The signal is off and running — flung from the antenna, tearing across open space at the speed of light. You might think the hard part is over. It isn't. A radio wave doesn't sail along at full strength until it arrives; it fades, and it fades far faster than your everyday sense of distance expects.
We're used to a shout getting gradually fainter across a field. Radio gets fainter too — but the arithmetic of how is steep enough to shape almost everything about the way radios are built.
Spreading thin
Here's the reason, and it's pure geometry. An antenna usually doesn't fire its wave in a tight beam; it sends it outward in every direction at once, like a balloon of energy swelling away from the source. The energy the antenna pours out is fixed. But the balloon it's smeared across only grows.
Picture that energy painted onto the surface of the balloon. While the balloon is small — close to the antenna — the paint is thick and bright. Let it double in size and the very same paint must now cover four times the area, so it ends up only a quarter as thick. Triple the distance, and it's stretched over nine times the area. None of the energy is lost; it's just spread thinner and thinner over a bigger and bigger surface — and a ball's skin grows with the square of its size, so each doubling of the distance quadruples the area the light has to cover.
One honest aside before we go on: nearly every wave we've drawn in this book has been a 1-D wiggle on a line or a 2-D ripple on a page — flattened to keep it easy to picture. A real radio wave fills all three dimensions of space at once, and here, at last, is the first place that difference truly bites.
Counting patches on a curved ball is fiddly, so picture just one small patch of that sphere, flattened out. Move twice as far from the source and the same light must stretch across a square twice as wide each way — four little squares where there was one; three times as far, nine.
The inverse-square law
That gives us one of the most important rules in all of radio, with a name to match its reach: the inverse-square law. Move twice as far from a transmitter and the signal isn't half as strong — it's a quarter. Three times as far, a ninth. Ten times as far — ten times ten is a hundred — a hundredth. Distance doesn't nibble at a radio signal; it devours it.
Drag the receiver below toward the transmitter and away again, and watch how violently the bars react. The first few steps away cost you dearly; out in the faint fringes, the signal is a ghost of what it was right beside the mast.
drag the receiver · signal here 83% — the phone rounds that to 4 bars
Why your bars drop
This is the law behind the little story your phone tells all day. Those bars in the corner are, more or less, a readout of how much signal is reaching you — and as you wander away from the nearest tower, that signal falls off as the square of the distance. It's why stepping a few feet can rescue a dropped call, and why cities bristle with towers spaced just close enough to keep you covered. It's even why a space probe billions of miles away — its radio whisper stretched almost to nothing — needs a receiving dish the size of a house just to be heard.
Out in the wide open, the quarter-for-every-doubling rule holds cleanly. In the real clutter of walls, hills, and weather the fall-off is usually even harsher — but that's a story for a later page.
So a signal arrives faint — often almost unimaginably faint — having spent nearly all its strength on the simple act of spreading out. And here is what makes faintness dangerous: a signal never arrives into perfect silence. You've already met the enemy — the crackle on a weak AM station, the static between FM channels. That hiss is everywhere: the air, the sky, even the receiver's own electronics simmer with it, and a signal that sinks below it is lost for good. Where all that hiss actually comes from — and just how faint a whisper a good receiver can still haul out of it — is where we head next.