Every Path Takes the Same Time
Sound leaves one focus of an ellipse in every direction, bounces once, and every single copy arrives at the other focus having travelled exactly the same distance. Step five centimetres off it and half the effect is gone.
What it is
An elliptical hall with a click going off at one focus. Every line is one direction the sound went, drawn out to the same total distance.
They all end up in the same place. That isn’t a drawing decision, it’s what an ellipse is.
Drag the source away from the focus and watch the fan stop agreeing with itself. Listen to what happens to the click while you do it.
How it works
An ellipse is the set of points whose two distances from the foci add up to a fixed number,
and that number is 2a, twice the long semi-axis. So a ray from one focus, off the wall, to
the other focus, travels 2a whichever direction it set out in. Here that’s 18 metres,
for all 360 directions, to six decimal places.
Same distance means same time, so all the copies land together. They add up instead of smearing, and a whisper survives the trip.
The sound isn’t an imitation of that. The page builds the room’s impulse response out of the actual path lengths, one delayed copy per ray, and convolves the click through it. When the copies coincide the response is a single spike and you hear a click. When they do not, it’s a smear and you hear a wash.
Which makes the interesting number the spread between the first arrival and the last:
| standing off the focus | spread | first to last | what survives at 2.2 kHz |
|---|---|---|---|
| 0 cm | 0.0 mm | 0.000 ms | 100% |
| 5 cm | 100 mm | 0.292 ms | 45% |
| 20 cm | 400 mm | 1.166 ms | 12% |
| 50 cm | 1000 mm | 2.915 ms | 5% |
| 110 cm | 2200 mm | 6.414 ms | 1% |
| 250 cm | 5000 mm | 14.577 ms | 0% |
What surprised me
Five centimetres costs more than half of it. I expected the focus to be forgiving, because an ellipse is a smooth thing and nothing about it looks sharp. It isn’t forgiving at all. Stand a hand’s breadth off and the arrivals already smear across 0.29 milliseconds, which at 2.2 kHz is most of a period, and the sum has lost more than half its height.
That’s the difference between a whispering gallery being a famous trick and being a thing nobody would ever notice. It only works if you put your ear in the right place, and the right place is small.
The spread is exactly twice how far off you stand. Every row of that table: 5 cm gives 100 mm, 20 gives 400, 50 gives 1000, 110 gives 2200, 250 gives 5000. A factor of two, holding across a fifty-fold range, with no rounding drama anywhere in it.
I went looking for that after noticing the first two rows and it held everywhere I tried. It falls out of the geometry rather than out of the model, which is the good kind of number: the page didn’t put it there and can’t avoid it.
It’s a whispering gallery and not a rumbling one. The last column is frequency dependent, and that’s the whole reason the effect has the name it does. A 0.29 ms spread is most of a period at 2.2 kHz and almost nothing at 250 Hz, so a whisper survives a sloppy position that would leave a low voice untouched but also unremarkable. The trick needs the high frequencies to be worth noticing, and the high frequencies are exactly what a whisper is made of.
What I would do next
Let it bounce more than once. Everything here is a single reflection, which is the clean case and the one the geometry guarantees. A real hall keeps going, and the second and third bounces from a focus do something worth seeing: they come back to the focus too, alternating between them, so the sound rattles between the two points rather than arriving once.
The other thing is the shape. An ellipse is the only closed curve with this property, and the interesting version of that claim is watching it fail: nudge the wall out of shape and the convergence should fall apart faster than the amount you nudged it, which is a measurement I would like to have and do not.