Illustrated Physics
physics, drawn
Concept

Huygens principle — where it appears

The construction in which every point of a wavefront is a source of secondary wavelets, and the front a moment later is built from them. As Huygens stated it the wavelets are a locus and carry no amplitudes, so the construction gives directions correctly and says nothing about how bright anything is.

Named by 10 essays across one field — each of them below, with the objects they name alongside it.

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 Cornu spiral, and the chords that are amplitudes. Fresnel's two integrals plotted against each other, traced from v = -4.2 to 4.2. Arc length along the curve is v — the distance along the wavefront in units of √(λL/2) — and the amplitude arriving from any stretch of that front is the straight chord between the stretch's two endpoints, not the length of curve between them. The two eyes at ±(½, ½) are where the far parts of an unobstructed front pile up, so the whole open front is the chord between them, of length 1.4142. A straight edge blocks half of it and leaves a chord of 0.7071 — exactly half the amplitude, so a quarter of the intensity, at the geometrical shadow's edge. The chord drawn here runs to v = 1.217, is 1.6556 long, and is the longest chord from the eye that exists: its square over the open chord's square is 1.370, which is why the brightest fringe outside a shadow is brighter than no obstacle at all.

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.

waves · Huygens 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 One sharp kick, heard 6 pulse-lengths away. The signal arriving at a fixed distance from a point source that emits a single short pulse, in one, two and three dimensions, each normalised to its own peak. In three dimensions the arriving signal is the emitted pulse, unchanged in shape: it arrives, and then there is silence. In two dimensions the same kick arrives at the same moment and then keeps arriving — a tail falling as one over the time, which is still at a tenth of its peak 12.1 pulse-lengths after the front has passed. In one dimension it never comes back down at all; the medium is left displaced. The wave equation is the same equation in all three, and the source is the same source; what differs is only how many dimensions the disturbance has to spread into. Sharp arrival is the exception rather than the rule — it happens in three dimensions, and in five, and in seven, and in no even number of them — and every argument that treats a wavefront as the whole of the signal is an argument that has quietly used the fact that we live in three.

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.

waves · Wave motion A step, a wave from the rim, and what they make together. The pattern behind a straight edge, split into the two things Young said it was: the incident wave where the edge does not block it, which is a step, and a wave that appears to come from the rim itself. The step is discontinuous at the shadow boundary by the whole incident amplitude. The edge wave's magnitude is continuous there, to 0.0e+0 — it changes phase by π instead of changing size — and it is exactly half the incident amplitude on both sides, which is why the total intensity on the geometrical shadow boundary is 0.250000 of the unobstructed value rather than a half. Everywhere else the two add, and their sum reproduces the directly computed field to 0.0e+0. The fringes in the lit region are the interference of the two, which is why they are fringes at all: a monotonic decay has nothing to beat against. In the shadow there is only the edge wave, so there is nothing to interfere with and the intensity falls smoothly.

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.

waves · Huygens Three ways for a wavelet to be strong, and what each leaves behind. On the left, the strength of a secondary wavelet against the angle from the forward direction, for three candidate rules. Huygens' construction as stated has no such rule: a wavelet is spherical and equally strong in every direction. On the right, what each predicts when the wavelets over a whole plane are added up, on the axis, in front of the plane and behind it. All three reproduce the incident wave in front, which is the part of the construction that has always worked. Only the rule that falls to exactly nothing at a hundred and eighty degrees leaves nothing behind, and that rule is not a repair invented for the purpose — it comes out of solving the wave equation.

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.

waves · Huygens One slit, four distances, one multiplication. The intensity across the beam behind a slit 5 wavelengths wide, at distances of 0.5, 5, 25, 100 wavelengths, each computed by multiplying the slit's plane-wave spectrum by the phase each wave accumulates and transforming back — no approximation about angles. Close to the slit the pattern is the slit's own shape with ripples at its edges; further out the ripples move inwards and the beam develops a bright centre; far away it spreads into the diffraction pattern. The travelling part of the field keeps its power to 10⁻¹⁰, running it back 100 wavelengths recovers it to 5 × 10⁻¹⁴, and at 100 wavelengths the result matches a direct Fresnel integral to 2.75 per cent rms. Near field and far field are not two theories; they are one multiplication at different distances.

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.

waves · Huygens The focus that gets sharper when the wave is scattered. The refocused amplitude against distance across the source, in wavelengths, for the same one-sided mirror and pulse, with the space between empty (dashed) and with a slab of 200 lossless point scatterers, 15 wavelengths thick, filling the space between source and mirror (solid). Through the slab the wave reaching the mirror is a jumble — scattered many times, arriving from every direction the slab can send it — yet time-reversed and sent back through the same slab, it refocuses on the source, in a spot 1.5 wavelengths wide against 4.8 in empty space. The scatterers, bouncing waves in from a wide range of angles, act as a lens much wider than the mirror: disorder that would ruin an ordinary lens makes a time-reversal mirror sharper.

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.

waves · Huygens Three arrivals, and the one that overtakes. Arrival time against distance from the shot for a 35 km crust at 6 km/s over a mantle at 8 km/s: the direct wave, a line through the origin with slope 1/v₁; the reflection from the base of the crust, a hyperbola; and the head wave, a straight line of slope 1/v₂ that begins at the critical distance, 79 km, where it touches the reflection. Close to the shot the direct wave arrives first. The head wave starts later but travels most of its path at the faster speed, and beyond the crossover distance, 185 km, it arrives first and stays first. At 300 km it beats the direct wave by 4.8 s, though it has travelled 26 km further.

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.

waves · Huygens The shadow a thunderclap's sound cannot enter. Sound rays from a point 3 km up a lightning channel, on an afternoon with 25 °C at the ground and the air cooling 6.5 K per kilometre upwards, so that sound travels slower the higher it goes; the vertical scale is stretched 3.5 times. Every ray curves upwards. Rays aimed below the horizontal reach the ground, reflect and climb away; the last ray to touch the ground grazes it 23.1 km from the channel. Beyond that, shaded, no ray arrives at all: a listener's ears 2 m up are inside the shadow from 23.7 km. The flash is visible there and the thunder is not, though the air between is perfectly clear and absorbs the low notes of thunder hardly at all.

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.

waves · Huygens

Named alongside it

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

DiffractionWavefrontObliquity factorSuperpositionInterferencePhaseEnvelopeFresnel zonesRefractionWave equationWave speedAcoustic shadow

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