Two pebbles
To find out what happens where waves cross paths, back to the pond we go — but this time, drop in two pebbles, a hand's width apart. Two families of rings glide outward, and here is the first surprise: each one spreads as if the other did not exist. The rings slide straight through each other and carry on, unhurried and unharmed — no crash, no tangle, no pile-up. Waves are very polite that way. Cross two flashlight beams in a dark room and you'll see the same manners: light is a wave too, and the beams pass through one another without so much as a flicker.
But politeness is only half the story. Pick one spot on the pond — one little patch of water where a ring from the left and a ring from the right happen to arrive at the same moment. Both waves are telling that patch of water what to do, at the same time. And the water there does something neither pebble ordered on its own.
The only rule: add them up
What it does could not be simpler. At every instant, at every spot, the water's height is wave A's height plus wave B's height. That's it. That's the whole rule. If one wave says "up two" and the other says "up one", the water rises three. If one says "up two" and the other says "down two", they add to zero — and the water lies perfectly flat, even while two waves are racing straight through it.
No averaging, no fighting, no taking turns. Just adding — the kind you learned before you learned to tie your shoes. When waves meet, their heights simply add. Try it below: two waves, and underneath them, the wave they add up to.
Perfect agreement, perfect argument
The figure opens on a puzzle: two perfectly healthy waves, and underneath them a sum that is a flat line. Let's take the two buttons in order and see why.
Press In step (0°) first. Now wave B starts at exactly the same moment as wave A. Every up meets an up, every down meets a down, and the sum is one proud wave twice as tall. Engineers say the two waves are in phase (and call the doubling constructive interference, which is a grand name for "they helped each other").
Now press Half a beat late (180°). Wave B is the very same wave, just half a wiggle behind — and remember the little fact we filed away at the end of chapter 2: two waves 180° apart never agree about anything. Whenever A says "up two", B says "down two". Up two plus down two is zero — at every spot, at every instant, forever. Two perfectly good waves, and the sum is nothing at all. This is the superpower we promised: with nothing but a half-beat delay, one wave can erase another completely.
In step — the two waves sit right on top of each other, and their sum (amber) is twice as tall
Half a beat late — every up meets a down, and the sum (amber) is dead flat
And erasing waves is not just a party trick — you may have worn it on your head. Inside a pair of noise-canceling headphones sits a tiny microphone that listens to the roar of the airplane or the rumble of the bus. The instant it hears the noise, the little speaker by your ear plays the very same wave, half a beat late. Roar plus anti-roar adds to nothing, right inside your ear. The headphones don't block the sound. They cancel it. Silence, manufactured out of more sound.
Dead spots and sweet spots
Cancellation doesn't always wait to be invited, though. Sometimes it just happens — to you.
A wave can often reach you two ways at once: straight from where it was made, and bounced off something on the way — a wall, a ceiling, a hillside. The bounced copy is the same wave, but its path is longer, so it arrives a little late. And you now know exactly what "a little late" can do. In some spots the two copies land in step and shake hands: extra loud. A step or two away, the bounce shows up close to half a beat behind, the copies argue, and the wave all but disappears.
That is why your Wi-Fi — one of those invisible electromagnetic waves from the last chapter, remember — can mysteriously die in one corner of the kitchen and spring back to life one step to the left, and why a car radio sometimes crackles and fades under a bridge, where bounced copies come at it from every direction. Nothing is broken. The signal is still flooding the room; it is simply canceling itself in that one unlucky spot. (Engineers call this whole business multipath, because the wave took multiple paths.) Dead spots and sweet spots, painted invisibly across every room — all by the same little rule of adding.
Keep it in your pocket
One rule, then, small enough to keep in your pocket: when waves share a spot, their heights simply add. In step, they build each other into something twice as tall. Half a beat late, they erase each other into silence. That's the whole chapter.
It doesn't look like much — but this little rule quietly explains an astonishing amount of what is coming in this book, and near the very end it performs one more magic trick that we won't spoil here. First things first, though. Look at what you now hold: the last chapter handed you the carrier, and this one taught you that waves add. But a perfect, steady hum says nothing at all — so in the next chapter, we finally make a wave talk, by taking the first of the three knobs, amplitude, and wiggling it on purpose.