The chain that swings in Bessel's notes

A heavy chain hanging from a hook, pushed at its lower end, swings back and forth almost like a pendulum. Pushed more cleverly — shaken at its top at the right rate — it can be made to swing in an S, its middle going one way while its bottom goes the other, with a still point a little way above the bottom. Shaken faster still, it swings in three and then four parts. Every stretched string does this, and only some notes fit found the rule for a string: its modes have frequencies one, two, three times the lowest, because whole numbers of half-wavelengths must fit between its fixed ends.

A hanging chain breaks the rule. Its second mode is not twice its first but 2.295 times it; its third, 3.598 times. The numbers are the zeros of a function, $J_0$, which Daniel Bernoulli derived for exactly this problem in 1732 and which Friedrich Bessel, studying planetary orbits ninety years later, met again and gave his name to. The hanging chain is the first problem in physics whose answer was a Bessel function, and it is the simplest place to see why a function that is not a sine can describe a standing wave.

A string with tension that grows #

A string’s harmonics depend on its tension being the same everywhere, so that waves travel at one speed along it. A hanging chain’s tension is not. At any height, the chain below that point has to be held up, and the tension there is the weight of everything below: zero at the free end, the chain’s whole weight at the top. With height $y$ measured up from the free end and the chain’s mass per length $\rho$, the tension is $\rho g y$.

A sideways wave on a string travels at the square root of tension over mass per length, so on the chain it travels at

$$c(y) = \sqrt{gy},$$

fast near the top and slow near the bottom, and zero at the free end itself. The medium decides the speed found that a wave’s speed is set by the medium it is in; here the medium changes continuously along the wave’s path, and every result below comes from that.

Writing Newton’s law for a short piece of the chain, the sideways displacement $u(y, t)$ obeys

$$\frac{\partial^2 u}{\partial t^2} = g\frac{\partial}{\partial y}\left(y\frac{\partial u}{\partial y}\right),$$

the wave equation with the tension inside the derivative. Looking for motions in which every point swings at one frequency $\omega$ with a fixed shape $U(y)$, and changing variable to $z = 2\omega\sqrt{y/g}$, turns the equation into Bessel’s equation of order zero, and the solution that stays finite at the free end is

$$U(y) = J_0\!\left(2\omega\sqrt{y/g}\right).$$

Shapes with uneven nodes #

The top of the chain is fixed to the hook, so the shape must vanish there: $J_0(2\omega\sqrt{L/g}) = 0$. The zeros of $J_0$ are at 2.4048, 5.5201, 8.6537, 11.7915 and so on, and the frequencies are

$$\omega_n = \frac{j_{0,n}}{2}\sqrt{\frac{g}{L}} = 1.2024,\ 2.7600,\ 4.3269,\ 5.8958\ldots \times \sqrt{\frac{g}{L}}.$$

The shapes look like the shapes of a string drawn on a distorted scale. The first swings as a whole, its bottom moving most and its top not at all, like a pendulum that bends. The second has one still point, but it is not halfway up: it sits at 0.190 of the length above the free end. The third and fourth have their nodes crowded down towards the bottom, the top part of each shape a long, gentle curve and the bottom part tight. The waves are short where they travel slowly, near the free end, and long where they travel fast, near the top, exactly as the variable $\sqrt y$ in the solution says: the chain’s shapes are evenly spaced not in height but in the square root of height.

That the nodes come in order — one in the second mode, two in the third, and so on — is guaranteed whatever the tension does. The count that cannot be cheated found that for any equation of this type, the $n$-th mode crosses the axis exactly $n - 1$ times, a theorem of Sturm’s from 1836. The hanging chain obeys it, with its crossings moved, and it is the cleanest example of the theorem holding while everything else about the string picture changes.

Not harmonics #

The frequencies themselves are not in whole-number ratios.

The zeros of $J_0$ become evenly spaced at large order, a distance $\pi$ apart, so the chain’s high modes are evenly spaced too — but by 1.306 times the fundamental, not by the fundamental itself, and offset from zero. The sequence is not a harmonic series, and its tones share no common pitch. The drum that has no harmonics found the same of a round drum, whose modes are also Bessel functions, for a related reason: in both, the geometry makes the wave’s effective speed or its room to spread vary across the vibrating object, and the regular spacing of a string’s modes depends on that not happening. A struck hanging chain does not ring with a note; it clatters.

