The two pendulums that will not stop swapping

Two pendulums of equal length hang from one beam with a light spring between them. One is pulled aside and released. Over the next half minute its swing dies away to nothing while the other, never touched, builds up to the amplitude the first one started with — and then the process reverses, and goes on reversing for as long as the pair is left alone.

The description almost everybody reaches for is that energy is travelling back and forth along the spring. It is wrong, and the correct one is stranger.

Everything below is where those two frequencies come from, what their difference does, and what becomes of the argument when two masses become twenty-four and then infinitely many.

Two masses, and the only two motions that keep their shape #

Any system sitting in a potential minimum oscillates at a frequency fixed by the curvature of the well. With one coordinate that is one number. With two the second derivative is a matrix, and the equation of motion is

$$\ddot{\mathbf{x}} = -\mathsf{M}^{-1}\mathsf{K}\,\mathbf{x},$$

an eigenvalue problem wearing the clothes of a differential equation. For two equal masses tied to walls by springs of stiffness $k$ and to each other by a coupling spring $k_c$, the stiffness matrix has $k+k_c$ on the diagonal and $-k_c$ off it. Its eigenvectors are the patterns in which every mass accelerates in proportion to its own displacement, and there are exactly two.

The first has $x_1 = x_2$: both masses go the same way at the same instant, so the distance between them never changes and the coupling spring is never stretched or compressed. A spring at its rest length exerts no force, so $k_c$ cannot enter the restoring force, and cannot enter the frequency:

$$\omega_1 = \sqrt{k/m}.$$

That is the frequency either mass would have with the other removed and the coupling cut. The second mode has $x_1 = -x_2$. The gap between the masses changes by twice the displacement of either, so the coupling contributes $2k_c x$ to the restoring force on each:

$$\omega_2 = \sqrt{(k+2k_c)/m}.$$

That factor of two is the whole asymmetry between the frequencies, and it is geometry rather than dynamics. The coupling spring’s extension is a difference of displacements, so the mode that zeroes that difference is blind to the spring and the mode that maximises it feels the spring twice. The ratio of the frequencies is $\sqrt{1+2k_c/k}$ and nothing else.

Two facts there are worth separating. The shapes are forced by symmetry: exchanging the masses leaves the system unchanged, so each mode must be preserved or reversed by the exchange, and $(+A,+A)$ and $(+A,-A)$ are the only such patterns. The frequencies are not fixed by symmetry, and carry all the information about how strongly the two are tied together. Measuring the shapes says nothing; measuring the splitting measures $k_c$.

Every motion is a mixture, and the exchange is arithmetic #

The pair has two coordinates, so two independent motions, so any motion whatever is a combination of them. That is a statement about a basis of a two-dimensional space rather than about springs, and it is why the swapping needs no mechanism.

Pull the first mass aside and hold the second still. That configuration, $(A, 0)$, is an equal mixture: half of it is $(A/2, A/2)$ and half $(A/2, -A/2)$. Release it and each half proceeds at its own frequency, forever, without change of amplitude. What the first mass does is then the sum of two steady oscillations at nearby frequencies, which is a beat.

Two steady tones in the ratio 11:10, added, give a rapid oscillation at the average and a swelling envelope at the difference — so the beat rate is set by the gap between the two frequencies and by neither of them separately. That ratio is the mode pair of a system at $k_c/k = 0.105$, and the resulting superposition is the motion of one of the masses. The exchange is not a separate phenomenon added to the modes; it is what two modes at slightly different frequencies look like.

A trigonometric identity does the rest: $\cos\omega_1 t + \cos\omega_2 t = 2\cos(\bar\omega t)\cos(\tfrac{1}{2}\Delta\omega\, t)$, with $\bar\omega$ the average and $\Delta\omega = \omega_2-\omega_1$. The fast factor is what an observer reads as the swinging, the slow one as the amplitude dying away, and the second mass gets the same components with one reversed in sign — so its envelope is the complement of the first’s.

The beat period is $2\pi/\Delta\omega$, and for weak coupling $\Delta\omega \approx (k_c/k)\sqrt{k/m}$, so the interval between handovers is inversely proportional to the coupling. A coupling ten times weaker makes the swap roughly ten times slower and changes nothing about how either pendulum swings. At $k_c/k=0.02$ the mode frequencies differ by 1.98 per cent — which no observer would catch by watching either pendulum — and the consequence is a complete reversal every fifty-one swings.

That traffic is worth setting beside the exchange inside a single oscillator, which runs at a quite different rate and is constantly confused with it.

