The equation that lets a shape travel

A pulse sent down a rope keeps its shape and arrives at a fixed speed. That much is an observation, and the first rung of this ladder is content to make it.

What follows is the reason it could not have happened any other way: three lines of Newtonian mechanics applied to a piece of string a millimetre long, whose answer gives away more than it was asked for.

Everything below is where those three lines come from, what they had to assume, or what the form of the answer is quietly announcing.

The element, and Newton’s second law #

Let a string of tension $T$ and mass per unit length $\mu$ be displaced sideways by a small $y(x,t)$, and consider a short piece of it from $x$ to $x+dx$.

Two forces act on it, both tensions of magnitude $T$ along the string — backwards along the tangent at the left end, forwards along the tangent at the right. The two tangents point in slightly different directions, so the pulls do not cancel, and what is left over is transverse. The transverse component of a tension inclined at $\theta$ is $T\sin\theta$, and for small angles $\sin\theta \approx \tan\theta = \partial y/\partial x$, so the net transverse force is

$$T\left[\left.\frac{\partial y}{\partial x}\right|_{x+dx} - \left.\frac{\partial y}{\partial x}\right|_{x}\right] = T\,\frac{\partial^2 y}{\partial x^2}\,dx.$$

The element’s mass is $\mu\,dx$ and its transverse acceleration is $\partial^2y/\partial t^2$. Newton’s second law, with the $dx$ cancelling from both sides:

$$T\,\frac{\partial^2 y}{\partial x^2} = \mu\,\frac{\partial^2 y}{\partial t^2}.$$

That is the whole derivation, and the middle step is the part worth dwelling on. The force on the element is a difference of two slopes — the second derivative, which is to say the curvature. A piece of string tilted at forty-five degrees but perfectly straight has no net transverse force on it; a bent piece does, in proportion to how bent it is. The restoring influence in a string is not proportional to displacement, since a displaced but straight string is in equilibrium; it is proportional to curvature. That is why a wave is not a row of independent oscillators, but a row of oscillators each pulled about by its neighbours.

Divide through by $\mu$ and read the coefficient in front of the spatial derivative. Tension is in newtons, mass per unit length in kilograms per metre, and $\mathrm{N}/(\mathrm{kg/m}) = \mathrm{m^2/s^2}$: a squared speed, before any wave has been mentioned and before anything has been solved.

A tuning fork, a bow, a hammer and a plucked fingernail appear nowhere in the equation, and therefore nowhere in the speed.

What the two derivatives cost #

The step from $\sin\theta$ to $\tan\theta$ is the only approximation in the argument, and it is worth pricing exactly.

The ratio of the two is $\cos\theta$, so the derivation overstates the restoring force by $1/\cos\theta$, and the fractional error is of order the slope squared. At a slope of 10° that is 1.5 per cent; at 30° it is 13 per cent. The equation is exact only in a limit no illustration can depict, which is the same bargain the pendulum strikes with its small angle — the first term of an expansion, kept because the rest are smaller.

A second assumption hides inside the first. The tension was taken as equal at both ends, which requires $T\cos\theta$ to be constant along the string — so the tension itself rises as $1/\cos\theta$, and at 30° the steep parts carry 15 per cent more than the flat ones. A coefficient that varies along the medium is no longer the equation that was written down.

The third cost changes the kind of equation it is. A displaced string is longer than a straight one — the excess is $\tfrac14 s^2$ of its length for a peak slope $s$ — so the material is stretched further and the tension rises with amplitude. Steel music wire works at a strain of roughly $5\times10^{-3}$; at a peak slope of 0.05 the extra strain is $6.3\times10^{-4}$, which is 12 per cent of that, so the tension rises 12 per cent and the speed 6. A hard-plucked guitar string therefore starts sharp and falls to pitch as its amplitude decays, and it cannot be tuned out.

D’Alembert’s two functions, and why there must be two #

The equation is second order in position and second order in time, and that symmetry decides its solutions completely. Change variables to $u = x - vt$ and $w = x + vt$: it becomes $\partial^2 y/\partial u\, \partial w = 0$, whose general solution is

$$y(x,t) = f(x-vt) + g(x+vt),$$

with $f$ and $g$ any twice-differentiable functions whatever. Jean d’Alembert obtained this in 1747, and the content of it is not that waves travel — it is that two arbitrary functions are needed, one for each direction.

