quarter cycle nobody mentions

Push an oscillator at its own frequency and the response grows enormously. The reason is not that the push is in step with the motion — at resonance it is a quarter cycle out, and that is precisely why it works.

The medium decides the speed, and the source only decides the note

A wave's speed is not chosen by whatever made it. It is a property of the material the wave is crossing, fixed before the wave arrives, and the wavelength is whatever is left over after the division.

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.

Every front is a source

Treat each point of a wavefront as though it were a little source of its own, and take the envelope of what they produce. That one rule gives straight-line propagation, reflection, Snell's law and diffraction — and its most famous failure is what told everyone what was missing from it.

What happens where the medium changes

Two conditions at a join — the displacement is continuous, and so is the transverse force — fix the reflected and transmitted amplitudes completely. What decides them is one quantity, the impedance, and not the stiffness or the density separately: two quite different media with the same impedance are, to a wave, the same medium.

How a wave thins out

A wave gets weaker with distance for two quite different reasons, and only one of them is a loss. Geometry alone fixes the first exactly — three exponents for three dimensions, with nothing about the medium in them — and whatever is left over is the medium eating the wave.

The equation that lets a shape travel

Newton's second law applied to a piece of string a millimetre long gives T·y″ = µ·ÿ, and the derivation never once asks what the string is made of. Two things fall out immediately: the speed is √(T/µ) and belongs to the medium, and the general solution holds two arbitrary functions rather than one. The second of them is the reflection, which is why a boundary condition can be met at all.

The drum that has no harmonics

A string's allowed frequencies are 1, 2, 3, 4 times its lowest, because counting half-wavelengths is arithmetic. Clamp a membrane round a circle and the same reasoning returns 1.000, 1.593, 2.136, 2.295 instead — zeros of Bessel functions, not integers. And those numbers belong to the shape of the boundary alone, which raises a question nobody could answer until 1992.

The swing that is pumped, not pushed

Nobody pushes a swing they are sitting on. They stand up at the bottom and sit down at the ends, which changes the pendulum rather than forcing it — and does so twice per period. The equation that describes it has no forcing term at all, so standing still is always a solution, and what the pumping changes is whether standing still is stable.

The correction that took a century

Newton derived the speed of sound in 1687 and got 290 metres a second against a measured 340. The arithmetic was right; the assumption was not. Heat cannot cross a wavelength in a period, so the compressions are adiabatic — and the factor that repairs the answer turns out to be a count of the ways a molecule can move.

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.

The layer that makes a reflection vanish

A wave meeting a step in impedance reflects, and nothing can be done about the step. Put a third medium between the two, a quarter of a wavelength thick and of exactly the intermediate impedance, and the reflection stops existing — not reduced, cancelled.

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.

The spiral that says how much light arrives

Huygens' construction says where a wave has got to and refuses to say how bright it is, because an envelope is a locus and a locus has no amplitude. Adding the wavelets with their phases instead of taking their envelope turns the whole subject into one curve, and every near-field pattern there is becomes a chord of it.

The mass that makes another stand still

Bolt a small mass on a spring to a machine that is shaking itself apart, tune it to the frequency that is doing the damage, and the machine stops moving. Not moves less — stops, exactly, at that one frequency. The price is two new resonances either side of it, and the whole device is a bet that the drive stays where it was put.

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.

The gap a repeat opens

Stack two transparent materials in alternating layers and there is a band of frequencies the stack will not carry — not weakly, not with loss, but not at all. Nothing has been absorbed and neither material has a resonance there. What forbids those frequencies is the repeat itself, and the width of the band has a closed form containing only the ratio of the two indices.

The channel with no walls

A pipe will not carry a note below its cutoff, and no length of pipe helps. Replace the walls with nothing but a region where the wave travels slightly slower, and the cutoff disappears — however weak the contrast and however thin the channel, at least one mode is bound. The difference is not a matter of degree; it is the difference between a boundary condition and a potential well.

The mode that lives in the mistake

A perfect stack of alternating layers refuses a whole band of frequencies — not weakly, not with loss, but not at all. Break the repeat once, by inserting a single layer of the wrong thickness, and exactly one frequency inside that band passes through the whole stack with a transmittance of one. The useful thing about a forbidden band turns out to be not what it excludes but what a single flaw is thereby allowed to hold.

