A plucked string keeps its corners

Fourier’s series were built to solve the flow of heat, and heat is the most forgetful process in physics. Each harmonic of a temperature profile decays as $e^{-m^2t}$, so fine detail is not blurred but deleted, the profile is infinitely smooth an instant after it starts, and running the flow backwards is hopeless.

Change a single derivative in the equation and every one of those statements reverses. A string under tension obeys

$$\frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2},$$

with the second time derivative where heat has the first. The sines are still the natural shapes, the coefficients are still the same projections, and the recipe is still “split, evolve each harmonic, add back”. But now the evolution of each harmonic is an oscillation rather than a decay. Nothing is lost, the sharp corner of a pluck survives for ever, and the whole motion comes back to its starting shape exactly, over and over.

One derivative, and decay becomes oscillation #

Try a single harmonic, exactly as for heat. A harmonic that turns rather than shrinks is most naturally written as $e^{im\pi t}$, a point going round a circle, and the real part of that turning point is the cosine that follows. On a string of length one pinned at both ends, the shapes that keep their form are $\sin(m\pi x)$, and putting $u = A(t)\sin(m\pi x)$ into the equation gives

$$A''(t) = -m^2\pi^2 A(t).$$

For heat the same substitution gave $A' = -m^2\pi^2 A$, whose solution is an exponential that dies. With a second derivative the solution is $A(t) = A(0)\cos(m\pi t)$ for a string released from rest — a cosine, which swings between $A(0)$ and $-A(0)$ and never gets smaller.

That is the entire difference, and it deserves to be stated as bluntly as the heat essay stated its own: each harmonic of a string keeps its size for ever and merely turns over, at a rate proportional to its frequency. The $m$-th harmonic completes $m$ swings in the time the first completes one. Harmonics do not interact, so the general motion is the sum of all of them swinging independently, each at its own pace, none of them fading.

The figure is the comparison worth carrying. The starting tent has a corner, so its coefficients fall away only as $1/m^2$ and it contains harmonics of every size. Under heat, each of those is multiplied by $e^{-m^2\pi^2 t}$: by $t = 0.02$ the tenth harmonic has shrunk by a factor of about $4\times 10^8$, and what is drawn is already a smooth hump. Under the wave equation each is multiplied by $\cos(m\pi t)$, which is never smaller than $-1$ or larger than $1$. At every time the string is built from harmonics of exactly the sizes it started with, only with some of them upside down. A corner is what harmonics of those sizes add up to, so a corner is what is drawn.

Two travelling halves #

There is a second way to solve the string, found by d’Alembert in 1747, before anyone had written down a Fourier series, and it explains the picture more directly than the harmonics do.

Any shape at all, slid sideways at unit speed, satisfies the wave equation: if $u = F(x - t)$ then both second derivatives equal $F''(x - t)$. So does any shape slid the other way. A string released from rest in the shape $f$ therefore moves as

$$u(x, t) = \tfrac12\big(F(x - t) + F(x + t)\big),$$

half of the starting shape travelling right and half travelling left, where $F$ is $f$ continued beyond the ends of the string. The ends are pinned, so the continuation has to make $u$ vanish at $0$ and at $1$ forever: $F$ is $f$ reflected upside down across each end — an odd shape, repeating every $2$.

Now the flat tops and running corners are obvious. The two halves each carry a corner. Where the corners have passed, the two sloping halves add to a straight line, and where they overlap, the halves’ slopes cancel into a flat top. When a corner reaches an end it meets its own upside-down reflection coming the other way, and it bounces back inverted — which is why at $t = 1$ the string holds the tent upside down and mirrored, and why at $t = 2$ both halves have travelled one full repeat of the pattern and the pluck is back, point for point.

The two solutions are the same solution. Each harmonic, $\sin(m\pi x)\cos(m\pi t)$, is itself half of a wave moving right plus half of one moving left — that is the product-to-sum identity for sine and cosine — so summing harmonics sums travelling waves and summing travelling waves sums harmonics. The figure checks the agreement numerically at every time drawn: the travelling halves and the truncated series differ by no more than the size of the terms left off.

A pluck that is not in the middle #

A pluck at the middle is too symmetric to show what the corners are doing, because the two split corners are mirror images and so are their reflections. A pluck a fifth of the way along is not.

Every shape in the sequence is a polygon of straight pieces, because it is always the sum of two straight-sided halves. The steepness of the sides never changes, only which pieces are present; a string plucked this way has, at every instant, a stretch with the slope of the short side of the original tent or of the long side, or half of each. There is no instant at which anything has rounded off. The corners are points where two slopes meet, and they are carried along the string at exactly the wave speed for as long as the string vibrates.