The immediate consequence is that a wave problem takes two conditions rather than one. The shape of a string at $t=0$ does not determine what it does next; the initial transverse velocity is needed too, because the two together fix $f$ and $g$ separately. A string released from rest has zero initial velocity, which forces $f$ and $g$ to be equal, each half the initial shape — so a plucked string’s triangular kink instantly becomes two copies of half the height, one running each way.

The second function is not bookkeeping. It is the reflection.

A string clamped at $x=0$ requires $y(0,t)=0$ for every $t$, so $f(-vt) + g(vt) = 0$, so $g$ is the mirror image of $f$ with the sign turned over. That is satisfiable only because the general solution already contained a family running the other way. Had the equation been first order in time there would have been one function, no second family, and no way to impose anything at an end. Reflection is not bolted onto travelling waves; it is what the second time derivative was for.

The two time derivatives have one further consequence. Replacing $t$ by $-t$ leaves the equation untouched, so a converging pulse that collapses to a point and re-emerges is as legitimate a solution as a diverging one. That is not true of the equation that describes diffusion, which is first order in time and runs one way only — the difference between an echo and a stain.

Superposition, and the standing wave as two travelling ones #

The equation is linear — $y$ and its derivatives appear to the first power, with no products — and it is homogeneous, so the sum of any two solutions is a solution.

Two identical waves in step add to twice one of them, ordinate by ordinate — and half a cycle apart they cancel completely, everywhere, at every instant. Both statements follow from the equation being linear, and neither is obvious: no energy has gone anywhere in the cancelling case, because the two waves are drawn as though they occupy the same string and a real superposition of two travelling waves carries the sum of their energies wherever it is not cancelling.

Superposition delivers the standing wave with no new assumption. Two equal counter-propagating waves of the same wavelength add to

$$\sin(kx-\omega t) + \sin(kx+\omega t) = 2\sin kx\,\cos\omega t,$$

in which position and time have separated. The shape $\sin kx$ does not move, the whole string oscillates in place at $\cos\omega t$, and the nodes are its fixed zeros. A standing wave is not a different kind of wave: it is $f$ and $g$ of equal size, and the reflection argument above guarantees it in any clamped string.

Imposing $y = 0$ at both ends of a length $L$ requires $kL = n\pi$, so only certain wavelengths survive — and that is the entire origin of a discrete list of notes from a continuous medium. The condition is a boundary condition rather than anything about the wave equation, which is why the same arithmetic governs a string, an organ pipe and a particle in a box.

The same equation, with the names changed #

Read the derivation back and look for the point at which it asked what the string was made of. There is no such point. Two ingredients were used and nothing else: a restoring influence proportional to curvature, and an inertia opposing acceleration. Any system with that pair obeys the same equation and has the same solutions.

The consequences are not analogies. Sound in air replaces $T$ with the adiabatic bulk modulus $\gamma P = 1.42\times10^5$ Pa and $\mu$ with the density 1.20 kg/m³, giving 343 m/s. Shallow water replaces them with gravity and the depth: an ocean 4 km deep carries a wave 100 km long at $\sqrt{gh} = 198$ m/s, or 713 km/h, so a tsunami crossing the Pacific is this equation obeyed almost exactly. A solid has two versions of the pair, for compression and for shear, giving the ground a P wave at 6 km/s and an S wave at 3.5. A coaxial cable uses inductance and capacitance per unit length, giving $v = 1/\sqrt{LC}$.

And then the case with no medium at all. Maxwell’s two curl equations, combined, give $\nabla^2\mathbf{E} = \mu_0\varepsilon_0\,\partial^2\mathbf{E}/\partial t^2$ — this page’s equation with $1/\mu_0\varepsilon_0$ where $T/\mu$ was. Permittivity plays the inertia and permeability the stiffness, the speed comes out at $2.998\times10^8$ m/s, and that number had already been measured with batteries and coils before anybody connected it to light. Nothing is displaced and nothing stretched, and every result derived from the string survives unchanged. The search for an ether was the reasonable response of physicists who took the derivation’s setting to be part of its content, when the setting was the part that cancelled.

The two constants in the two places give the speed for the case with nothing in between, and it is worth recording how the numbers arrived. Fizeau timed light through a toothed wheel in 1849; Weber and Kohlrausch measured the ratio of electrical units in 1856 — seven years and one continent apart, neither of them looking for the other’s answer — and the two agreed. That agreement is what turned a wave equation into a claim about what light is.

