The two normal modes of a coupled pair at kc/k = 0.1

At its defaults it draws 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.

normal-modes is one function in lib/figures/mechanics.js — motion, force, energy and rotation. 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.

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.

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 fraction of the energy that has crossed from one oscillator to a second one coupled to it, against time, for four amounts of frequency mismatch between them — in units where the coupling is one. With the two exactly in tune, all of the energy crosses and comes back. With them detuned, only part of it ever crosses, and the fraction is 1/(1 + (Δ/2V)²): 100.0 per cent at a detuning of 0, 80.0 per cent at a detuning of 1, 30.8 per cent at a detuning of 3, 5.9 per cent at a detuning of 8 — measured by integrating the two coupled equations and checked against the closed form to three parts in a thousand. This is the whole mechanism the previous figure implies. A mismatch does not prevent two modes from exchanging energy; it caps how much of it ever crosses and makes the exchange periodic, so nothing accumulates. The exchange also gets faster as the detuning grows, which is the counterintuitive half: the beat frequency is the root of the squared detuning plus four times the squared coupling, so a badly matched pair swaps a little energy quickly and a well matched pair swaps all of it slowly.

How spread the energy is across the modes after four hundred periods of the longest mode, against the least three-wave mismatch that chain has, for six chains from sixteen masses to ninety-six — logarithmic in the mismatch. Two things are held fixed and the work is in holding them: the energy per mass, chosen for each chain by bisection so that a longer chain is not simply given more energy, and the run length in units of each chain's own longest period, since a longer chain is slower. What is left varying is the mismatch, which falls as the cube of the length. The spread rises from 0.651 at 16 masses to 0.900 at 96, and the line through the points slopes the way the mechanism requires. This is a prediction the empirical curve could not have made: it says nothing about chain length, and the observation that a long chain thermalises more readily at the same energy density follows from the resonance condition and from nothing else.

How spread the energy is at the end of a run of 160 periods, against the energy per mass the chain was given, on a logarithmic scale. At a density of 2.3e-3 it reaches 0.33; At a density of 9.3e-3 it reaches 0.40; At a density of 2.1e-2 it reaches 0.45; At a density of 4.7e-2 it reaches 0.54; At a density of 9.9e-2 it reaches 0.66; At a density of 1.9e-1 it reaches 0.75; At a density of 3.4e-1 it reaches 0.87. The curve is flat and low at small densities and climbs steeply over about a decade. What that means is that the failure of equipartition is not permanent and not universal: it is a statement about a timescale. Below the threshold the chain would eventually share its energy out, on a time longer than the run — and the time grows so fast as the density falls that it passes any patience. Above it, the sharing happens while anyone is watching. The original calculation ran for a few thousand periods at a density well below this curve's knee, which is why it found a recurrence rather than a thermal chain. Running the same problem for long enough, or hitting it harder, gives the answer equipartition predicts.

How spread the energy is across the chain's modes — the entropy of its distribution, one being equipartition — against time multiplied by a power of the amplitude, for four runs of a thirty-two mass chain differing only in how hard it was struck. The exponent is not assumed: it is scanned for, and the value that lines the four curves up is 1.75, which reduces the spread between them by a factor of 3 against plotting them against time alone. That a single power works at all is the result — it says the four runs are the same process at different speeds rather than four different ones. What the figure cannot reach is the regime the original calculation was in. These amplitudes are above the knee where the sharing becomes fast enough to watch; below it the rate is predicted to fall as a much higher power of the nonlinearity, and no run of a length anybody will wait for gets near it. The exponent here belongs to this range and is not the exponent of the weakly nonlinear theory.

physicscheck asserts something about normal-modes 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.