There are two exchanges going on at once and they run at completely different rates. Kinetic and potential energy trade within one oscillator twice per swing, at $2\bar\omega$; the two oscillators trade with each other once per beat, at $\omega_2 - \omega_1$. At $k_c/k = 0.02$ those differ about a hundredfold, which is why the slow one looks like transport and the fast one looks like the oscillation itself.

N masses, a band, and a ceiling #

The argument above used nothing about the number two. A system with $N$ coordinates has an $N\times N$ stiffness matrix, $N$ eigenvectors and $N$ frequencies, and every motion of it is a sum of those $N$. What that list looks like when the masses are arranged in a line is where a wave comes from.

The closed form for the chain is

$$\omega_n = 2\sqrt{k/m}\,\sin\!\left(\frac{n\pi}{2(N+1)}\right),$$

and it is a dispersion relation: mode number is a wavenumber in disguise, since mode $n$ of a chain of length $L$ fits $n$ half-waves into $L$. Read that way it has two regimes, both visible in the figure.

At small $n$ the sine is its own argument and $\omega$ is proportional to $n$. Proportionality is the condition for a medium to carry a wave without spoiling it: every wavelength travels at the same speed, so a shape made of many wavelengths holds together as it moves. At large $n$ the sine flattens and the medium turns dispersive — wavelengths travel at different speeds and a packet spreads and outruns its own crests. The crossover is gradual: the curve has fallen a tenth below the line by mode 4.5 of eight and mode 12.5 of twenty-four, halfway up the list either way.

Then the surprise, which is the flat top read literally. The chain has a highest frequency it can carry at all. No motion of it, however violently or cleverly driven, oscillates faster than $2\sqrt{k/m}$, because the sine cannot exceed one. That is a property of neither the masses nor the springs, which appear only in the prefactor setting the scale. It is a property of there being a spacing: the shortest wave the chain can hold has neighbouring masses moving oppositely, half a wavelength per spacing, and there is no way to ask for less when nothing lies between one mass and the next to be displaced.

A continuum has no such limit, and matter is not a continuum, so the ceiling is real and measurable. Written as $\omega_{\max}=2v/a$, with $v$ the long-wave speed and $a$ the spacing, it can be checked against materials. Copper’s longitudinal sound speed of 4.76 km/s over a nearest-neighbour spacing of 2.56 Å gives $3.7\times10^{13}$ rad/s, or 5.9 THz, against a phonon spectrum measured to end near 7 THz. Diamond’s 18 km/s over a bond length of 1.55 Å gives 37 THz, against the 39.9 THz of its Raman line at 1,332 reciprocal centimetres. A chain of beads locates to within ten per cent the frequency at which a crystal’s vibrational spectrum stops.

The continuum limit, where the wave equation comes from #

The chain does not only resemble a string. It becomes one, exactly, and the limit takes three lines.

Hold the length $L=(N+1)a$ fixed and let the spacing $a$ go to zero. A string of tension $T$ and mass per unit length $\mu$, chopped into segments of length $a$, gives each bead a mass $m=\mu a$ and each segment a stiffness $k=T/a$. Then $\sqrt{k/m}=\sqrt{T/\mu}\,/a$, which diverges — as it must, since the cutoff runs away — while the sine’s argument $n\pi a/2L$ goes to zero. The product is finite:

$$\omega_n \to \frac{n\pi}{L}\sqrt{\frac{T}{\mu}} = \frac{n\pi v}{L}.$$

Those are the modes of a clamped string, evenly spaced with no ceiling, and the equation whose solutions they are is the wave equation. So the equation that lets a shape travel is not an independent law that strings happen to obey: it is what the coupled-oscillator problem becomes when the oscillators are many and the wavelengths far exceed the spacing. That is why a rope carries waves, a rope being a chain of atoms.

In the continuum limit the chain becomes a string, whose first four modes sit at frequencies exactly 1 : 2 : 3 : 4. The twenty-four-bead chain gives 1 : 1.996 : 2.984 — the second mode low by two parts in a thousand, the third by half a per cent — so the harmonic series is what the discreteness is converging to. The discreteness of the list of modes survives the limit, because it comes from the boundary conditions; the discreteness of the medium does not, and takes the high-frequency cutoff with it.

The continuum approximation is therefore excellent at the bottom of the band and worthless at the top. That asymmetry is what makes continuum mechanics usable: audible sound in a solid sits at $10^4$ Hz against a cutoff of $10^{13}$, nine decades down the straight part of the curve.