The frequency is set by the source and the speed by the medium, so the wavelength is whatever the two of them leave over. The same 50 Hz on two strings differing only in mass per unit length travels at 283 and 141 m/s and arrives at wavelengths of 5.66 and 2.83 m — which is worth stating because it is the one relation among the three that is not a choice, and because it is where a wave equation stops being an abstraction and starts predicting a number.

The cleanest way to see how little the medium matters is to build one out of nothing but masses and springs. Let masses $m$ sit at spacing $a$, each coupled to its neighbours with stiffness $k$. Newton’s law for mass $n$ is $m\ddot y_n = k(y_{n+1} - 2y_n + y_{n-1})$, and the bracket is the second difference — $a^2$ times the second derivative, once $a$ is small compared with the scale of the disturbance. So $\ddot y = (ka^2/m)\,y''$, and identifying $T = ka$ and $\mu = m/a$ recovers $\sqrt{T/\mu}$ exactly. A chain of oscillators becomes a continuous medium in the limit, which makes two masses swapping their motion the smallest instance of this page’s subject. Lagrange took the limit this way in 1759, to avoid arguing about arbitrary functions.

The quantum case needs care, because the resemblance is partial. A massless field obeys this equation as written, which is why photons travel at one speed. Schrödinger’s equation is first order in time, so it is not this equation, and matter waves are dispersive as a direct result. What is shared is the spatial half: for a state of definite energy it reduces to $\psi'' = -k^2\psi$, the string’s spatial equation with $\psi$ for $y$, which is why a particle in a box has the harmonics of a string.

Where the model stops #

Dispersion is usually introduced as a property some media happen to have. It is better read as a statement about this equation: a medium is dispersive exactly when it is not obeying it.

Substituting $y = A\cos(kx-\omega t)$ gives $\mu\omega^2 = Tk^2$, so $\omega = vk$ with $v$ constant. A straight line through the origin in the $\omega$–$k$ plane is not one dispersion relation among many; it is the only one the equation permits.

The equation demands a straight line between frequency and wavenumber, and that is where it stops being true. Shallow water gives $\omega = kc$ exactly, so a pulse holds its shape; deep water does not, and a pulse spreads. Every medium that disperses is a medium the plain wave equation does not describe — which is most of them, and the reason this equation is the start of the subject rather than the whole of it.

Four ways for a real medium to depart from it:

Bending stiffness. A real string resists being bent as well as stretched, adding a term in the fourth derivative: $\mu\ddot y = Ty'' - EI\,y''''$. The relation becomes $\omega = vk\sqrt{1+EIk^2/T}$, so short waves run fast. On a piano this is an inharmonicity coefficient $B$, with the $n$th partial at $nf_0\sqrt{1+Bn^2}$; at $B = 2\times10^{-4}$ the eighth partial is 11 cents sharp of eight times the fundamental. Tuners stretch the octaves to match, so every piano is tuned around a term this derivation does not contain.

Atoms. The chain of masses obeys the equation only for $ka \ll 1$. Solved exactly it gives $\omega = 2\sqrt{k/m}\,|\sin(ka/2)|$, below the straight line by 1.0 per cent at $ka = 0.5$, by 10.0 per cent at $ka = \pi/2$ — a wavelength of four atomic spacings — and by 36 per cent at the shortest wavelength the lattice supports, where the packet speed falls to zero. So the price of never asking what the medium is made of is that the answer fails once the wavelength approaches whatever it is made of: a nanometre in a solid, which is terahertz. Air fails at roughly the mean free path, 68 nm, so continuum sound is good to some 5 GHz.

Amplitude. Once the tension depends on the displacement the equation is not linear and superposition fails, so waves of different amplitude travel at different speeds. Crests overtake troughs and a sine wave steepens into a sawtooth; ocean waves break for this reason and a loud enough sound becomes a shock.

Damping. A loss term $b\dot y$ makes the wavenumber complex, and — the part that is easy to miss — it also makes the medium dispersive. The Kramers–Kronig relations of 1926 and 1927 tie absorption and dispersion together, so anything that absorbs at any frequency has a frequency-dependent speed, and a perfectly non-dispersive medium would have to be perfectly lossless. Glass, which disperses enough to make a rainbow, is the ordinary case; an ocean swell sorting itself by wavelength is the extreme one.

