In chapter 4 we found the one rule for what two waves do when they share a spot: their heights simply add. Adding is friendly, and a child can do it. But it is only half of what you can do with two waves at once. There is a second thing, and it looks, at first, like a slightly odd request. Instead of adding wave A's height to wave B's, at every instant, multiply them. Height times height, moment by moment.
Why on earth would anyone want to multiply two waves? Because, quietly, it is the move behind a huge amount of this book. It is how AM presses a voice onto a carrier. It is how a receiver slides a station down the dial to a comfortable speed. It is how the un-adding machine of the last appendix reads a wave's secret recipe. All three are the same act of multiplication, wearing three different coats, and once you have seen it once you will spot it everywhere.
The usual promise for these appendices holds here too: pictures only, no algebra, nothing to memorize, and nothing later in the book leans on it. Just one lovely fact, and a playground to feel it in.
Adding keeps, multiplying makes#
Here is the surprise, right up front, because everything else follows from it. When you add two different waves, nothing new is ever born. A low note plus a high note is just the low note and the high note, sounding together, a little chord your ear can pick straight back apart. That is the whole message of the recipe chapter: a sum is only its ingredients, stacked. You can add waves all day and never invent a speed that was not already in the room.
Multiplying breaks that rule wide open. Multiply two waves and the two you started with vanish, and in their place appear two brand-new waves that were nowhere in what you fed in: one wiggling at the sum of the two speeds, faster than either, and one at their difference, slower than either. Two speeds go in; two different speeds come out. That is the entire secret of this appendix. Everything below is just watching it happen.
You can already see the difference between the two operations in a single picture. Both panels start from the very same pair of waves, one slow and one fast.
Add them, and the fast wave just rides up and down on the slow one: both are still plainly there.
Multiply them, and the slow wave pinches the fast one to nothing wherever it crosses zero: something new.
On the left, adding lets the fast wave ride up and down on the slow swell like a boat lifting on a wave. Both are still plainly there. On the right, multiplying makes the slow wave squeeze the fast one, choking it to nothing wherever the slow wave passes through zero. Same two ingredients; the adding stacks them, the multiplying makes a shape neither one had on its own.
Where the two new waves hide#
Let's watch the two new waves appear, and learn to point at them. In the playground below, wave A holds a steady speed and you set wave B's. The bottom band is their product, drawn point by point. Look at how it is built: wherever both waves are up, up times up is up, so the product rides above the line. Wherever both are down, a negative times a negative is up again. And wherever the two disagree, one up and one down, the product dips below. (Hold on to that up-times-up, down-times-down business; it comes back with a vengeance.)
The product looks like a tangle. But it is secretly just two clean waves added together, and the figure draws the slower of them, the difference wave, in amber, straight through the middle of the mess. The busy dark product is nothing more than that calm amber wave with a second, faster wave, the sum, wiggling along it.
Now for the one thing to feel in your own hands. Drag wave B's speed slowly toward wave A's, and keep your eye on the amber difference wave. As the two speeds close in, the amber wave stretches out and slows down, because the difference between them is shrinking. Bring them to exactly the same speed and the difference is zero: the amber wave stops wiggling altogether and freezes into a flat, steady lift. A wave of speed zero, after all, is just a level, a wave that never gets around to wiggling. Keep that frozen lift in mind. It is about to explain two different chapters.
The slow wave is the whole point#
That slow difference wave is not a curiosity. It is one of the most useful things in all of radio, and you have met it before under a workaday name. Back in chapter 14, a receiver caught a station wiggling millions of times a second, far too fast to handle, and slid it down to one easy speed by "combining" it with a wave the radio makes for itself. Now you know precisely what "combine" meant: multiply. The station times the radio's homemade wave makes a difference wave, and if the radio picks its own speed just so, that difference lands exactly on the one comfortable speed the rest of the machine is built around. Every station in the sky, multiplied down to the same little workbench.
Engineers call that slow difference wave the beat , and multiplying two waves to make it is called mixing. You can even hear it. Two guitars tuned a hair apart give you the sound of this very wave: the slow wah-wah-wah swelling and fading is the beat between them, throbbing at exactly the difference of their two pitches. A mixer just does that trick on purpose, and hands you the slow wave as a real signal to keep and use.
And when the speeds match exactly#
Go back to that frozen amber lift, when the two speeds matched. Something quiet and wonderful happens there. With the speeds equal, the product never dips below the line at all: up-times-up and down-times-down, everywhere, so the whole product humps up above zero, and its average comes out a solid positive number instead of the nothing you would otherwise get. That is a detector.
Which is the whole trick of the un-adding machine from the last appendix, now with its lid off. Hand it a mystery wave, multiply by a clean probe, and average. If the probe's speed matches an ingredient hiding in the wave, the product lifts and the average rings out loud. If it matches nothing, the difference wave just wiggles above and below the line in equal measure and averages away to zero. Multiply and average: a matched multiply makes a steady lift, and a mismatched one makes a wave that cancels itself. That is why the un-adding machine can pick a single ingredient out of a lump, and it is the same up-times-up, down-times-down you were told to hold on to.
One last coat: AM's hidden chord#
Now push wave B the other way, down to a slow crawl beside A. The two new waves, the sum and the difference, settle in close on either side of A's speed, one a touch faster, one a touch slower. A little chord of three: the original carrier and two near companions. Those companions have a name you met in chapter 5, the sidebands , and here they are being born, live, from a single multiplication.
Because that is all AM ever was. Chapter 5 said the message multiplies the carrier's height, and a multiplication of two waves always makes that pair of new waves flanking the carrier. The voice you hear on an AM station is riding in those two conjured companions, called into being the instant the message and the carrier were multiplied together. The chord AM chapter promised you was hiding in the wave is exactly this: the two children of a multiplication.
The one thing to remember#
Adding two waves keeps them both, and you can always un-hear the chord. Multiplying two waves throws them away and makes two new ones, at the sum and the difference of their speeds. Slide the two speeds together and the difference wave slows to a crawl, then to a standstill, a steady lift. Nearly everything a radio does to move a signal around, pressing a voice on, sliding a station down, reading a wave's recipe, is that one act: multiply, and reach for the difference. Adding built the world of chapter 4. Multiplying builds most of the rest of the book.