The string whose end will not keep still

The dent that raises the note moves a wall of a resonator slightly and slowly, and finds that each mode simply retunes, by an amount set by where that mode keeps its energy. The change there is small and still. This essay asks what happens when neither holds — when the wall moves back and forth quickly, on the timescale of the vibration itself. The answer is not a larger version of the same retuning. A wall moving at the right frequency does not shift a mode; it pumps energy into the string, gathers it into pulses, and drives it up through the modes without limit.

The system is as plain as possible: a string of length $L_0$, fixed at one end, whose other end is moved back and forth by one per cent of the length, $L(t) = L_0(1 + \varepsilon \sin \Omega t)$ with $\varepsilon = 0.01$. The string starts swinging gently in its fundamental. Everything that follows is computed exactly. A wave on a uniform string is the sum of two shapes travelling in opposite directions, and with one end fixed it can be written $u = F(t + x) - F(t - x)$ in units where the wave speed is one. The moving end imposes one condition, $F(t + L(t)) = F(t - L(t))$: every point of the shape reflected from the moving end is carried from one argument to another, and its slope is multiplied by the Doppler factor of the wall at the instant of reflection. No modes are truncated and no amplitude is assumed small; the pictures are that rule applied, round trip after round trip.

A smooth swing that sharpens into pulses #

The first ten round trips look like almost nothing: the arch leans a little and its energy has risen by nine per cent. By twenty the arch has become a hump, by thirty a spike, by forty a needle. The height of the displacement never changes — the rule carries values of $F$ from place to place without altering them — but the distance over which the string rises to that height shrinks steadily, and the energy, which depends on the steepness of the string rather than its height, grows with it: 3.15 times its start at twenty round trips, 10.9 at thirty, 38.3 at forty.

The needle is really two pulses, one travelling each way, drawn at the moments they cross in the middle. Half a round trip later they are at the two ends, one reflecting from the fixed end and the other from the moving one. That the two ends are reached at the same moment, and that the moving end is at a particular point in its cycle when it happens, is the key to the whole effect.

Every part of the wave drifts to the moment the wall comes in #

A piece of wave that reflects from the moving end while the end is advancing into the string comes back compressed and steepened, exactly as light reflected from an approaching mirror comes back blueshifted in the shift a mirror gives twice. A piece that reflects while the end is retreating comes back stretched. If each piece met the wall at a random phase of its motion, the gains and losses would average away. The question is whether they do.

They do not. A piece of wave that strikes the wall while the wall is further out than its middle position has a longer trip to the fixed end and back, and arrives next time at a later phase of the wall’s cycle; a piece that strikes while the wall is further in has a shorter trip and arrives earlier. Just before the wall passes its middle on the way in, it is still further out, so pieces striking then are pushed later, towards that passing; just after, it is further in, so pieces striking then are pulled earlier, towards it again. The drift converges on one phase, 180 degrees — the instant the wall passes its middle moving inwards, which is also the instant it moves inwards fastest — and holds every piece there. After forty returns all twenty-four rays in the drawing, started at phases spread evenly round the cycle, arrive together at that instant.

So the pieces of wave sort themselves. They crowd together, which is why the smooth arch becomes pulses. And every piece in the crowd meets an advancing wall on every return, which is why the pulses keep sharpening. The wall does not have to aim at the wave. The wave finds the moment when the wall is moving towards it, and stays there.

The same drift appears in the wall that moves while the ball is in flight, where a single ball bounces between walls and gains energy from a wall moving towards it. There the ball is one point; here the wave is a continuum of points, and the drift gathers them.

The Doppler factor per round trip #

Once every part of the wave meets the wall at its greatest inward speed $v$, each reflection multiplies the slope of that part by $(1+v)/(1-v)$ and compresses its length by the same factor, and the energy of the part — slope squared times length — rises by that factor once per round trip.

On a logarithmic axis the three resonant curves become straight lines once the gathering is complete, and their slopes can be predicted from nothing but the wall’s speed. At twice the fundamental the wall’s greatest speed is $\varepsilon\Omega L_0 = 0.01 \times 2\pi \approx 0.063$ of the wave speed, and the Doppler factor $(1.063)/(0.937)$ is 1.134: thirteen per cent per round trip. The computed energy grows from 10.9 to 38.3 between the thirtieth and the fortieth round trip, a factor of 3.5 in ten, or 13.4 per cent per trip — the Doppler factor, to the accuracy the growth can be read. At three times the fundamental the wall moves half as fast again, the factor is 1.21 per trip, and the energy climbs to 961 times its start. At the fundamental itself it is 1.065, and the energy creeps to six times.

