Concepts explained
The Wave Equation — It Pulls Back as Hard as It Is Bent
An equation that began with a violin string ended up writing sound, light, earthquakes and electrons in one language. What it says is startlingly simple: the more sharply something is bent, the harder it snaps back.
Take one point on the string. What pulls on it is the string on either side.
Where the string is straight, the pull from the left and the pull from the right are exactly opposed and cancel. Nothing happens.
But where the string is bent at that point, the two pulls are slightly misaligned. What is left over drags the point back.
The overshoot follows too. By the time the point reaches its rest position it is already moving. The restoring pull is zero there, but the speed is not — so it sails past to the other side, bends the other way, and gets pulled back again. That repetition is the vibration.
High notes and low notes
So much for guitar strings. This same equation went on to answer what light is, and then what an electron is.
A little further in
In the mid-eighteenth century mathematicians argued for decades about one question: pluck a string fixed at both ends, and what shape does it take as it moves?
Practical for musicians, novel for mathematicians. Until then an equation usually asked for a number. This one asked for an entire shape — and a shape changing in time.
What the equation says
Take one point on the string. What force acts there? The tension of the string on either side.
Where the string is straight, the two pulls are exactly opposed and cancel. Where it is bent, the two directions are slightly misaligned and a restoring force remains. The sharper the bend, the greater the misalignment and the stronger the pull.
D'Alembert set the equation down in 1747 and solved it. His answer was elegant: any wave is the sum of one shape travelling left and one travelling right. Euler soon generalised it.
The quarrel
Then came the argument. D'Alembert held that the string's initial shape must be a smooth curve. But pluck a real string and the point held by the finger is a corner. At a corner the slope is undefined, so his solution does not apply.
Bernoulli offered another answer: any shape can be built by superposing smooth sine waves — a fundamental plus its overtones. Euler and d'Alembert refused it, holding that a corner cannot be a sum of smooth curves.
Half a century later Fourier ended it. Working on the spread of heat, he showed that any shape at all — cornered, even broken — can be written as a sum of sine waves. Bernoulli had been right.
From string to light
When Maxwell reduced electricity and magnetism to four equations and combined them, out came exactly the form of the wave equation — meaning there are waves in which electric and magnetic fields generate each other as they advance.
And the equation gave the speed of those waves. Put the numbers in and it matched the already-measured speed of light.
We can scarcely avoid the inference that light consists in the transverse undulations of the same medium.Maxwell, 1865
An equation built to settle a violin string had answered what light is. And that this speed comes out the same for every observer became, forty years later, the starting point of special relativity.
And on to matter
When de Broglie proposed that matter has wave properties, Schrödinger asked the obvious question: if it is a wave, where is its wave equation?
The equation he wrote differs in form — one time derivative rather than two, and an imaginary unit in it. But the skeleton is the same: how sharply the thing is bent in space sets how it changes.
What is bent physically moves — the string's height, the field's strength. Quantities you can measure.
What is bent is still argued over. Born read it as probability.
Schrödinger himself disliked that reading. He wanted his equation to describe something actually spread out in space. But the calculations only worked when it was read as probability.
The question that remainsOne equation from a violin string writes sound, light and electrons in the same shape. When unrelated things obey the same equation, is that a fact about nature — or about how few equations we know how to solve?