Illustrated Physics
physics, drawn
Concept

Wave speed — where it appears

The rate at which a disturbance travels through a medium, fixed by the medium's stiffness and inertia and not by the wave itself. It is the square root of a restoring quantity over a density in every case — tension over mass per length on a string, bulk modulus over density in a fluid.

Named by 15 essays across 6 fields — each of them below, with the objects they name alongside it.

A travelling wave, caught at one instant. A sine wave plotted against position at a fixed moment. The wavelength is the distance between repeats. The ghosted curve is the same wave a moment later.

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.

waves · Wave motion Refraction from n = 1 into n = 1.5. A ray crossing a boundary between media of refractive index 1 and 1.5, bending by the amount Snell's law requires.

The bend at the boundary, and what it is really about

Light changes direction when it changes speed. Snell's law is the geometry of that statement, and it can be derived without knowing anything about light at all.

optics · Refraction Wavefronts from a moving source. Circles showing where each crest has reached, centred on where the source was when it emitted them. Ahead of the source the crests are closer together and the frequency heard is higher; behind, they are spread out and it is lower.

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.

waves · Doppler 50 Hz on two strings: 5.66 m and 2.83 m. The same 50 hertz note driven onto 2 strings at the same tension of 80 newtons but different thicknesses. The frequency is identical — it is the source's — while the wavelengths are 5.66 metres and 2.83 metres, because each string carries the wave at its own speed.

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.

waves · Wave motion 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 Why a straight front stays straight. A plane wavefront, with 9 points on it treated as sources and a wavelet of radius vt drawn about each. The envelope of those circles — the curve touching all of them — is a second straight line, parallel to the first and displaced by exactly vt. That is the whole of straight-line propagation: nothing else has to be assumed, and in particular nothing has to be said about rays, which are afterwards defined as the normals to these fronts. The construction is drawn with the wavelets left in, because they are the part that does the work.

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.

waves · Huygens The two normal modes of a coupled pair at kc/k = 0.1. Two equal masses, each held to a wall by a spring of stiffness k and to each other by a coupling spring of 0.1k. Above: the in-phase mode, in which both masses move the same way by the same distance, the coupling spring never changes length, and the frequency is therefore 1.0000√(k/m) — the coupling does not appear in it at all. Below: the out-of-phase mode, in which the coupling spring changes length by twice the displacement, so each mass feels k + 2kc and the frequency rises to 1.0954√(k/m), a ratio of 1.0954. Both displacement patterns are the eigenvectors of the pair's stiffness matrix, obtained from its trace and determinant and checked against those two square roots. The red arrows are the force each mass is pulled back by, computed as −Kx: 1.00kA in the first mode against 1.20kA in the second, a factor of 1.20, which is the square of the frequency ratio because ω² is a stiffness over a mass. Any motion of the pair whatsoever is a sum of these two and nothing else.

The two pendulums that will not stop swapping

Coupled oscillators joined by a weak spring appear to hand energy back and forth. Nothing is handed anywhere: the system has only a pair of motions that keep their shape, at √(k/m) and √((k+2kc)/m), and the apparent traffic is the beat between them — 51 swings from one handover to the next at a coupling of one part in fifty. Extend the same arithmetic to N masses and it produces a dispersion relation with a hard ceiling, near 7 THz in copper.

mechanics · Harmonic approximation Reflected upside down. A pulse arriving at a join where the impedance rises by a factor of 3, drawn at three moments. The amplitudes are read off the marched wave: the reflected pulse is -0.500 of the incident one and the transmitted pulse is 0.500, against (1−Z₂/Z₁)/(1+Z₂/Z₁) = -0.500 and 2/(1+Z₂/Z₁) = 0.500 from the two matching conditions. The reflection is inverted, which is the same fact as a pulse on a string flipping when it reaches a wall: a wall is a medium of infinite impedance, and the inversion is what keeps the displacement at the join equal to zero. Note that the transmitted amplitude exceeds one where the second medium is lighter, and that this is not a violation of anything: amplitude is not energy.