That is not a mathematical idealisation that a real string quietly ignores. A photograph of a bowed violin string, taken with a strobe, shows exactly this — a single sharp corner running round a lens-shaped envelope — and Hermann von Helmholtz described and drew it in the 1860s. Real strings do smooth their corners eventually, because real strings are stiff and lose energy to the air and to the bridge. But the loss is slow: a guitar string rings for seconds, and in those seconds the corner goes round thousands of times.

Nothing is lost, and the motion runs backwards #

The heat equation could not be run backwards because running it backwards multiplies the $m$-th harmonic by $e^{+m^2t}$, and any error in the fortieth harmonic is magnified by a number with seven hundred digits. The string has no such problem. Running it backwards replaces $\cos(m\pi t)$ with $\cos(-m\pi t)$, which is the same number: the motion of a string released from rest is identical played forwards and backwards. More generally the wave equation is unchanged when $t$ is replaced by $-t$, since a second derivative does not care about the direction of time, and the past of a string can be recovered from its present as accurately as the future.

The exact return is the arithmetic of the frequencies. The $m$-th harmonic swings with period $2/m$, so after time $2$ every one of them has completed a whole number of swings at once and is back where it started. That happens because the string’s natural frequencies are whole-number multiples of the lowest one, which is also why a string sounds like a single note: its overtones are the harmonic series, $1, 2, 3, \ldots$ times the fundamental, and the ear hears a harmonic series as one pitch with a colour rather than as a chord. A drumhead’s overtones are not whole-number multiples — they are ratios of zeros of Bessel functions, $1$, $1.59$, $2.14$, $2.30$ and so on — so a drum never repeats its shape, and it sounds like a thud with a pitch in it rather than a note.

What is conserved is energy. The quantity $\tfrac12\int\big(u_t^2 + u_x^2\big)\,dx$, motion plus stretch, is constant in time for the wave equation, and in the harmonic picture it splits into one constant share per harmonic, each sloshing between kinetic and potential as its cosine turns. Heat conserves only the average temperature and destroys every other share — which is also why the density of a random walker, which obeys the heat equation, forgets where the walk began, while a string remembers its pluck for ever. That is the sense in which one of these flows is reversible and the other is not, and it comes down to the single difference between $e^{-m^2t}$ and $\cos(m\pi t)$.

A string is the limit of a chain of beads #

The whole-number frequencies are a property of the continuous string, and the quickest way to see how special they are is to break the string into pieces. Put $n$ equal beads on a weightless thread, equally spaced, with the thread’s ends fixed. Each bead is pulled by its two neighbours in proportion to how far it sits from their average, so the accelerations are given by a matrix with $2$ down the diagonal and $-1$ on either side — the same matrix whose frequencies interlace when a bead is clamped.

Its natural shapes are the directions that matrix leaves alone, and they turn out to be the string’s sines sampled at the beads: the $k$-th shape puts bead $j$ at height $\sin\big(k\pi j/(n+1)\big)$. The chain does not lose the string’s shapes. It loses the string’s frequencies. The squared frequency of the $k$-th shape is $2 - 2\cos\big(k\pi/(n+1)\big)$, so the frequency itself is $2\sin\big(k\pi/(2(n+1))\big)$, which for small $k$ is almost exactly $k\pi/(n+1)$ — proportional to $k$, like the string — and for large $k$ bends over and flattens, since a sine never exceeds one.

So a chain of beads has overtones that are nearly whole-number multiples of its fundamental at the bottom and increasingly flat at the top. It never returns exactly to its starting shape, because no single time is a whole number of swings for every overtone at once, and a corner travelling along it does not stay a corner: its high harmonics fall behind its low ones, and it smears into a ripple. That is the discrete version of the stiffness that makes a real piano string’s overtones slightly sharp. The perfect return of the ideal string is the limit of infinitely many beads, and it is the ratio $\sin(\theta)/\theta \to 1$ that makes the limit exist.

Where the string is plucked decides the harmonics #

The coefficients of a tent with its peak at $a$ come out, by the usual projection, as

$$b_m = \frac{2\sin(m\pi a)}{m^2\pi^2\,a(1 - a)}.$$

Two things in that formula are audible. The $m^2$ in the denominator is the signature of a single corner — a jump in the function itself would give $1/m$, as the square wave’s does, and a jump in the slope costs one more power of $m$. And the numerator vanishes whenever $m\pi a$ is a multiple of $\pi$: when $ma$ is a whole number, which is when the $m$-th harmonic’s shape has a node exactly where the string was pulled.

