Episode 10 - Waves: How patterns travel

Tomlin Hawthorne

A loudspeaker is playing a single steady note. It's fairly loud.

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I bring in a second loudspeaker and switch it on, playing the same note at the same volume.

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Is it possible for the room to get quieter?

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Not the same. Quieter. Can adding a sound to a sound give you less sound?

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Yes. And if you own a pair of noise-cancelling headphones, you've paid good money for exactly that. They listen to the drone of the aeroplane, manufacture a second drone, and play it into your ear. Two noises go in. Silence comes out.

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Nothing made of stuff can do that. You can't add a pile of sand to a pile of sand and get a hole. Today is about the things that can, and why almost everything in the second half of this series turns out to be one of them.

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Last time, a calculator. You started with 0.3, and again with 0.3001, and applied the same simple rule fifteen times. If you did it, you watched two lists of numbers walk side by side for nine or ten steps and then part company completely. A difference in the fourth decimal place, grown until it was the whole answer. That's chaos, on a phone.

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This is the last episode of Part Two, and it's a toolkit episode, like the one on estimating. One idea, the wave, which we'll need for light, for atoms, for the Big Bang, and for the strangest experiment ever done. Today's question: what is a wave?

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Let's start somewhere beautiful. The theatre at Epidaurus, in the Peloponnese, built in the fourth century BC. It seats about fourteen thousand people in a great stone fan on a hillside. And the guides there do a trick for the tourists. They stand in the middle of the circular stage and drop a coin, or tear a sheet of paper, or strike a match. And you can hear it in the back row, sixty metres away and twenty-odd metres up.

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Nobody is quite sure how much of that the builders intended. One modern study suggests the rows of limestone seats themselves act as a filter, damping the low rumble of wind and crowd and passing the higher frequencies of the voice. The Greeks may have found a good design by trial and error and then copied it. Either way, an actor in a mask, with no amplification, could make fourteen thousand people hear a whisper.

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So something travels from the stage to the back row. What?

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Not air. That's the first thing to get straight, and it's the key to everything today. The air next to the actor's mouth does not fly up the hillside to the cheap seats. If it did, the audience would feel a breeze every time Oedipus spoke.

Tomlin Hawthorne

What travels is a disturbance. The actor's voice gives the nearby air a little squeeze. That squeezed air pushes on the air next to it, and relaxes back. That air squeezes the next bit. And so on, up the hill. Each bit of air just jiggles back and forth where it is. The squeeze travels.

Tomlin Hawthorne

The best picture of this is a stadium wave. You've seen one, possibly been in one. A ripple of standing, arm-waving people sweeps round the stands at about twenty-five miles an hour. But nobody leaves their seat. Each person stands up and sits down. The wave is a pattern passed from neighbour to neighbour.

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Or think of a rumour. It crosses a city in an afternoon, and no one has gone anywhere.

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That's a wave. A pattern that moves through something without carrying the something along. It does carry two things: energy and information. The sound from the stage can rattle your eardrum and tell you who killed Laius. And it brings no air with it.

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We need three words, and then we can play.

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Picture ripples on a pond, seen from the side. A row of humps and dips.

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The distance from one hump to the next is the wavelength.

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If you stand still and count how many humps pass you each second, that's the frequency.

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And the height of the humps is the amplitude.

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For sound, frequency is what you hear as pitch. More squeezes per second, higher note. The A that an orchestra tunes to is four hundred and forty squeezes a second. And amplitude is loudness.

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Now, the first person to connect any of this with number, as far as the tradition goes, was Pythagoras. There's a famous story that he was walking past a blacksmith's forge and noticed the hammers ringing in harmony, went in, weighed them, and found their weights were in simple ratios. It's a lovely story and it can't be true, because hammers don't work that way. The pitch of a hammer blow doesn't go by its weight like that.