The distance that takes the treble out

Spreading treats every frequency alike; absorption does not. The loss per cycle is roughly fixed and the number of cycles per metre goes as the frequency, so absorption climbs as f² and a sound gets duller with distance as well as quieter — which is the whole account of why a nearby thunderclap cracks and a distant one rumbles.

The cone the source leaves behind

Take the Doppler construction past the speed of the wave and the wavefronts acquire an envelope. Its half-angle obeys sin θ = 1/M, an expression with no pressure, no density and no shape of the object in it — so a photograph of the cone is a speedometer. And the bang is not an event at the moment of crossing — it is a signature dragged along the ground for the whole of the flight.

The width that is a lifetime

Hit a bell and time how long it rings; drive it and measure how narrow its response is. The two numbers are the same number, and they cannot disagree — not because the physics conspires but because a decay and a linewidth are one function seen in two coordinate systems.

The taper that matches every note

A quarter-wave layer cancels a reflection at one wavelength and only near it. Spread the same change of impedance over a distance instead, and the reflection vanishes for every wavelength shorter than about twice that distance — not by cancelling one echo against another, but by leaving no step anywhere for an echo to come from.

The answer that cannot come first

A medium absorbs at some frequencies and bends light at all of them, and those look like two separate facts to be measured separately. They are one fact. The requirement that a material respond after it is asked rather than before ties the absorption at every frequency to the refraction at every other, by an integral — and a material given the two halves independently answers before the question arrives.

The arrival that keeps arriving

A clap heard across a field arrives and stops. The same clap in two dimensions arrives at the same instant and then goes on arriving for ever, fading as one over the time — and the difference is not absorption, or echo, or scattering. It is the number of dimensions, and sharp arrival happens in three of them and in no even number at all.

How many ways there are to vibrate

A drum has infinitely many modes, and below any given frequency it has a finite number of them. That number turns out to depend on the drum's area and the length of its rim and — to the accuracy anybody uses — on nothing else about its shape. Almost every result in thermal physics that involves waves is an application of that count.

The front that steepens until it cannot

In a linear medium every wave keeps its shape, because every part of it travels at the same speed. Let the speed depend on the height by even a little and the crest overtakes the trough, the front leans forward, and after a time that can be written down the wave demands two values at one place — which is where the description ends and a shock begins.

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.

The wave that comes from the rim

A shadow's edge is not a boundary between light and no light, and the fringes on either side of it are not a smudge. The whole pattern is the sum of two things — the light nothing blocked, and a single wave that behaves in every respect as though the rim of the obstacle were radiating it — and splitting it that way is exact rather than a picture.

The wave a surface is enough to hold

A pipe guides with walls and a fibre guides with a slower core. A solid needs neither: one free surface binds a wave that is not a bulk wave bouncing but a separate solution, travelling slower than any wave in the material, dying away exponentially into it, and carrying its energy round a circle instead of over a sphere — which is why it is the part of an earthquake that knocks buildings down.

The count that cannot be cheated

A string with a lump in it has no harmonics, no symmetry and no obvious order to its modes. It has one thing left: the nth mode crosses the axis exactly n−1 times, whatever the string is made of. Ordering by frequency and ordering by node count turn out to be the same operation, and in two dimensions the equality quietly becomes an inequality.

The shift a mirror gives twice

A moving reflector is a receiver and a source in one, so it shifts a wave twice — and the two shifts are different functions of the speed, which is why the round trip is not the square of either. Everything a speed radar does follows from that, including the two things it cannot do.

Swap the ends and nothing changes

Put a source at one end of the most complicated arrangement of materials anybody can build and a detector at the other, then exchange them. The reading is identical — not approximately, not on average, but to every digit the arithmetic has. Two things in physics break it, and neither of them is a shape.

What adding does to the energy

Waves add their amplitudes and detectors read squares, so two waves together do not deliver the sum of what each delivers. Where the two get dimmer, the natural question is where the energy went — and the answer depends entirely on whether the sources can feel each other.

