Lightning Channel Length from the Duration of the Rumble

Why does thunder rumble instead of banging? Because a lightning channel is kilometres long, the whole of it heats and expands at essentially the same instant, and you are not the same distance from all of it. Sound from the nearest point of the channel reaches you first; sound from the farthest point reaches you last. The roll you hear is the flash being delivered to your ear a slice at a time, and its duration multiplied by the speed of sound is a length.

Three seconds of rumble is about a kilometre. Ten seconds is about three and a third. Those are useful numbers to carry, because they immediately tell you that the thirty-second rumbles people describe are not thirty seconds' worth of channel.

The answer is a lower bound, and the reason is geometry rather than measurement error. What the duration actually measures is the spread in range across the channel — how much further away the far end is than the near end. A channel pointing directly toward or away from you converts its entire length into duration. A channel lying across your line of sight, every part of it at the same distance, produces almost no duration at all, and this page will report a long flash as a short one. Real discharges are branched, tilted, and partly horizontal inside the cloud, so the honest reading of the answer is "at least this much lightning, oriented at least this much along my line of sight". Two observers standing a kilometre apart will hear the same flash rumble for different lengths of time, and both are measuring correctly.

Two things stretch the roll that are not channel at all. The first is echoes: thunder reflects off hillsides, buildings, and — importantly — off the cloud base and off atmospheric layers, so sound keeps arriving after the direct wave has finished. In mountain country the reverberation can double or triple the apparent duration. The second is that most cloud-to-ground flashes are not one stroke but three or four, travelling down the same channel over a few tenths of a second, each with its own acoustic signature. Both effects push the computed length high, which is worth setting against the geometric effect that pushes it low.

Why thunder sounds the way it does at different ranges follows from the same picture. Close by, you are near one end of the channel and the arrival times are compressed, so you get a sharp crack — and high frequencies survive a short path, which is what makes it a crack rather than a boom. Far away, the arrival times are spread out and the atmosphere has absorbed the high frequencies preferentially, so the same physical event arrives as a low, drawn-out roll. The change in character with distance is therefore a real cue, and a more robust one than most people expect.

The underlying acoustics are worth a sentence. The return stroke heats the channel to something like 30,000 K in a few microseconds, which is roughly five times the surface temperature of the sun, and the resulting pressure — tens of atmospheres in a channel a few centimetres across — drives a genuine shock wave. That shock decays into an ordinary sound wave within a few metres, which is why the far-field problem is plain acoustics and why a simple speed-times-time relation works at all. What survives to your ear from a kilometre away is a sound wave, not a shock, and it obeys the same $v = 331.3 + 0.606\,T_C$ as everything else.