Tomlin Hawthorne

But the real discovery behind the legend is true, and you can check it on any guitar. Pluck a string. Now press it down at exactly the halfway point and pluck again. The note is the same note, an octave higher. Half the length, double the frequency. Press at two-thirds of the length and you get what musicians call a fifth, the most consonant interval there is after the octave. Three-quarters gives a fourth.

Tomlin Hawthorne

Simple whole-number ratios sound good together. Complicated ones grate. The Pythagoreans thought this was the most important fact they'd ever found, and I'm not sure they were wrong. It was the first time anyone had shown that a human experience, in this case beauty, the sense that two notes belong together, follows a mathematical law. They concluded that number is the secret structure of the world. Twenty-five centuries on, physics is still more or less proceeding on that assumption.

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Right. Back to the two loudspeakers.

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Here's the property that makes waves utterly different from objects. When two waves meet, they pass straight through each other. And while they overlap, they simply add.

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If a hump from one arrives at the same place and time as a hump from the other, you get a double hump. Twice as high.

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If a hump from one arrives with a dip from the other, the hump fills the dip. They cancel. You get nothing. Flat water. Silence.

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This is called interference, and I'd like you to remember that word, because it's going to matter more than any other in the series.

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So: the noise-cancelling headphones. A microphone on the outside picks up the engine drone. A chip produces the same sound, flipped over, every hump turned into a dip. That goes into your ear along with the real drone. Hump meets dip. Hump meets dip. And you're left with near silence, into which you can then pour a podcast.

Tomlin Hawthorne

And this is the test. If you ever want to know whether something is a wave, ask whether it can do that. Can two lots of it add up to none? Bullets can't. Sand can't. Water waves can, sound can. Keep the test in your pocket.

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Now a second thing waves do, and it's the source of all music.

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Take a guitar string again. It's fixed at both ends. It can't move there. So when it vibrates, it can't make just any shape. It has to make a shape that's still at both ends.

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The simplest is one big bow, the whole string swinging up and down together. That gives the lowest note.

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The next simplest has a still point in the middle too, with the two halves swinging opposite ways. That's the octave above.

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Then three sections. Then four. And that's all. One, two, three, four. There's no shape with two and a half sections, because it wouldn't be still at the ends.

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So a string, which you'd think could wobble any way it liked, is only allowed a certain list of patterns, a ladder of them, with nothing in between. They're called harmonics. And when you pluck a real string, you get a blend of several at once.

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That blend is why a violin and an oboe playing the very same note sound nothing alike. Same bottom rung, different recipe of the rungs above. It's the tone colour of the instrument. Your ear is doing an astonishing piece of analysis every time you recognise a friend's voice in one syllable down a bad phone line.

Tomlin Hawthorne

I want to put a flag here, a large one. The lesson of the guitar string is this: when you trap a wave, when you confine it between fixed ends, it's forced to come in separate, countable steps. One, two, three. Smooth wave in, discrete ladder out.

Tomlin Hawthorne

In episode twenty-eight we're going to ask why atoms only have certain energies, and nothing in between, and it's going to seem like the most mysterious thing in nature. And the answer is going to be: it's a guitar string. That's the heart of quantum mechanics, and you've just met it.

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One more behaviour, with a good story.

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You know how a siren drops in pitch as an ambulance goes past. Nee-naw coming toward you, and then a lower nee-naw going away. That's called the Doppler effect, after an Austrian who predicted it in 1842.

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The reason is simple once you picture it. The ambulance is sending out humps at a steady rate. But as it drives toward you, it's chasing its own waves. Each hump is launched from a bit closer than the last, so they arrive bunched up. Shorter wavelength, higher pitch. Once it's past, it's running away from its waves. Each one is launched from further off. They arrive stretched out. Lower pitch.

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In 1845 a Dutch scientist with the excellent name of Buys Ballot decided to test this, and he did it in style. He borrowed a steam locomotive and an open wagon on the line near Utrecht. He put a group of horn players on the wagon, and told them to hold one steady note. And he stationed musicians with perfect pitch beside the track, to write down what note they heard as the train thundered by at forty miles an hour.