The resonance with a zero in it

Where a resonance is the only way through a system, the response is a symmetric peak. Where there is also a smooth path that does not care about the frequency, the two add before anything is squared — and the result is lopsided, with a frequency at which nothing gets through at all.

The backward wave Huygens had to remove

Every point of a wavefront is a source of a spherical wavelet, and a spherical wavelet goes in every direction — so the construction predicts a wave travelling backwards as well as forwards. Nothing of the kind exists, and the repair is a factor that Huygens' geometry has no room for.

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.

The node that is not standing still

A wave meeting a perfect reflector makes a standing wave with real nodes. A partial reflection makes something that looks the same and is not: the minima are not zeros, energy flows steadily through them, and the depth of the pattern is a measurement of the load that caused it.

The mode that will not turn a corner

Bend a waveguide and the field has to go round with it, which means the part furthest from the centre has to travel faster. Past a certain distance it would have to travel faster than the surrounding medium allows, and everything out there radiates away — which is why bend loss is exponential in the radius and arrives all at once.

The dent that raises the note

Push a wall of a resonator inwards and the pitch goes up or down depending entirely on where the wall is pushed. A mass added at a node changes nothing at all; the same mass at an antinode changes as much as it can. One rule covers a loaded string, a tuned microwave cavity and a bead drawn through a resonator to read out its field.

Where the loudness goes

An absorption coefficient removes energy from a wave, and energy removed has to appear somewhere. It appears twice, from the same coefficient: as heat, and as momentum. So a beam of sound has a weight — a watt absorbed in water weighs sixty-eight milligrams — and acoustic power is measured by putting an absorber on a balance. The ratio of the force to the heating contains no intensity at all.

The steps in an absorption curve

Every absorption law up to this point is smooth — a fixed loss per cycle, a relaxation, a power of frequency. Take the photon energy up to where it can eject an electron from an atom and the curve acquires steps, and they go the wrong way: the material becomes suddenly more opaque to a harder photon. Iodine absorbs four times as strongly at 33.4 keV as at 33.0, and that discontinuity is why it is injected into people.

The end that knows how the middle was cut

A chain whose links alternate, strong and weak, has the same bands whichever kind of link is counted as inside a cell. The infinite chain cannot tell the two choices apart. A finite chain can: cut it so that a weak link is outermost and each end holds a state at exactly zero energy, in the middle of the gap; cut it the other way and it holds none. What decides is not anything at the ends but a whole number counted from the bulk — how many times a loop winds round a point.

The mirror that works from every direction

A stack of alternating transparent layers reflects nearly all the light of one colour arriving straight on, and less as the light tilts, because tilting moves the forbidden band. A structure that repeats in only one direction ought therefore to be a mirror for only a range of directions. It is not, if the layers differ enough: light arriving from air cannot bring enough sideways momentum to escape the forbidden band at any angle, for either polarisation, and a flat stack of plastic and tellurium reflects every angle over a band of frequencies almost half as wide as its centre.

The resonance that refuses to leak

A resonance that sits at a frequency where waves can escape has a width, because it leaks. Put two such resonances into the same channel and let them talk to each other, and at one precise detuning one of their combinations stops leaking altogether — a mode with no width, surrounded by a continuum it could escape into and does not. It cannot be seen from outside, it traps whatever energy lands in it, and if a symmetry is what forbids the leak, it survives any change that keeps the symmetry.

The mismatch no network can remove

A quarter-wave layer or a taper can match a resistance to a resistance as well as anyone likes. Put a capacitance across the load and that stops being true for every network that could ever be built from lossless parts: Bode and Fano proved that the total amount of match available is fixed by the load's resistance and capacitance, so a network can only move it about — and a flat match across a band can never be better than e to the minus π over the load's time constant times the band.

The fan of plane waves inside every beam

Huygens added up wavelets from every point of a front. The same content can be written as a sum over plane waves travelling in every direction, and then propagation stops being an integral and becomes a multiplication: each plane wave picks up a phase in proportion to the distance. One square root in that phase holds all of diffraction, near field and far field alike — and when the square root turns imaginary, it holds the reason no instrument a wavelength away can see detail finer than half a wavelength.