What it costs #

Two conditions, or nothing. Every numerical wave solver must be handed both the displacement field and the velocity field at the start. A code given only the shape produces two waves where one was intended, each of half the amplitude — not a crash, but a plausible answer at half scale.

A number every simulation meets. Because the solution propagates along the characteristics $x \pm vt$, a finite-difference scheme is unstable unless the wave cannot cross a cell in one step: $v\,\Delta t/\Delta x \le 1$, the Courant condition, from Courant, Friedrichs and Lewy in 1928. Modelling air on a one-centimetre grid caps the time step at 29 µs, or 34,000 steps per second of simulated sound, and the limit belongs to the equation rather than to the machine.

Every echo instrument is d’Alembert’s second function. Ultrasound, sonar, seismic reflection surveying and time-domain reflectometry all send $f$ out and read $g$ back; the timing gives the distance, the sign and size the mismatch. The same fact is a nuisance in reverse for anyone laying cable, which is why coaxial line is sold at 50 or 75 ohms.

A plucked string can be simulated with two delay lines. Because the solution is a pair of functions sliding in opposite directions, a digital waveguide model implements a string as two shift registers with a filter where the bridge is. Physical-modelling synthesis of a guitar therefore runs in real time on modest hardware: the partial differential equation costs two memory shifts per sample.

The dispute that became Fourier analysis #

D’Alembert published the two-function solution in a 1747 memoir on the vibrating string for the Berlin Academy, and immediately restricted it: he required $f$ and $g$ to be given by a single analytic expression, which excludes a plucked string’s corner.

Euler objected in 1748 that the initial shape of a string is whatever a hand puts there, kink included, and that the solution must accept it. Daniel Bernoulli then argued in 1753 for something different in kind: that the general motion is a sum of sinusoidal modes, $\sum a_n \sin(n\pi x/L)\cos(\omega_n t)$, on the ground that each mode is a thing a string demonstrably does and the coefficients are free.

D’Alembert and Euler both rejected that as a general solution, and the objection was not obtuse. A sum of sines is smooth and periodic; a plucked string’s initial shape has a corner in it. That infinitely many smooth periodic functions could add to a function with a corner looked like a category error, and it took Fourier’s work on heat, presented in 1807 and published in 1822, to justify the expansion — longer still to settle the sense in which it converges, which is in value but not in slope.

The dispute changes how the physics reads, because both sides were describing the same solution set and both languages are still in daily use. On an unbounded line, d’Alembert’s pair of arbitrary functions is natural and a Fourier sum is a detour. For a clamped string, an organ pipe, an optical cavity or an atom, the modes are natural and the two travelling functions are the detour. The choice is decided by the boundary — by where the medium stops being uniform.

What the picture cannot show #

Every figure here is drawn at slopes of order one half, and the derivation holds only for slopes much smaller than one. Drawn honestly — at the 3° where the approximation costs a tenth of a per cent — the wave would be a barely perceptible waver on a straight line. The illustrations are all outside the regime they illustrate.

The figures are also one-dimensional, and the step to more dimensions is not cosmetic. The same equation with $\nabla^2 y$ in place of $y''$ governs a membrane, and its allowed frequencies stop being integer multiples of anything: a drum has no harmonic series, for reasons that live in the shape of the boundary. Stranger still, the number of dimensions changes the character of the solution. A sharp pulse in three-dimensional space stays sharp, which is why speech is intelligible and why treating every point of a wavefront as a new source works; in two dimensions the same pulse leaves a tail that never quite dies.

Nothing here draws a loss, so every figure shows a wave that never fades, while a real wave thins out geometrically and is absorbed besides. And the displacement is one number per point, which is right for a rope and wrong for light, for sound in a solid, and for anything with a polarisation to lose.

The ladder from here #

Later rungs on this anchor: the equation in three dimensions, and the spherical solution whose amplitude must fall as $1/r$; the energy and momentum flux read out of the equation rather than assumed; the inhomogeneous equation with a source, and the retarded Green’s function that answers it; the short-wavelength limit in which the equation hands back ray optics; and the non-linear corrections, which produce shocks in one direction and solitons in the other.

The neighbouring ladders are what happens where the coefficients change, which is this equation with a discontinuity in it, standing waves, which is this equation with two boundaries, and wave packets, which is what has to be said once it stops holding exactly.