The growth is exponential and does not saturate. Nothing in the rule stops it: each part of the wave that is gathered keeps being compressed, and in an ideal string it can be compressed forever. At 2.05 times the fundamental — five per cent off — there is no gathering phase at all. The pieces of wave drift round the wall’s cycle, meeting it sometimes advancing and sometimes retreating, and the energy rises and falls by a few tens of per cent and comes back.

The energy climbs through the modes #

The same story can be told in the language of modes, and there it becomes something a single oscillator cannot do.

The swing that is pumped, not pushed shows that an oscillator whose stiffness is modulated at twice its frequency grows exponentially: parametric resonance. Moving the end of a string at twice its fundamental modulates the fundamental’s frequency at twice that frequency, and the fundamental is pumped in exactly that way. But a string has more than one mode, and they are evenly spaced — the harmonics of only some notes fit, at one, two, three, four times the fundamental. The wall’s frequency, $2\omega_1$, is therefore not only twice the first mode’s frequency but also the gap between the first and the third, between the third and the fifth, and between every mode and the one two above it. A moving end couples modes to one another as well as modulating each one, and a coupling at the gap frequency is resonant with every pair. The energy pumped into the first mode passes to the third, from the third to the fifth, and on up, with no pair anywhere in the sequence where the resonance fails.

After ten round trips the energy has spread over a few modes; after twenty its mean sits at mode 13; after thirty, at mode 47, and the tail reaches past the eightieth mode with a per cent of the energy still in it. The narrowing pulses and the climbing spectrum are one fact seen twice: a feature that narrows by a factor of fifty needs modes fifty times higher to draw it.

The cascade depends entirely on the even spacing. In a drum, whose modes are the non-integer ratios of the drum that has no harmonics, or in a stiff piano string whose upper partials run sharp, twice the fundamental is not the gap between higher modes, the chain breaks at the first link, and a moving boundary pumps the fundamental alone, like a swing. Evenly spaced modes turn a parametric pump into a cascade. The same even spacing makes a wave on a string, or a coherent state in the return a classical cloud never makes, reassemble itself periodically; here it lets the energy climb without ever meeting a gap it cannot cross.

Narrow windows at every whole multiple #

The resonances sit at every whole multiple of the fundamental, not only at twice it, because the condition for gathering is that the wall’s cycle fits a whole number of times into a round trip: then a piece of wave that meets the wall advancing meets it advancing again next time. At three or four times the fundamental the wall passes through several cycles per round trip, and the wave gathers at one advancing phase of one of them. Higher multiples grow faster only because a wall moving through the same one per cent more often moves faster, and its Doppler factor is larger.

The windows are narrow — a few hundredths of the fundamental wide at one per cent amplitude — and widen in proportion to the wall’s amplitude, like the instability wedges of a parametrically driven swing. Between them, the wall’s motion does nothing lasting: the energy wobbles and returns, as it did at 2.05.

The window is the wall’s own speed #

How wide the windows are follows from the drift that gathers the wave. Let a piece of wave strike the wall when the wall is at phase $\varphi$ of its cycle. It travels to the fixed end and back, a distance of about $2L_0$, and the wall’s displacement at the two reflections lengthens or shortens the trip: to first order in $\varepsilon$, the next strike comes at phase

$$\varphi' = \varphi + \delta + 2\varepsilon\,\Omega L_0 \sin\varphi,$$

where $\delta$ is how far the wall’s cycle is from fitting a whole number of times into a round trip. On resonance $\delta = 0$, and the map has two fixed phases, where $\sin\varphi = 0$. At zero the wall is moving outwards, and a small departure grows; at 180 degrees it is moving inwards, and a small departure shrinks by the factor $1 - 2\varepsilon\Omega L_0$ on every return. That is the gathering in the second figure, derived rather than observed.