The one medium that was supposed to add exactly

Superposition holds because an equation is linear, and every material stops being linear at some amplitude. Empty space was the exception: Maxwell's equations are linear exactly, and two beams cross with no interaction of any kind. Quantum electrodynamics says otherwise — light scatters light, and a strong field makes the vacuum birefringent — at a field of 1.3 × 10¹⁸ volts per metre, which no laboratory has come within four decades of.

Two tails that swap everything

Bring two guides close enough for their evanescent tails to overlap and they do not leak a little power into each other. They exchange all of it, and then exchange it back, over a length fixed by the splitting between two modes that belong to neither guide — so a coupler is cut to a length rather than tuned to a ratio, and the length depends exponentially on a gap of a few hundred nanometres.

The ripple that counts the neighbours

An absorption edge is drawn as a step and it is a step with a ripple on it — a modulation of eleven per cent in copper, five in a zinc site buried in a protein. The ripple is the ejected electron's own wave, scattered back onto the atom that emitted it, so its period is a distance. It is the only way of measuring where an atom's neighbours are that does not need a crystal.

Below the gap, where there is nothing to absorb

A semiconductor is supposed to be transparent below its band gap, and it is not. The absorption falls exponentially instead, over seven decades, along a straight line whose slope is an energy of a few millielectronvolts — and the description that produces that line has no states in the gap at all. What the slope measures is how much the gap itself is moving about.

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.

The coupler that does not care about the colour

Two identical guides side by side swap their light back and forth, and a coupler cut to the length of one swap works perfectly at one wavelength and badly at every other. Make the guides unequal, and sweep the inequality from one sign to the other along their length, and the light no longer swaps — it follows a single mode of the pair as that mode moves from one guide to the other. The transfer is then nearly complete across hundreds of nanometres of wavelength and immune to the widths being made wrong, and it costs length.

The phases that turn a glow into pulses

A laser oscillates on many frequencies at once, spaced by the time light takes to go round its cavity. Add those waves with random phases and the output is a steady glow with noise in it. Add exactly the same waves with their phases equal and the output is a train of pulses, each N² times brighter than one wave and a fraction 1/N of the round trip long. Nothing about the frequencies, the amplitudes or the average power changes — only the phases, and they decide everything a detector fast enough would see.

The lattice no phase can bend

Cross a few laser beams of one colour and they paint a crystal of light in the space where they overlap. The beams' phases drift whenever a mirror trembles, and it would seem the pattern must tremble with them. Whether it does depends on a count. Three beams in a plane, or four in space, give a lattice whose shape no change of phase can alter — the phases can only slide it. One beam more and the phases decide the shape, and must be held still.

The wiggle faster than any wave in it

Light passing through a lens carries no detail finer than half a wavelength, because no travelling wave in it varies faster than that. It would seem to follow that the light itself cannot vary faster anywhere. It can. A sum of slow waves can be arranged to wiggle faster than any of them over a stretch of any length chosen — and to focus to a spot as narrow as wanted — provided the stretch sits between side lobes that grow exponentially with every extra wiggle.

The absorber whose bandwidth is its thickness

A metal plate reflects everything, and a surface that must swallow a radar wave instead has to be a match between free space and a short circuit. A resistive sheet a quarter-wave in front of the metal does it perfectly at one frequency. Widening the band takes more sheets and more depth, and a theorem says why that is not a failure of design: for any absorber backed by metal, the total absorption over all wavelengths is fixed by its thickness, and by nothing else except its permeability.

The string whose end will not keep still

Move one end of a vibrating string slowly and each mode simply retunes. Move it quickly, at twice the note the string is sounding, and something else happens: the smooth swing gathers into pulses, the energy grows by the Doppler factor of the moving end on every round trip, and it climbs through the modes without stopping — because the modes of a string are evenly spaced, and a wall that couples one pair couples them all.

The field that turns with nothing turning

A coil fed with alternating current makes a magnetic field that grows, shrinks and reverses along the coil's axis and never leaves it. Put three such coils at 120° to one another and feed them currents a third of a cycle apart, and their fields add to one of constant strength that turns steadily round the centre — a rotating magnet made of nothing that moves. The trick is superposition read backwards: a field pulsing along a line is already two fields turning opposite ways, and the three coils are arranged so that one pair cancels and the other adds. Everything that goes wrong afterwards is a field turning the wrong way.

