Strum three guitar strings at once. Three strings, three notes, one pleasant chord. Now think about the air on its way to your ear. Back in chapter 4 you learned the one rule of meeting waves: their heights simply add. So the three ringing strings don't send three polite, separate wiggles. They add up into a single lumpy one, and that single wiggle is all your eardrum ever feels. The air at your ear can only be at one pressure at a time.
And yet you hear three notes. Clearly, separately, three. Your ear takes that one lumpy wiggle and, somehow, un-adds it: it splits the stack back into the pure tones it was made of. A feat your head performs every moment of every day without being asked.
One wiggle, three notes#
That little marvel points at something enormous. If a chord's single wiggle can be split back into pure tones, maybe other wiggles can too. Two hundred years ago a French mathematician named Joseph Fourier showed just how far the idea goes, and the answer is: all the way. Every wave that repeats, no matter how lumpy, is a stack of plain sine waves, added together.
Which means any repeating wave can be written down like a recipe: this much of this sine, plus a pinch of that one, plus a dash of a third. The ingredients aren't random, either. They are the wave's own base rate of repeating (the fundamental) and its multiples: twice as fast, three times, four times. Those multiples are called harmonics , and they are the only ingredients any repeating wave ever needs.
A claim that bold deserves a hard test. So let's pick the least sine-like wave we can imagine: a square wave, all flat tops and knife-edge corners, the very opposite of round, and try to build it out of nothing but sines. The recipe turns out to be almost comically simple: take a sine, add a third as much of one wiggling three times as fast, a fifth as much of one at five times, a seventh at seven times, and keep going. Drag the slider and watch the stack sharpen.
One sine: perfectly smooth, and not square at all.
One sine is hopeless. Three start to square the shoulders. Eight are square enough to fool anyone. And notice which ingredients did the sharpening: every step toward crisper corners came from adding a faster sine. That's a rule worth keeping: sharp corners are expensive, and the currency is fast ingredients. A perfectly square corner would take ingredients faster than any limit you set, which is why nothing in the real world is ever perfectly square. Tuck that away; it comes back in the next chapter wearing a disguise.
The recipe card, drawn as bars#
The bar chart under the slider deserves its own name. Engineers call a wave's list of ingredients its spectrum : which frequencies it contains, and how much of each. You met that word back in chapter 3, where the spectrum was a highway of stations, each in its own frequency lane. This is the very same picture, the same axis of slow-to-fast, only zoomed in: instead of many stations sharing the dial, it's one signal's ingredients sharing the chart. A spectrum is just a recipe card, drawn as bars.
And now the sounds from last chapter stop being mysterious scribbles. Each one is a recipe, and each recipe card explains its waveform at a glance.
Whistle
one ingredient: a pure sine has nothing else in it
Hum — “mmm”
a small stack: the pitch, plus fading multiples of it
Hiss — “sss”
grass everywhere: every ingredient at once, none of them loud
The hiss card is the one to stare at. A hiss has no favorite frequency, so its recipe is a little of everything: grass growing right across the axis. That is what the noise floor is made of, and it's why noise is so hard to escape. You can dodge a station by moving to another lane; you cannot dodge something that lives in every lane at once.
Watching a recipe live#
Recipe cards are snapshots, but sounds move, so engineers built a camera that films the recipe itself. It is called a spectrogram: time slides across the picture, pitch runs up it, and brightness shows how much of each ingredient the sound contains at that instant. A pure whistle films as one thin, steady line. A buzzy hum films as a ladder of harmonics. And a siren, which has no fixed recipe at all, films as a snake, its one ingredient forever on the move. (How does the camera do that? How does anything un-add a wave? There is a machine inside, and it is lovely: Appendix B, the un-adding machine, whenever you're curious.)
And you don't have to take the camera's word for any of it. There is a microphone button under the picture. Press it, pucker up, and whistle: one bright line, because a whistle is a single pure ingredient. Now hum low and steady, and watch the ladder of harmonics stack up out of your own throat. You have been a walking stack of sine waves all along; this is just the first time you get to watch.
A shelf, not a line#
Here is where the recipe stops being a party piece and starts running the world. A recipe takes up width. A rumbly voice needs ingredients from a low growl up to around three thousand wiggles per second; that span of the frequency axis is the signal's bandwidth , the width of shelf its recipe occupies. A phone call keeps only that 3 kHz shelf, which is why phone voices sound thin but perfectly understandable. Music, with its cymbals and overtones, wants a shelf several times wider. Video, painting millions of dots many times a second, wants wider still.
Now recall chapter 5's parting secret: an AM signal is secretly a chord, the carrier plus faint sidebands just above and below it. This chapter is the payoff of that promise. When a carrier is made to wiggle in time with a message, the message's whole recipe gets smeared into a band of ingredients around the carrier's frequency. A broadcasting station is therefore not a thin line on the dial. It is a shelf: the carrier in the middle, the message spread out on both sides, the shelf as wide as the message's recipe demands.
And that is the real reason the radio dial is divided the way it is. Lanes on the invisible highway can't be spaced any old way; each lane must be wide enough for a whole shelf, or neighbors overlap and garble each other. An AM voice is the tidy case: a lane about twice as wide as the speech comfortably holds it. FM asks for far more room, and here is the subtlety: its music is nowhere near that wide. FM swings its frequency much further than the bare message needs, and that swing is exactly what buys the static-resistance of chapter 6. FM stations end up spaced 200 kHz apart. Wi-Fi channels, carrying torrents of data, are about 20 MHz wide, a hundred times wider again. Bandwidth is the true currency of radio: the more a signal has to say each second (or the more ruggedness it buys), the wider the shelf it must rent.
A list of numbers#
Step back and look at what a recipe card actually is. A handful of bars; each bar a frequency, an amount, and (to pin the wave down exactly) a starting point too. In other words: a short list of numbers that describes a wave completely. Not a sketch of it, not an impression: the wave itself, exact, rebuildable any time from the list.
Hold on to that thought, because it is a clue to something big. If a wave can become numbers, then a wave can be copied the way numbers are copied: perfectly, every time, forever, without wearing out. In the next chapter we stop admiring that idea and do it for real: we take a living, wiggling wave and turn it into numbers.