What it costs #

A spectroscopist is handed $3N-6$ numbers and must decide what each one is. A molecule of $N$ atoms has $3N$ coordinates, six of them rigid translation and rotation, leaving $3N-6$ vibrational modes — three for water, thirty for benzene. Infrared and Raman spectra give their frequencies and nothing else, so assigning a peak to a motion is a calculation rather than an observation.

A structural engineer designs against a list of modes, not against a force. A building of $N$ storeys has $N$ lateral modes, the lowest with a period near $N/10$ seconds, and an earthquake or a crowd excites whichever it happens to match. The London Millennium Bridge closed two days after opening in June 2000 because a lateral mode near 0.5 Hz sat at half the pedestrian footfall rate, and reopened after some ninety dampers had been fitted. A tuned mass damper — the 660-tonne sphere in Taipei 101 — is a deliberately built second oscillator, coupled and slightly mistuned, doing on purpose what the second pendulum in the hero figure does.

The cutoff sets a clock speed for anybody simulating atoms. A molecular-dynamics integrator must resolve the fastest mode present, and a carbon–hydrogen stretch near 90 THz has a period of eleven femtoseconds, so the timestep is about one femtosecond. A microsecond of simulated time is $10^9$ steps, and the cost of every biomolecular simulation ever run is set by a ceiling that comes from the spacing between two atoms.

A solid’s heat capacity is a mode count with the cutoff included. Counting standing waves in a continuum gives infinitely many modes and, with half a $k_\mathrm{B}T$ apiece, an infinite heat capacity. A chain has exactly $N$ modes per direction. Debye’s 1912 theory imposes the ceiling by hand to keep the count at $3N$; the chain produces it.

Where the model stops #

Small oscillations, and “small” is smaller than expected. The mode decomposition is exact for a linear system, and a real pendulum is not one.

Divide three wells by their own curvature and compare them against one parabola: a spring never leaves it, a pendulum leaves it by one per cent at 0.35 radians — twenty degrees — and an asymmetric chemical bond leaves it at 0.01. Above those amplitudes the stiffness matrix depends on where the system is, so the modes stop being fixed patterns and begin to feed one another. The clean exchange described here is a small-amplitude phenomenon, and it is small-amplitude by a stated number.

The trouble arrives sooner than that one per cent suggests, because what matters is not whether each pendulum is harmonic but whether the two are identical. A pendulum’s period grows with its amplitude by $\theta_0^2/16$, so one swinging at 45° is 3.9 per cent slower than one at rest — and during a swap the two are at very different amplitudes almost all the time. The pair detunes itself as it works.

Identical oscillators, to a tolerance that can be written down. For two oscillators whose natural frequencies differ by a fraction $\varepsilon$, the largest share of the energy that ever reaches the second is

$$F = \frac{1}{1+\left(\varepsilon k / k_c\right)^2},$$

the expression that governs mixing in any two-state system. Transfer is complete only while the mismatch is small compared with the coupling. At $k_c/k = 0.02$, pendulums differing in length by one per cent differ in frequency by half a per cent, and 94 per cent of the energy transfers, leaving 6 per cent that never crosses; a four per cent difference in length halves it. The same formula applied to the amplitude effect above, with $\varepsilon = 0.037$, gives roughly 23 per cent — so the demonstration barely works at 45° and works beautifully at 5°. Weak coupling, which makes the beat slow and dramatic, is what makes it fragile.

No damping anywhere. With dissipation the beat decays, and the two modes decay at different rates, because they strain the coupling differently. For a full handover to be visible the beat period must be shorter than the decay time, which needs a ringing time of order $\pi k / k_c$ oscillations — about 160 swings at $k_c/k=0.02$ and 31,000 at $k_c/k=10^{-4}$. That ratio is what a resonance’s sharpness measures, and belongs to that neighbouring subject.

The coupling is a spring here, and often it is not. Two clocks on a shared beam couple through the beam’s recoil, an inertia rather than a stiffness, and the sign structure inverts: the out-of-phase motion leaves the support still and loses nothing to it, while the in-phase motion shakes the beam and is the one shifted. Which mode is the quiet one is a fact about the mechanism, not about coupling in general.

