Illustrated Physics
physics, drawn
Concept

Greens function — where it appears

The response of a system to a point disturbance, from which the response to any disturbance follows by superposition. Its symmetry under exchanging the source and the observation point is the reciprocity theorem, and it is the object every wave calculation eventually reduces to.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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 Nine slabs, no symmetry, and one transmission. A stack of 9 slabs of random wavenumber and random thickness, with no symmetry anywhere in it. Send a wave in from the left and the amplitude that emerges on the right is 0.6694240937; turn the stack round and send it in from the right and the amplitude is 0.6694240937. The two agree to every digit the arithmetic has. This is not a property of the stack — it survives any arrangement, any number of layers, any amount of internal reflection — but of the equation, which is unchanged when the sign of time is reversed, and the phase is protected as well as the modulus, which is the stronger statement and the one an interferometer would notice. The reflections are a different matter. Their moduli are equal too, at 0.742880, but only because nothing here absorbs; their phases differ by 1.621 radians, because the wave meets a different first surface from each side. What the theorem protects is the pair of ends, and not what the wave does on its way between them.

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.

waves · Wave motion A pulse that comes out of the average. The cross-correlation of the two stations' records against lag, in seconds: for single sources at 0°, 45° and 90° from the line through the stations, each a copy of the noise's own ten-second wavelet shifted to that source's arrival difference — 20.0 s, 14.1 s and 0 s — and the average over all 720 sources round the ring, scaled up 9-fold to share the axes. The single-source correlations are wherever their geometry puts them. The average has peaks at ±20 s, the time a wave takes to cross 60 km at 3 km/s, one for travel from A to B and one from B to A, and little between them. That pulse is the signal a seismometer at B would have recorded had an earthquake happened at A: the Green's function between the stations, recovered from noise that never came from either.

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.

waves · Guided waves

Named alongside it

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

Time reversalAntennaCross correlationDetectorDiffuse fieldDimensionalityDispersionHuygens principleImpulse responseIsolatorLinearityNonreciprocal medium

All concepts