The layer too thin to see

A seismic survey sends a pulse of sound into the ground and listens to the echoes from every boundary where the rock changes. A thick layer returns two echoes, one from its top and one from its base, and the time between them gives its thickness. A thin one returns one echo, and below about a quarter of a wavelength the timing says nothing more: every thin layer looks the same, just weaker as it thins. It has not become invisible. It has become unmeasurable in the one way that was being used, and its thickness has moved into the echo's strength and its spectrum.

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.

The clock that is pulled into step

In 1665, ill in bed, Christiaan Huygens noticed that two of his pendulum clocks hanging from the same beam were swinging in exact opposition, and that however he disturbed them they returned to it within half an hour. A clock is not a passive resonator waiting to be pushed; it keeps its own time, by its own mechanism. What a weak push can do to it is not change how strongly it swings but pull its phase — and within a range of frequencies set by the push's strength, pull it all the way into step. The same arithmetic locks lasers, pacemaker cells, power stations and the body's daily clock.

The sandbars that reflect the sea

A stack of alternating layers reflects the one colour whose half-wavelength matches the repeat. A row of sandbars on a gently sloping seabed is the same arrangement for ocean waves, and it does the same thing. Waves twice as long as the bar spacing are sent back out to sea, one small reflection per bar, all adding in step. A patch of eight bars can return most of the swell that meets it. The reflected wave, added to the incoming one, makes a standing pattern whose nodes are spaced exactly like the bars. Sand gathers in that pattern, so a field of bars can build its own next bar.

The loss that falls as the frequency rises

Every metal conductor loses more at higher frequencies, because the current crowds into a thinner skin and meets more resistance. One wave in a round metal pipe breaks the rule. The TE₀₁ mode's electric field circles the axis and touches the wall nowhere. As the frequency rises it becomes more and more nearly a plane wave running down the middle of the pipe, and it drives less and less current in the metal. Its loss falls with frequency, to about half a decibel per kilometre at 60 GHz in a pipe 60 mm across. In the 1960s it was the best way anyone knew to send a hundred thousand telephone calls across a continent. The catch was that the pipe also carried seven hundred other modes.

The pigment that absorbs less for being packed

Dissolve a dye evenly in water and it absorbs a certain fraction of the light passing through. Gather the same dye into grains — the same molecules, the same amount, in the same volume — and it absorbs less. The light has not changed and neither have the molecules. The grains shadow their own insides, and the light that would have been caught by the shaded molecules slips through the gaps between grains instead. The effect is called packaging, and it is why the chlorophyll inside large algae is worth less per molecule than the chlorophyll in small ones, and why an ocean's colour does not say directly how much chlorophyll is in it.

The lever that lets air speak to water

Sound in air striking a surface of water is almost entirely reflected: 99.9 per cent bounces off, which is why a swimmer underwater hears the world above as a muffled murmur. The inner ear is filled with fluid, and the eardrum faces air, so every land animal has the problem of getting sound across that boundary. The answer is three tiny bones and a difference in area, which together multiply the pressure about twenty-fold and turn a thousandth of the power into nearly half. The middle ear is an impedance transformer, the mechanical twin of the one in every radio, and it works best over just the band where speech carries its meaning.

The mirror that sends a wave back to where it began

Record a wave as it arrives at a row of microphones, then play each recording backwards through the loudspeaker beside it, and the sound converges on the place it came from — however twisted the route it took. Huygens said every point on a wavefront acts as a source; a time-reversal mirror takes him literally, making every point of a recorded front re-emit its own history in reverse. The result has a property no lens has. Put a thicket of scatterers between the source and the mirror, so that the arriving wave is an unrecognisable jumble, and the reversed wave still finds its way back — to a sharper focus than it could reach in empty space, because the disorder acts as a lens wider than the mirror.

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.

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.