The same decomposition everywhere else #

Water is the tidiest example, and it is the hero figure with an oxygen atom for the coupling spring. Its two O–H bonds are two oscillators sharing an atom, and the molecule’s stretching modes are their in-phase and out-of-phase combinations, at 3,657 and 3,756 reciprocal centimetres. Their ratio is 1.0271; setting that equal to $\sqrt{1+2k_c/k}$ gives $k_c/k = 0.027$, almost exactly what the exchange figure is drawn at. The two stretches would hand their vibration back and forth every thirty-odd oscillations if nothing interrupted them.

The same 2×2 problem appears wherever two nearly degenerate things are weakly connected: coupled optical waveguides pass light to each other over a coupling length, which is how a directional coupler works, and two tuned circuits sharing a capacitance give the twin-peaked response every band-pass filter is designed around.

The best generalisation is upward rather than sideways, and it is the electronic twin of this essay. Two atomic wells brought close together split one energy level into two, one symmetric and one antisymmetric — the in-phase and out-of-phase modes of an electron rather than of a mass. Bring $N$ wells together and $N$ levels appear, spread into a band with edges, for the identical reason that $N$ masses give $N$ frequencies between zero and a cutoff. That solids conduct or insulate is the same arithmetic as that beads on a thread have a fastest possible vibration.

History where it earns its place #

Christiaan Huygens, ill in bed in February 1665, noticed that two of his pendulum clocks hanging from one wooden beam had fallen into opposite swings and stayed there, and reported it to the Royal Society as “an odd kind of sympathy”. It is the first recorded observation of coupled oscillators, and not quite the phenomenon described here: an escapement-driven clock is a self-sustaining nonlinear oscillator, and such oscillators lock rather than beat. When the experiment was reproduced in 2002 the coupling proved to run through the beam’s tiny recoil, the antiphase mode surviving because it leaves the beam at rest.

The linear theory came from the loaded string. Johann Bernoulli treated a line of $n$ equal beads on a stretched thread in 1727 and found its modes sinusoidal; Lagrange, in a memoir on the propagation of sound published in 1759, took that chain to the limit of infinitely many beads and obtained the continuous string’s modes — the derivation of the section above, two hundred and sixty years earlier and in the direction nobody teaches it. His Mécanique analytique of 1788 generalised it, and Rayleigh’s Theory of Sound of 1877 gave the normal-mode theorem in the form still used.

The cutoff waited until 1912, when Born and von Kármán built lattice dynamics on precisely this chain and Debye published his heat-capacity theory with the ceiling put in as an assumption. That the two arrived in one year is a fair summary: the ceiling was needed, and only one derivation explained it. Direct measurement came in the 1950s with neutron spectrometry, work that took Bertram Brockhouse a share of the 1994 Nobel prize.

What the picture cannot show #

The exchange figures plot energy, which is a slow quantity. Nothing in them shows the pendulums swinging: the carrier at $\bar\omega$ has been deliberately averaged away, and each smooth hump conceals fifty-one fast oscillations at the weaker coupling. A reader taking those curves for a picture of the motion sees a flow where there is a beat — the error this essay exists to correct, reintroduced by its own illustration.

The chain figure plots frequency against mode number, a list rather than a motion, so it cannot show anything travelling. Whether a disturbance holds its shape depends on the phases of the modes as well as their frequencies, and phases are not on the axes. The mode-shape panels err in the other direction: a smooth sine drawn through the beads invites reading the chain as continuous, when the point is that the sine exists only at the beads.

The chain is also one-dimensional with one atom per cell. A real crystal has three polarisations at every wavenumber, two transverse and one longitudinal with different speeds, and a cell holding two different atoms splits the band into acoustic and optical branches with a forbidden gap between them. Nothing drawn here has a gap. And every amplitude is a nominal $A$: how energy is actually distributed over the modes — thermally, by a pluck, or by whatever crosses the boundary — is what all of applied vibration turns on, and appears nowhere here.

The ladder from here #

Later rungs on this anchor: the general $N$-coordinate problem with unequal masses, where $\mathsf{M}^{-1}\mathsf{K}$ is no longer symmetric; damped normal modes, whose eigenvalues go complex and stop being displacement patterns; forced response mode by mode, where modal analysis earns its keep; two atoms per cell, the optical branch and the gap between branches; and the quantised chain, where each mode’s energy comes in units of $\hbar\omega$ and those units acquire the name phonon.

The neighbouring ladders are the harmonic approximation, which supplies the matrix diagonalised here; the wave equation, which is what the chain becomes; standing waves on a string and on a drumhead, where the frequencies stop being whole-number multiples for a reason of shape rather than spacing; and electronic bands, this argument with an electron in place of a mass.