A quarter cycle lost at the free end #

The offset in that spacing has a meaning. For a uniform string fixed at both ends, the $n$-th mode fits exactly $n$ half-wavelengths into the length. The chain’s high modes fit, counting by the travel time of a wave, $n - \tfrac14$ of them: the zeros of $J_0$ approach $(n - \tfrac14)\pi$. A quarter of a half-cycle — an eighth of a cycle each way, a quarter in the round trip — has gone missing at the free end.

That loss is exactly what the quarter cycle a turning point costs found for a quantum particle approaching a classical turning point, and for the same reason. At the chain’s free end the wave speed falls to zero, so a wave travelling down slows, its wavelength shrinks to nothing, and it turns back not at a hard wall, which would reflect it with a half-cycle shift, but in a region where the medium itself changes smoothly — a soft turning point. A wave reflected from a soft turning point loses a quarter of a cycle, and the approximation that treats a slowly varying medium wave by wave, counting phase along the way, gives the chain’s high frequencies as $(n - \tfrac14)\pi/(2\sqrt{L/g})$ — within a fraction of a per cent of the exact zeros from the third mode on. The chain is a classroom example of the semiclassical quantisation rule, with a chain in place of a particle.

The chain as a drum seen edge-on #

The change of variable that turns the chain’s equation into Bessel’s is $z = 2\omega\sqrt{y/g}$: the square root of height plays the part of a radius. It is more than a trick. The symmetric modes of a round drum — the ones in which the membrane moves in concentric rings, with no nodal diameters — satisfy Bessel’s equation in the radial distance from the centre, with the rim fixed. The hanging chain’s modes are those drum modes, with the drum’s centre at the chain’s free end and its rim at the hook, and with radius measured as the square root of the height. A drum’s centre is where its symmetric modes move most, and the chain’s free end is where its modes do too; the drum’s rings crowd towards its rim, the chain’s nodes towards its free end, because the square-root stretching moves the rings’ positions. One equation, written in two coordinates, describes two objects that look nothing alike.

A ripple that speeds up #

The variable speed has a simple consequence for a single pulse. Flick the bottom of a hanging chain sideways and a kink runs up it. At height $y$ it is moving at $\sqrt{gy}$, so it starts from rest at the free end and speeds up as it climbs. Solving $dy/dt = \sqrt{gy}$ gives $y = gt^2/4$: the kink rises with a constant acceleration of $g/2$, exactly half the acceleration of a falling stone.

The kink takes $2\sqrt{L/g}$ to reach the top: 0.90 seconds for a 2-metre chain, 1.56 seconds for a six-metre climbing rope in a gymnasium, which, pushed by its bottom end, swings back and forth once every 4.1 seconds — not quite the 4.9 seconds of a simple pendulum on a light cord of the same length, because the rope’s own weight is spread along it rather than gathered at its end. A kink sent down from the top does the reverse, decelerating as it approaches the free end and arriving — in principle — with zero speed. The amplitude of a wave travelling into a region where it slows grows, by the same rule how a wave thins out applied in reverse, and a kink sent down the chain whips the free end, which is why shaking a hanging rope from the top makes its bottom fling about far more than its top moves.

The fundamental frequency and the travel time are related, as they are for any string. A string’s fundamental has a period of twice the time a wave takes to cross it and come back. For the chain, the round trip is $4\sqrt{L/g}$ and the fundamental’s period is $2\pi/(1.2024\sqrt{g/L}) = 5.23\sqrt{L/g}$; the two do not match exactly, because a wave in a non-uniform medium is partly reflected all along its path, not only at the ends.

Between a chain and a pendulum #

The hanging chain swinging in its lowest mode is very nearly a pendulum, and the comparison is worth making precisely. A rigid uniform rod pivoted at its top swings at $\sqrt{3g/2L} = 1.2247\sqrt{g/L}$, a familiar result from the length nobody has to measure, which found the equivalent simple pendulum of any rigid swinging body. The chain swings at $1.2024\sqrt{g/L}$, two per cent slower. It can bend, its lower part lagging behind its upper part as it swings, and a body that can bend is always a little floppier than the same body held rigid.

Hang a weight on the chain’s end and it moves the other way.

With a weight equal to the chain’s own mass, the frequency falls to 1.0564; with a weight a hundred times the chain’s, the chain is the light string of a simple pendulum and the frequency is $\sqrt{g/L}$ to within half a per cent. The curve is the continuous passage between two idealisations, the massless string of the pendulum and its small lie and the chain with no weight, and every real pendulum sits somewhere on it — a clock pendulum’s rod is not massless, and its mass slightly raises the frequency the simple formula gives. It is the same correction the third of itself a spring carries made for a mass on a spring whose own mass is not negligible, where a third of the spring’s mass is added to the load; a chain adds its own mass in a more complicated way, because the tension it carries varies along it.

