Illustrated Physics
physics, drawn
Concept

Beats — where it appears

The slow waxing and waning heard when two close frequencies add, at a rate equal to the difference between them. The beat frequency is nowhere in the spectrum of the sum — the ear hears an amplitude modulation of a single average tone, not a third note.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

Two sources 3 wavelengths apart. Circular wavefronts from two sources, with the lines along which they arrive in step drawn through the pattern. Those lines are where the path difference is a whole number of wavelengths.

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.

waves · Superposition A packet on deep water, ω = √(gk), 1.6 s apart. A wave packet built from a Gaussian spread of wavenumbers about 1.57 per metre, drawn at two times 1.6 seconds apart, with its computed envelope ghosted around it. Between the two frames the envelope's peak moves 2.01 metres and the marked crest moves 3.95 metres, so the packet travels at 1.26 metres per second and the crests at 2.47 — a ratio of 0.51.

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.

waves · Wave packets 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 Two sources 4 wavelengths apart. Circular wavefronts from two sources, with the lines along which they arrive in step drawn through the pattern. Those lines are where the path difference is a whole number of wavelengths.

Why two lamps never interfere

Adding amplitudes is unconditional; fringes are not. What decides is whether the phase difference holds still for longer than a detector takes to record it — and a 10 nm slice of white light holds it for 100 femtoseconds, across a path of 30 micrometres.

optics · Coherence How far apart the two paths can be. Fringe visibility against the difference between the two path lengths, for light at 550 nm with a bandwidth of 100 nm. Each curve is the modulus of the Fourier transform of its own line shape, summed over the spectrum here rather than taken from a standard result, and the three shapes have the same width at half height. The conventional coherence length λ²/Δλ is 3.02 µm for this light, and what the curves show is that the convention is a rounding of three genuinely different behaviours: a flat band halves at 1.83 µm, a Gaussian line halves at 1.33 µm, a Lorentzian line halves at 0.68 µm. The flat band comes back — a rectangle's transform rings — and the Lorentzian's tails keep a little visibility very much further out than its width suggests. Nothing here is about the apparatus: the fade is the source forgetting its own phase.

How far a wave can remember

Split a beam, delay one half, and put them back together. The fringes are bright while the delay is short and fade as it grows, and the distance at which they die is fixed by nothing but the width of the source's spectral line. Watching them fade is reading the line shape.

optics · Coherence The narrow packet is the one that spreads. Three packets on deep water, ω = √(gk), all built on the same 4 m carrier and differing only in bandwidth — 10%, 18%, 28% of the carrier wavenumber. Each curve is the width of the emitted envelope, measured as the second moment of its intensity about its own centroid in a frame moving at the group velocity, divided by that width at the start. The starting widths are 4.50 m, 2.50 m, 1.61 m, and the order of the curves is the reverse of the order of the widths: the shortest packet, which is the one with the widest spectrum, is the one that comes apart first. The dashed curves are √(1 + (t/τ)²) with τ = σ₀²/|d²ω/dk²| — computed from the dispersion relation, not fitted. They agree with the measured widths to 0.3% at 10% bandwidth, 3.3% at 18% bandwidth, 8.5% at 28% bandwidth, and that ordering is the second thing the figure says: the closed form keeps only the curvature of ω(k), so it is exact for a narrow spectrum and starts to fail for a wide one, by about as much as the cubic term is worth. A packet with no bandwidth would never spread at all, and would also never begin or end.

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.

waves · Wave packets The cross term, and the fact that it averages to nothing. The intensity of two waves of amplitude 1 and 0.7 added together, against the phase difference between them, in turns. A detector reads the square of the summed amplitude, which is the sum of the two intensities plus a cross term that swings between plus and minus twice the product. At no phase difference the reading is 2.89 and at half a turn it is 0.09; the flat line is what the two would give with no interference, 1.49, and it is exactly the average of the curve over a whole turn. Interference redistributes and does not create — which answers the question of where the energy goes at a dark fringe by saying that it never left.

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.

waves · Superposition Two guides, and the two solutions they have. The transverse field of the two modes a pair of identical slab guides supports, against position across them, for guides half a micrometre wide separated by a gap of 300 nanometres at a wavelength of 1550 nanometres. A single guide has one fundamental mode; two guides side by side have two, and neither of them lives in one guide. The symmetric one is a single hump spanning both, the antisymmetric one has a node exactly between them — checked here to be exactly zero rather than nearly so — and they have slightly different propagation constants because the symmetric one has more of its field in the high-index gap region. That difference, computed from the slab's own dispersion condition, is 4.60e-2 per micrometre. Everything else about a coupler follows from it: a wave launched into one guide alone is the sum of the two supermodes in equal parts, they run at different speeds, and the interference between them moves the power from one guide to the other and back. There is no leakage in the account anywhere — only two solutions beating.

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.

waves · Guided waves

Named alongside it

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

SuperpositionInterferenceCoherencePath differenceWave packetBandwidthWavelengthBoundary conditionsDispersion relationEnvelopeFourier transformGroup velocity

All concepts