This is Thomas Young’s law, from 1800: a string plucked or struck at a node of one of its harmonics does not sound that harmonic. Plucking at the middle gives a hollow, clarinet-like sound, missing every even partial; plucking close to the bridge gives a bright, thin one, rich in high partials, because a pluck near an end puts a node for none of the low harmonics and makes $\sin(m\pi a)$ grow with $m$ for a long stretch. A string struck rather than plucked starts straight but moving, and the same projection applied to the initial velocity of a narrow hammer blow gives amplitudes $2\sin(m\pi a)/(m\pi)$ — falling only as $1/m$, one power slower than the pluck’s, because a blow puts the corner in the velocity rather than in the shape. That is part of why a struck string sounds brighter than a plucked one of the same pitch. Guitarists choose where to pluck for exactly this reason, and piano makers place the hammers about a seventh or an eighth of the way along the string — a choice traditionally explained as silencing the seventh harmonic, which sits badly against the scale, although what a real felt hammer does is more complicated than a clean node.

The quarrel over whether a corner is allowed #

The plucked string is where Fourier analysis began, and the origin is instructive because the mathematicians involved were arguing about the very object in the first figure.

d’Alembert solved the string with travelling waves in 1747. Daniel Bernoulli, in 1753, argued on physical grounds that every motion of a string is a superposition of its harmonics — that the general solution is a sum of sines. Euler agreed that d’Alembert’s solution was right and denied that Bernoulli’s could be general: a sum of sines was a smooth, analytic expression, and a plucked string had a corner, so a plucked string could not be a sum of sines. The argument ran for decades and involved Lagrange as well, and at bottom it was an argument about what a function is.

Both sides were partly right, which is why it could not be settled. A finite sum of sines is smooth, so no finite sum can have a corner. But an infinite sum can, and the tent is one: its sine series converges at every point, uniformly, to a function with a corner in it. What nobody in 1753 had was a notion of an infinite sum precise enough to say so. Fourier’s work on heat half a century later made the claim unavoidable — every reasonable initial temperature, corners and jumps included, has a sine series — and the effort to say exactly which functions have convergent series occupied Dirichlet, Riemann and Cantor and created much of modern analysis. The string’s corner was the first function in history that forced mathematicians to decide whether an infinite sum of smooth things could be rough.

What the pictures leave out #

The figures are of an ideal string, and three omissions matter.

They cannot show what makes real strings smooth their corners. Stiffness adds a fourth derivative to the equation, which makes high harmonics travel faster than low ones, so a corner slowly disperses into a ripple; air resistance and the bridge drain energy, faster from high harmonics than from low ones. Those effects are small against the tension, which is why the ideal picture matches a strobe photograph for many periods, but they are why the sound of a plucked string brightens for an instant and then mellows.

They cannot show a motion in two dimensions. The shapes drawn are of a string moving in one plane. A real string swings in two, and a plucked string whose two directions have slightly different frequencies — because the bridge supports it differently up and down than side to side — can trace an ellipse that slowly turns, which is part of why a guitar note’s loudness beats.

And the agreement of the two solutions is checked, not proved. The figures compare the travelling halves with four hundred terms of the series at hundreds of points and find them within the size of the neglected terms. That is evidence that the two constructions agree. The proof is the product-to-sum identity applied to every term, together with the uniform convergence of the series, and a figure can only illustrate it at the points it samples.

Where this leads: sums of sines that do not converge #

The tent’s series converges everywhere because its coefficients fall like $1/m^2$, fast enough to add up absolutely. The square wave’s fall only like $1/m$, and its series converges anyway, though with the Gibbs overshoot at every jump. So a natural question is whether the series of every continuous function converges, since continuous functions have no jumps at all.

It does not, and the reason is a single number attached to the partial sums: the most that adding up $N$ terms can amplify a function that never exceeds one in size. That number grows without bound, slowly, like the logarithm of $N$, and a continuous function built to exploit it has a Fourier series that diverges at a point. That is where the next essay goes. It is also the answer to the question Euler raised about the string: not every continuous shape is the sum of its own sines, point by point, although every one of them is the limit of the averages of those sums.

A second derivative in time #

Replacing $\partial u/\partial t$ by $\partial^2u/\partial t^2$ turns each harmonic’s decay $e^{-m^2t}$ into an oscillation $\cos(m\pi t)$, so a string loses nothing that heat destroys. Its corners split and travel, as d’Alembert’s two sliding halves show directly; its motion runs identically backwards; and because its frequencies are whole-number multiples of the lowest, every harmonic realigns after one period and the starting shape returns exactly.

The coefficients a pluck starts with are the ones it keeps, so where the string is plucked is what it sounds like: a harmonic with a node at the pluck is absent, and the rest fall like $1/m^2$, the price of one corner. That a sum of smooth sines could hold such a corner at all was the question that divided Euler from Bernoulli, and the answer turned out to need the whole theory of infinite sums.

When two equations differ by a single derivative, compare what each does to one sine — the whole difference in behaviour is in that one line.