Illustrated Physics
physics, drawn
Concept

Normal mode — where it appears

A pattern of motion in which every part of a system oscillates at one frequency and in fixed proportion. The whole system then behaves like a single oscillator, any motion at all is a sum of such patterns, and they are exactly independent only while the system is linear.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Where a small weight is felt, and where it is not. The fractional change in the frequency of harmonic 3 of a stretched string when a point mass of 0.06 of the string's own mass is placed at each position along it. The solid curve is the exact answer, found by solving the string's frequency equation for the loaded string at each position; the dashed curve is the first-order prediction, minus the mass fraction times the square of the mode shape. The two agree to 10.4 per cent at the antinode, where the shift is largest. Every node is a place where the exact shift is zero to better than a part in a thousand million, and that is not an approximation: a mass at a node is never moved by the mode, so it takes no part in the motion and cannot change its rate. Between the nodes the shift follows the square of the displacement, which is the square of the amplitude — the mass is felt in proportion to the kinetic energy the mode was already keeping there.

The dent that raises the note

Push a wall of a resonator inwards and the pitch goes up or down depending entirely on where the wall is pushed. A mass added at a node changes nothing at all; the same mass at an antinode changes as much as it can. One rule covers a loaded string, a tuned microwave cavity and a bead drawn through a resonator to read out its field.

waves · Standing waves The energy that goes and comes back. A chain of 32 masses with springs a few per cent nonlinear, started with all its energy in its longest mode, with the energy of the first five modes followed against time in units of that mode's own period. The first mode gives up most of what it has — down to 9 per cent by 104 periods — and the energy appears in the second, third and fourth. Then it comes back: at 154 periods the first mode holds 98 per cent of the total again. Equipartition would put an equal share in every one of the thirty-two modes and leave it there. What happens instead is that a handful of modes trade with each other and return almost exactly to where they began, and go on doing so. The total energy is checked against its starting value throughout and holds to 9.2e-5, so nothing here is the integrator losing track of what it was given.

The energy that refuses to be shared

Put all the energy of a chain of masses into its longest mode and add a few per cent of nonlinearity, and equipartition says it should spread out among all thirty-two modes and stay there. It does not. It leaks into three or four neighbours and then comes back — almost exactly — and goes on doing so, and the calculation that found this was expected to be a demonstration that it would not happen.

thermodynamics · Equipartition The condition three modes never quite satisfy. By how much three modes of a thirty-two mass chain fail to be in resonance — the sum of two mode frequencies minus the frequency of their sum, on a logarithmic scale, against the second of the two modes for four choices of the first. The leading nonlinear term couples modes in threes and the exchange accumulates only where this quantity is zero. It never is: the chain's dispersion is a sine, a sine is concave, and the sum of two of its values always exceeds the value at their sum. The smallest mismatch anywhere on the chain is at the two lowest modes and equals 2.156e-4 — which the scan finds and which is the cube of pi over four times the cube of one more than the mode count, checked here on chains from eight masses to two hundred and fifty-six. That closed form is the whole of why this is a finite-chain problem: the mismatch falls as the cube of the length, so a long enough chain is arbitrarily close to resonant and the continuum limit is exactly resonant, which is where the solitary waves come from.

The condition three modes never meet

Whether two modes of a chain can hand energy to a third is arithmetic on the dispersion relation, and for a chain of masses the answer is never: a sine is concave, so the sum of two frequencies always exceeds the frequency of their sum. The smallest shortfall anywhere on a chain of N masses is π³/4(N+1)³ — never zero, and never far from it — and a shortfall turns a transfer into a beat.

thermodynamics · Equipartition A smooth swing that sharpens into pulses. The shape of a string whose far end moves back and forth by 1% of its length at 2 times its fundamental frequency, started swinging in its fundamental, after 0, 10, 20, 30, 40 round trips of a wave. The smooth arch gathers into two ever-narrower pulses running in opposite directions, drawn at the moments they cross in the middle, with their height unchanged and their sides ever steeper; each reaches the moving end just as it moves inwards fastest. The energy has become 1.00, 1.09, 3.15, 10.9, 38.3 times the starting value. Every shape is computed exactly from the moving end, with no approximation to the wall's motion.

The string whose end will not keep still

Move one end of a vibrating string slowly and each mode simply retunes. Move it quickly, at twice the note the string is sounding, and something else happens: the smooth swing gathers into pulses, the energy grows by the Doppler factor of the moving end on every round trip, and it climbs through the modes without stopping — because the modes of a string are evenly spaced, and a wall that couples one pair couples them all.

waves · Standing waves The standing shapes of a lake and of a harbour. The water surface at its greatest tilt in the slowest standing waves of a basin of length L, exaggerated vertically. In a closed lake the water must be still at both ends' walls, which are antinodes of the surface: the first mode tilts the whole lake about a node in the middle, the second has two nodes. In a harbour open to the sea at one end, the sea holds the mouth's level fixed, so the mouth is a node: the first mode has a quarter of a wave in the harbour, the second three quarters. Their periods are 2L/(n√(gh)) for the lake, Merian's formula, and 4L/((2n + 1)√(gh)) for the harbour — a harbour of a given length rings at twice the period of a lake of the same length and depth.

The tide a lake makes by itself

On Lake Geneva the water at Geneva rises and falls by a few centimetres every seventy-three minutes, and the water at the far end does the opposite, as though the whole lake were a bath slopping back and forth. It is a standing wave, set ringing by wind and air pressure, and its period follows from the lake's length and depth alone. A harbour rings the same way with its mouth as a node, and it does something odder: narrowing its entrance, the obvious way to keep waves out, makes its resonance stronger.

waves · Standing waves A bounce that turns into a swing and back. The share of a swinging spring's oscillation energy that is in its sideways swing rather than its up-and-down bounce, against time in units of the swing's period over 2π, for a spring started bouncing with a stretch of 0.08 of its length and almost no tilt (0.02 rad). When the bounce frequency is exactly twice the swing frequency (red), the energy drains out of the bounce into the swing until 100 per cent of it is swinging, then drains back, over and over. At a ratio of 2.4 (blue) the swing never takes more than 1.5 per cent. The bounce pumps the swing parametrically, twice per swing, which works only when the two keep step.

The spring that swings when it was only bounced

Hang a weight on a spring and pull it straight down: it bounces straight up and down. Unless its bounce happens to be twice as fast as its swing would be as a pendulum. Then the bounce dies away and the weight starts to swing, wider and wider, until the bounce has gone completely — and then the swing dies and the bounce comes back. Two motions that look independent are trading their energy through a resonance, regularly and for ever at small amplitude. Give the spring more energy and the trading becomes chaotic: the same simple rule, with nothing random in it, produces motion that never repeats and that a hundred-millionth of a radian can redirect.

mechanics · Chaos

Named alongside it

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

ResonanceStanding waveBoundary conditionCavityEquipartitionErgodicityIntegrabilityNonlinearityNumerical experimentParametric resonanceRelaxationSpectral entropy

All concepts