The Doppler shift with nothing coming closer

Every Doppler shift so far has needed something to approach or recede. Light can be shifted in frequency by something that does neither: a surface spinning in place, or a polarising plate turning in a beam that passes straight through it. A beam whose wavefronts wind round its axis, ℓ times per wavelength, comes back from a surface spinning at Ω shifted by exactly ℓΩ, at every distance from the axis. A circularly polarised beam through a spinning half-wave plate comes out shifted by exactly 2Ω, because the plate takes two units of angular momentum from each photon and does work on it. The shift is tiny and perfectly measurable, and satellite navigation has to correct for it.

The note a bottle sings whatever its shape

Blow across the mouth of an empty wine bottle and it sounds a low note, about 109 hertz for a 750-millilitre bottle. Its wavelength is three metres, ten times the bottle's height, so it is not a standing wave in the bottle the way a flute's note is a standing wave in the flute. It is a mass bouncing on a spring: the plug of air in the neck, thirty milligrams of it, riding on the springiness of the air in the body. The note depends on the body's volume and the neck's size and on nothing about the body's shape, which is why a sphere, a cube and a tall cylinder of the same volume all sing the same note — and why a car driven with one window open throbs at eighteen hertz.

The wave that arrives first by going the long way

A shot fired at the ground sends a wave straight along the surface layer to a distant listener, and another down to a faster rock beneath, along the boundary and back up. The second path is longer, and beyond a certain distance it arrives first, because most of it is run at the faster speed. Huygens's construction shows what the second wave is: a front drawn by wavelets that the disturbance racing along the faster floor sends up into the slower layer, the Mach cone of a source moving faster than the layer can carry sound. Reading where it overtakes the direct wave gave Andrija Mohorovičić the thickness of the Earth's crust in 1909, and gives every refraction survey since the depth to whatever lies beneath.

The depth an ultrasound image pays for its sharpness

An ultrasound scanner sees finer detail at higher frequencies, because shorter waves make shorter pulses, and sees less deep, because tissue absorbs higher frequencies more strongly. Both effects scale as one over the frequency, so they cancel in the one number that matters: how many separate depths fit along a line of the image. For soft tissue, which absorbs nearly in proportion to frequency, that number is about four hundred whatever the frequency — set by the tissue and by the scanner's electronics and by nothing else. A scan of the abdomen at three megahertz and a scan of the skin at fifty are the same picture at different magnifications.

The photograph that keeps the spectrum

Light falling on a mirror adds to its own reflection, and the sum does not travel: it stands, with sheets of darkness every half-wavelength. Otto Wiener photographed those sheets in 1890 and found the film clear where it touched the mirror, which settled which of light's two fields acts on matter. A year later Gabriel Lippmann recorded the sheets through the depth of an emulsion and developed them into stacked layers that, lit with white light, send back exactly the colours that wrote them. His photographs store the spectrum itself, so they can tell apart two yellows that every colour film and every eye records as one.

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.

The tide a lake makes by itself

On Lake Geneva the water at Geneva rises and falls by a few centimetres every seventy-three minutes, and the water at the far end does the opposite, as though the whole lake were a bath slopping back and forth. It is a standing wave, set ringing by wind and air pressure, and its period follows from the lake's length and depth alone. A harbour rings the same way with its mouth as a node, and it does something odder: narrowing its entrance, the obvious way to keep waves out, makes its resonance stronger.

The wall that stops sound by its weight

A wall between two rooms keeps out sound not by absorbing it but by being hard to shake. The sound's pressure pushes the wall back and forth, the moving wall becomes a loudspeaker on the far side, and how much gets through depends almost entirely on the wall's mass per square metre against the frequency: six decibels more for every doubling of either. Read as impedance, a wall is a mass placed in series with the air, and the same reading explains the frequency at which a stiff pane of glass nearly vanishes and why two thin walls with a gap beat one thick one.

The pile of glass that stops every colour

A sheet of window glass reflects about eight per cent of the light that falls on it. A pile of a hundred sheets, George Stokes calculated in 1862, should still let through a tenth, because each extra sheet only adds a little more reflection to the total. That is right for sunlight, whose colours are too mixed to interfere. For a single colour, with every sheet and every gap a slightly different thickness, the same pile lets through a thousandth, and a pile of four hundred a hundred-millionth of a millionth. Interference with random phases does not average away into the intensity sum. In one dimension it traps every wave, at every wavelength, however weak each reflection is.

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.