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The horn players held their note. The listeners heard it about a semitone sharp as it approached and a semitone flat as it left. Doppler was right. It is my favourite experiment in the history of acoustics, and I regret that there is no recording.

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Why am I telling you this? Because the same thing happens with light. Light from something moving away from you is stretched, and shifted toward the red. In the 1920s astronomers pointed their instruments at distant galaxies and found that nearly all of them are shifted to the red. They're all leaving. That's how we found out the universe is expanding, and it gets all of episode forty-two. The evidence is an ambulance siren.

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Which brings me to light, and to the experiment that closes Part Two.

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For a century after Newton, most people thought light was a stream of tiny particles. Newton had said so, and you didn't lightly disagree with Newton.

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In 1801 an English doctor called Thomas Young did disagree. Young was the sort of person who makes the rest of us feel underemployed. He could read at two. He worked out how the eye focuses. He made a serious start on deciphering the Rosetta Stone, before Champollion finished the job. And he did an experiment with light that you could do on a kitchen table.

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He let light through two narrow slits, very close together, and looked at where it landed on a screen.

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If light were particles, you'd get two bright lines, one behind each slit. Like spray paint through a stencil.

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He got stripes. A whole row of them: bright, dark, bright, dark, bright.

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Think what the dark stripes mean. Those are places on the screen that light from the first slit can reach, and light from the second slit can reach. And with both slits open, there's darkness.

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Light plus light equals dark.

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That's our test. That's two loudspeakers making silence. Light is a wave.

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It was an unpopular result. One reviewer in Edinburgh savaged him so thoroughly that Young more or less gave up physics for some years. But he was right, and by the 1820s everyone knew it. Case closed.

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I'd like you to enjoy the feeling of that case being closed. Light is a wave. It's settled. We proved it.

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Because in episode twenty-nine we are going to come back to those two slits, with different equipment, and the result is going to be the most disturbing thing I can tell you.

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The myth of the week is the one we began with: that a wave carries stuff from here to there. It doesn't. A cork on the sea bobs up and down as the waves go by. It doesn't travel with them. The water in a wave that breaks on the coast of Ireland has not come from America. The pattern has.

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And the question to take away. It's a real puzzle, and musicians and physicists answer it differently.

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In the early 1800s a Frenchman, Joseph Fourier, proved something remarkable. Any sound at all, however complicated, a cough, a chord, a door slamming, can be built by adding together pure, simple tones in the right amounts. And any sound can be taken apart into them.

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So play a chord on a piano. Three notes together. Is that one thing or three? The air at your ear is doing just one thing: wiggling in one complicated way. There aren't three separate sounds travelling side by side. And yet you hear three notes, and Fourier says the three are in there.

Tomlin Hawthorne

Are the three notes really in the world? Or is the taking-apart something we do to the world, a way of describing it that happens to suit the way ears are built? Keep it in mind. When we get to quantum mechanics, somebody is going to ask exactly this question about a cat.

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Your challenge. Two things to hear.

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One. Find a tone generator. There are free ones for any phone, and websites that do it. Play a tone of 440. Then, on a second device, play 442 at the same time. You won't hear two notes. You'll hear one note that swells and fades, twice a second, a slow throb. That's the two waves drifting in and out of step. Hump on hump, then hump on dip. You're hearing interference. Piano tuners use exactly this, which takes us back to episode three.

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Two. Take a wine glass, a decent thin one. Wet your finger and run it slowly round the rim. It'll sing. That's a standing wave in the glass itself, the rim flexing in a fixed pattern. Put a little water in and the note drops.

Tomlin Hawthorne

That's the end of Part Two. We've built Newton's universe and found the cracks in the idea that it can be predicted. Next time we start Part Three, which is about the one question that I think matters most in all of physics. Why does time only go one way? And we begin with a Roman poet, a shaft of sunlight, and some dust.