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.

waves · Wave motion 6 modes of a drum, and their frequency ratios. Nodal-line diagrams for 6 modes of a circular membrane, each labelled with its frequency as a multiple of the lowest mode's. A mode (m, n) has m nodal diameters and n − 1 nodal circles, and the circles are drawn at the radii where the computed radial function J_m(j(m,n)·r/R) crosses zero — not at guessed fractions of the radius. The two tints are the two directions the head is moving in at that instant, and the lines between them are the parts of it that never move. The ratios are 1.000, 1.593, 2.136, 2.295, 2.653, 2.917: each one is a quotient of two zeros of Bessel functions, computed here from the power series and checked against their published values to 4.4e-7. Not one is a whole number, which is why a drum has no harmonic series and no pitch in the sense a string has one — and why these same ratios belong to every circular membrane ever stretched, whatever it is made of and however tightly it is pulled.

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.

waves · Standing waves Simultaneity at β = 0.866. Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other.

The pole that fits and does not fit

A twenty-metre ladder is carried through a ten-metre barn at 0.866 of light speed, and both doors shut behind it. In the ladder's own frame the barn is five metres long and there is plainly no room. Both accounts are correct, and the doors' closings are 57.8 nanoseconds apart in one of them.

relativity · Length contraction Newton's answer, Laplace's, and the measurement. The speed of sound in 4 gases at 273.15 K. The short bar is Newton's √(RT/M), which assumes the compressions stay at one temperature; the long one is the same multiplied by √γ, which is what they come to if no heat crosses between a compression and the rarefaction beside it; the upright mark is the measured value. Newton's is 15.5% low for air, 22.5% low for helium, 22.5% low for argon, 12.0% low for carbon dioxide, and the corrected one is right to 0.05% for every gas here. Turning it round: γ read off each pair of bars is 1.400 for air, 1.665 for helium, 1.666 for argon, 1.290 for carbon dioxide, which is 1 + 2/f with f = 5.0, 3.0, 3.0, 6.9 ways of holding energy — three for the monatomic gases, five for the diatomic ones, and nearly seven for carbon dioxide, whose bending modes have begun to take a share at this temperature and its stretch has not. A speed measured with a stopwatch counts the ways a molecule can move.

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.

waves · Wave motion An f² law that is right in shape and out by 30× in size. Two absorption curves for air against frequency, both logarithmic, in decibels per kilometre. The lower one is the classical Stokes–Kirchhoff result computed from air's viscosity and thermal conductivity alone, and it goes as f^2.000 — exactly two, because the loss per cycle is fixed and the number of cycles per metre is proportional to the frequency. The upper one is the measured atmospheric absorption at 20 °C and 50 per cent humidity, which fits f^1.42 and is 30 times larger at 1 kHz and 211 times at 125 Hz. The excess is not a correction to viscosity: it is nitrogen and oxygen storing energy in vibration and giving it back late, at a rate the water vapour sets, and it is the mechanism that actually removes the treble from a distant sound.

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.

waves · Attenuation A source moving faster than its own waves. Circles showing where each crest has reached, centred on where the source was when it emitted them. Ahead of the source the crests are closer together and the frequency heard is higher; behind, they are spread out and it is lower.

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.

waves · Doppler The speed of a wave that carries no pressure. Second-sound speed against temperature, computed from the two-fluid equations with the normal component treated as a phonon gas — which it is below about six-tenths of a kelvin. The upper line is ordinary sound at 238 m/s, which moves the two components together. The lower curve is the other mode, and at low temperature it sits at 137.4 m/s, which is 238/√3 to three figures: a result with no adjustable constant in it, and the reason to believe the two-fluid model rather than merely to use it. What oscillates in this wave is not the density — the two components move in opposite directions and their sum stays put — but the fraction that is normal, which is a temperature. So a temperature disturbance in helium II propagates, with a speed, a reflection and a resonance, where in every ordinary liquid it diffuses and has none of those. Above a kelvin the rotons take over from the phonons and the measured curve falls to about 20 m/s; the model here is the low-temperature one and it is drawn only where it holds.

The heat that arrives as a wave

Two fluids with two velocities give two wave equations, not one. In the first the components move together and the density oscillates, which is ordinary sound. In the second they move oppositely, the density stays put, and what oscillates is the temperature — so a heat pulse in liquid helium has a speed, a front and a reflection.

fluids · Superfluidity 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

Named alongside it

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

SuperpositionBoundary conditionsDispersionDispersion relationFrequencyStanding waveWavefrontsWavelengthContinuumDegrees of freedomDiffractionDiffusion

All concepts