The silence that fits in a tenth of a wavelength

Two sounds of equal strength and opposite phase add to nothing, so any noise can be cancelled by playing its inverse. At one point. A microphone and a loudspeaker can silence the air exactly where the microphone is, and nowhere else for certain: in the diffuse sound of a room the quiet zone is a few tenths of a wavelength across, under ten centimetres at a kilohertz, and beyond it the cancelling speaker only adds noise. That is why noise cancellation works in a headphone and not in a room, and why it works on the drone of an engine and not on a voice.

The noise that helps a weak signal through

A system that can sit in either of two states, nudged back and forth by a periodic push too weak to switch it, does nothing at all — until noise is added. Then it begins to switch, and with the right amount of noise it switches in step with the push, once each half-cycle, as though the push had been strong enough all along. Too little noise and it never switches; too much and it switches at random. The response peaks at a nonzero noise, which is why the effect is called a resonance, and why crayfish, paddlefish and human fingertips appear to use it.

The shadow that shows the core is liquid

Nobody has been more than a few kilometres into the Earth, and its interior was mapped from the timing of earthquake waves alone. The decisive observations were of waves that do not arrive. Seismometers between about 100° and 140° from a large earthquake record no direct compressional wave, because something deep inside bends the rays away; and over two-fifths of the planet beyond 100° no direct shear wave arrives at all, because that something cannot be sheared. A slow, liquid core casts both shadows, and a few faint waves inside the first one revealed a solid core within it.

The chain that swings in Bessel's notes

Hang a chain from a hook and set it swinging. It can swing as a whole, like a pendulum, or in shapes with still points along it, as a guitar string does — but its higher notes are not two, three and four times its lowest. They are 2.295, 3.598 and 4.903 times it, the zeros of a function that Daniel Bernoulli met for the first time in exactly this problem in 1732, a century before Bessel's name was attached to it. The reason is that a hanging chain is a string whose tension grows from nothing at its free end to its whole weight at the top, and a wave crossing it changes speed all the way.

The tube that sings when heated low down

Hold a wide metal tube upright, push a red-hot wire gauze a quarter of the way up inside it, and take the flame away. The tube begins to howl — a loud, steady note at its lowest resonance, lasting until the gauze cools. Put the gauze a quarter of the way down from the top instead and the tube is silent; lay the tube on its side and it stops. Heat is driving a sound, and whether it drives or damps depends on where the heat is released and when, in a rule Lord Rayleigh stated in 1878 that now governs rocket engines and power-station turbines.

The speeds a pulsed radar folds into one

A weather radar measures the wind inside a storm by the Doppler shift of its echoes, but it never hears the shift itself. It listens in pulses, and between pulses it notes only how far the echo's phase has turned — which is known only to within a whole turn. So speeds that differ by a fixed amount give identical echoes, and every speed is folded into a window. Make the window wider by pulsing faster, and the echoes from distant rain arrive after the next pulse has gone and land at the wrong range. The two limits multiply to a constant the designer cannot touch.

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.

The pattern two gratings make that neither contains

Lay one fine net curtain over another and broad, swirling bands appear that are in neither curtain. Two sets of lines a tenth of a millimetre apart, differing by one part in a hundred, make bands a centimetre apart. This is moiré: the beat of two gratings in space, as two notes a little out of tune beat in time. It magnifies whatever makes the gratings differ — a twist, a stretch, a shift — by the ratio of the bands to the lines, so it is used to measure strain in bridges and to align the layers of microchips. And two sheets of carbon, twisted by just over one degree, make a moiré on which electrons stop moving and the material superconducts.

The ripples that answer at half the shaking

Shake a dish of water up and down fifty times a second and nothing happens — until the shaking passes a threshold, when the whole surface erupts into a pattern of standing ripples, squares or stripes, each crest rising and falling twenty-five times a second, at exactly half the frequency of the shaking. Michael Faraday saw it in 1831. The size of the ripples is not set by the dish. It is chosen by the water itself, as the one wavelength whose natural frequency is half the drive's, and the threshold is set by how quickly the water's own viscosity would kill that wave.

