Two waves 180° out of step, and their sum
At its defaults it draws two waves 180° out of step, and their sum. Two sine waves differing in phase by 180 degrees, drawn faintly, with their sum drawn solid. The sum is computed point by point.
superposition is one function in lib/figures/waves.js —
travelling, standing, adding and shifting. Everything below came out
of it during this build, at parameters taken from the essays rather than invented for this
page. A figure here is the figure a reader meets in an essay, and if the generator changes,
this page changes with it.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
Two sine waves differing in phase by 180 degrees, drawn faintly, with their sum drawn solid. The sum is computed point by point.
A travelling wave, caught at one instant
The options are the ones A wave is a shape that travels, and nothing else does passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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 travelling wave, caught at one instant
The options are the ones A wave is a shape that travels, and nothing else does passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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 travelling wave, caught at one instant
The options are the ones A wave is a shape that travels, and nothing else does passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
A sine wave plotted against position at a fixed moment. The wavelength is the distance between repeats.
A travelling wave, caught at one instant
The options are the ones A wave is a shape that travels, and nothing else does passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
The path the tip of the field traces
The options are the ones The field that turns with nothing turning passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The tip of the total field over one cycle, for four arrangements of coils, with each coil's axis as a faint line and the field at one moment as an arrow. One coil: forward-turning part 0.50, backward-turning part 0.50; Two coils, 90° and a quarter cycle apart: forward-turning part 1.00, backward-turning part 0.00; Three balanced coils: forward-turning part 1.50, backward-turning part 0.00; Three coils, the third at 70 per cent current: forward-turning part 1.35, backward-turning part 0.15. One coil traces a line: equal forward and backward parts, a field that pulses and does not turn. Two coils at right angles a quarter cycle apart trace a circle of radius 1, three balanced coils a circle of 1.5, and a weakened third coil an ellipse, the sum of a large forward circle and a small backward one.
What checks it
physicscheck asserts something about superposition that
could fail — it draws it and measures the result against a value reached some other
way.
Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
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.
WavesThe 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.
WavesThe 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.
WavesThe 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.
WavesThe 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.
WavesThe 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.
WavesThe 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.
WavesThe 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.
ElectromagnetismThe term that made light
Ampère's law contradicts itself the moment a current stops being steady, and the contradiction is visible in one picture: two surfaces on the same loop, with a current through one of them and nothing through the other. The term that repairs it turns four equations into a wave, whose speed is two constants measured in a room with the lamps off.
WavesWhat 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.
WavesWhen 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.