Why the function appeared here first #

Bernoulli was not looking for a new function. He wanted the shapes in which a hanging chain can swing, and he wrote them as an infinite series in the height, the series that is now written as $J_0$, and he found its first few zeros numerically. Leonhard Euler extended the analysis, and the same series turned up again in Euler’s work on vibrating membranes and in Fourier’s on heat flow in a cylinder before Bessel, studying the perturbations of planetary orbits in 1824, made a systematic study of the whole family of functions. They now describe every problem in which something spreads or vibrates with circular symmetry, or along a line where a property grows steadily — a drumhead, the light diffracted by a round aperture, the flow of heat out of a pipe, and the tide in a basin whose depth rises linearly from its centre, the open-sea version of the tide a lake makes by itself. The hanging chain belongs to the second kind: its tension grows linearly with height, and the square root of height turns that into the radius of a drum.

Trying it with a hand #

The frequencies are low enough to drive by hand. A 2-metre chain’s fundamental is 0.42 hertz, a swing every two and a half seconds; its second mode is at 0.97 hertz and its third at 1.53, both easily reached by moving the top of the chain side to side a few centimetres. Shaking at 0.97 hertz makes the chain settle into the S of the second mode, with a still point about 38 centimetres above its bottom — not at the middle, as anyone expecting a string’s behaviour would guess — and shaking a little faster or slower makes the shape break up, because the frequency no longer matches a mode. The third mode, at 1.53 hertz, puts its nodes at 15 and 81 centimetres. The chain’s own mass decides all this. A heavier or lighter chain of the same length has exactly the same frequencies and node positions, because its mass cancels from the wave speed: both tension and inertia are proportional to it.

The same rules govern any long flexible line hanging under gravity — the hoist rope of a crane, the cable of a lift, the tether below a balloon. A crane’s hook swinging on its rope is, at small loads, the hanging-chain problem with an end weight, and at large loads a simple pendulum, and the controllers that stop a crane’s load from swinging after a move are tuned to the frequency the end-weight curve gives for the current rope length and load.

What the figures leave out #

The figures treat the chain as a perfectly flexible, inextensible line with no bending stiffness, swinging in small sideways motions in one plane, with no air resistance. Real chains have links that rub, damping the higher modes quickly; real ropes have some stiffness, which raises the frequencies of the higher modes slightly, as stiffness does for a piano string. Large swings make the equation nonlinear, and a chain shaken hard enough at its top whirls in a circle rather than swinging in a plane. The free end is assumed to be truly free; a chain whose end rests on the floor is a different problem, with a moving boundary where it leaves the floor. The domain is small, planar oscillation of a uniform, flexible chain hanging under gravity. Within it the theory is exact, and it is easy to put to the test: a bead chain of a couple of metres, shaken by hand against a metronome, shows where its second-mode node sits, and a ruler compares it with the 38 centimetres the zeros of $J_0$ predict.

Still open: whirling and the falling chain #

The small swings of a hanging chain have been understood since Bernoulli; its large motions have not finished surprising people. A chain whirled from its top settles into steady rotating shapes — a skipping rope’s arc, or shapes with nodes, like the swinging modes but spinning — whose stability depends on the rotation rate in ways mapped only in recent decades. A chain falling off the edge of a table, or pulled out of a beaker in the fountain that a chain of beads forms when it pours out of a jar, raises questions about how a flexible chain exchanges momentum at a point where it changes direction, and whether that exchange is conservative or dissipative depends on the chain’s links in ways still argued from experiments with high-speed cameras. The linear problem, with its Bessel functions, is the foundation all of these are measured against.

A uniform chain hanging under gravity is a string whose tension grows from zero at its free end, so waves climb it at $\sqrt{gy}$, accelerating at $g/2$, and its modes are $J_0(2\omega\sqrt{y/g})$, with frequencies $(j_{0,n}/2)\sqrt{g/L}$ — 1.2024, 2.7600, 4.3269 times $\sqrt{g/L}$, in the ratios 1 : 2.295 : 3.598 rather than 1 : 2 : 3 — and nodes crowded towards the free end; a weight on its end turns it continuously into a simple pendulum. The chain sings in Bessel’s notes because its own weight tunes it differently at every height.