The wall made heavy only at one note

A wall stops sound by being hard to shake, and how hard a plain wall is to shake is its weight: that is the mass law, and it makes quiet walls heavy. Hang small masses on springs inside a light panel, tuned to one frequency, and near that frequency the panel becomes many times harder to shake than it weighs, because the hidden masses swing against it. A two-kilogram panel then blocks a tone as a wall thirteen times heavier would. The price comes just above the tuning, where the panel's dynamic mass passes through zero and it lets sound through more easily than if the masses were not there.

The noise that warns a tube is about to sing

A resonator that is about to break into self-sustained oscillation — a combustor about to hum, a heated tube about to howl — is silent on average right up to the moment it starts. Its noise is not. As the margin of stability shrinks, the random hiss in the mode grows louder, remembers each kick for longer and narrows towards the coming tone, in proportions fixed by the margin itself. The instability can be heard before it exists, except in the case engines most often present, where noise starts the song before the warning has properly begun.

The thunder that passes over the listener's head

Lightning twenty-five kilometres away is plainly visible and usually silent, though the air between is clear and absorbs the low notes of thunder hardly at all. The thunder has not faded; it has gone over the listener's head. Daytime air cools upwards, sound slows as it climbs, and Huygens's wavelets, smaller at the top of every front than at the bottom, tilt each front towards the sky. Beyond the ray that just grazes the ground there is a shadow the sound cannot enter — and a hundred and fifty kilometres further out, the stratosphere can bend it back down.

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.

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.

The chirp a staircase sings back

A hand clap at the foot of the great stair of El Castillo at Chichén Itzá is answered by a chirp — a falling whistle that some have compared to the call of the quetzal bird. Nothing in the stone is tuned. Each of the ninety-one steps returns its own copy of the clap, the copies arrive as a train of clicks, and the ear hears a regular train of clicks as a pitch. The pitch falls because the steps are not evenly spaced as the listener sees them: low steps are seen almost level, a tread apart, and high ones almost along the slope, a whole step's diagonal apart. It is Bragg reflection from a grating whose period, seen from one spot, changes along its length.

The loudspeaker that goes round

The swirling sound of a Hammond organ comes from a loudspeaker cabinet in which the speaker's horn turns, six or seven times a second, and the swirl is the Doppler effect — the horn's mouth swings towards and away from the listener at about six metres a second, and every note it plays rises and falls by a third of a semitone. But a pitch that rises and falls periodically is not a note being shifted: it is a note split into a comb of new frequencies, spaced by the rotation rate and spread in proportion to the note itself, so the turning horn leaves the bass nearly pure and blurs the treble into a shimmer. In a room each wall returns its own copy, swinging out of step with the others, and the copies beat against each other several times a turn.

The echo hidden in random noise

A seismometer that is not recording an earthquake is not recording silence. It hears the Earth's hum: the pounding of ocean waves on every coast, arriving from all directions as surface waves with periods near ten seconds, indistinguishable from random noise. Two seismometers sixty kilometres apart hear different noise. Yet if their records are correlated and averaged over hours, a pulse appears at a lag of twenty seconds — the time a wave takes to travel from one to the other — exactly as if an earthquake had happened under one station and been recorded at the other. Noise from everywhere contains the echo between any two points, and since 2005 seismologists have mapped the Earth's crust with it.

The fork inside a watch

Every quartz watch keeps time with a tuning fork three millimetres long, cut from a crystal of quartz and ringing thirty-two thousand seven hundred and sixty-eight times a second. Each number in that sentence is a decision. The frequency is two to the fifteenth, so that fifteen halvings make one tick a second, and it sits just above the range of human hearing. The shape is a fork because two tines swinging against each other leave their base still and lose almost no energy into it. And the fork's rate drifts with temperature along a parabola whose vertex is set near twenty-five degrees, which is why a watch worn on a wrist keeps better time than one left in a cold drawer.

The oven that cooks in spots

A microwave oven is a metal box, and a metal box has only so many ways of ringing at a given frequency. At 2.45 GHz a kitchen oven allows a handful, and the field inside is the sum of those few standing patterns, with hot patches and nodes several centimetres apart. Food on a stopped plate shows them as a pattern of scorch and frost. A turntable can smear this only into rings, and frozen food has a further trap, because ice absorbs thousands of times less than water.

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.