Off resonance the fixed phases survive only while the drift term can cancel the mismatch, $|\delta| \le 2\varepsilon\Omega L_0$. Written in units of the fundamental, that is a window of half-width $\varepsilon$ times the multiple: ±0.02 around twice the fundamental at one per cent amplitude, ±0.04 around four times it. The computed windows sit just outside these edges because growth near an edge is slow but still growth, and a twofold threshold catches some of it. The width of each window is the wall’s greatest speed, measured against the wave’s, which is also what sets the growth rate inside it — the same statement as the pumped swing’s wedges, in the swing that is pumped, not pushed, and as the bands in which a tide can split into two of half its frequency: a parametric drive is effective in a band as wide as the modulation is deep.

How fast the wall has to move #

The numbers in the figures ask a lot of the wall. A guitar string 65 centimetres long sounding its fundamental at 110 hertz, with its bridge moved by 6.5 millimetres at 220 hertz, would have a wall moving at nine metres a second, about six per cent of the 143 metres a second its waves travel — the ratio used here, and achievable with a shaker. For light in a cavity the same ratio would need a mirror moving at six per cent of the speed of light, which no mirror can. That is why the optical version of this effect was long only a calculation.

It has been made to happen by moving not the mirror but the boundary condition. In 2011 a group at Chalmers University terminated a superconducting transmission line with a device whose inductance could be modulated at about ten gigahertz, which moved the line’s effective end back and forth at speeds no mechanical mirror could approach. Driven at twice a mode’s frequency, the line emitted pairs of microwave photons with nothing in it to begin with — the quantum version of what the figures draw classically, and the reason the phenomenon is called the dynamical Casimir effect.

Where the exact rule holds #

A uniform string with ideal ends. The rule $F(t+L) = F(t-L)$ is exact for a string whose wave speed is the same everywhere and for every frequency, with a perfectly fixed end and a perfectly moving one. Any dispersion — stiffness, a surrounding medium, a real cavity’s mode structure — spoils the even spacing and eventually stops the cascade, as the mode count of how many ways there are to vibrate is spoiled by anything that bends the relation between frequency and wavelength.

No losses. The pulses in the drawing narrow without limit because nothing dissipates them. A real string loses its high harmonics fastest, and the cascade ends where the loss per round trip among the highest modes it has reached matches the Doppler gain.

Small displacements. The string is linear: its tension does not change as it is deflected, and the pulses’ steepness is allowed to grow without the string’s own nonlinearity entering. A real string with steep enough pulses would stretch, change its wave speed, and behave differently.

The end moves along the string. The wall moves along the direction of the string rather than across it; moving an end sideways is ordinary driving, a different problem.

The seed a classical wave needs #

Every figure here is classical, and the classical picture needs something to amplify. Start the string perfectly at rest and the moving end does nothing at all: zero, compressed and blueshifted, stays zero. In a quantum field the string is never perfectly at rest, because every mode keeps its zero-point motion, and a moving boundary pumps that motion just as it pumps a classical wave — creating real quanta in pairs from what was empty. The figures show the amplification and not the seed, and the seed is what makes the quantum effect exist when the classical one would give nothing.

The figures also show the energy in the string and not where it comes from. The moving end does work against the pressure of the waves reflecting from it: a string full of pulses pushes harder on the end at the moments it meets it, and the drive must push back. The energy in the string is paid for by whatever moves the end, and the ever-rising cost is why the growth, in any real system, eventually stops.

Still open: the cascade in a real cavity #

For a perfect one-dimensional cavity the classical cascade is understood exactly. What happens in a real one is not settled. Three-dimensional cavities have unevenly spaced modes, so the cascade should break, but nearly degenerate modes and the width that losses give each mode can restore enough coupling for energy to leak upwards. How the photons created from the vacuum in the quantum version are distributed among the modes once the cascade is partly open, and how losses, the detuning of real modes and the finite speed of real walls combine to set how many are made, is still being worked out — and matters for the experiments that now try to see the effect in cavities rather than in transmission lines.

The habit worth carrying away is to look at the spacing of a spectrum before asking what a periodic push will do to it. A drive that is resonant with one pair of modes is resonant with every pair that shares the same gap, and a spectrum of evenly spaced modes shares one gap throughout. A single oscillator pumped at twice its frequency simply grows; a string pumped the same way grows, sharpens and climbs, because none of the gaps between its modes is one the pump cannot bridge.