The most repetitive echo

I listened recently to a very interesting interview with Professor Trevor Cox about the longest echo ever recorded, and other acoustic topics. Near the end of the interview (around 42:44), Professor Cox mentioned that he was still looking for “the most repetitive echo” — the place where you can shout “ECHO!” and hear “Echo! Echo, echo, …” repeated the greatest number of times.

This prompted me to write an email to the radio programme about how one might generate such a repetitive echo. Then, I decided that it might be worth putting the email here, for you to read. So what follows is a slightly edited version of that email, with added pictures.

I wonder if Professor Cox has considered looking for chambers in the shape of prolate spheroids. (He probably has; he seems like a clever guy.) A prolate spheroid is what you get if you rotate an ellipse around its major axis (which is the line cutting it in half length-wise).

Prolate Spheroid by Sam Derbyshire is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.
Creative Commons License
Prolate Spheroid by Sam Derbyshire is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.

The ellipse has two foci on the major axis, and these have the property that if you draw a straight line from one focus in any direction, bounce it off the ellipse wherever it hits (as if it’s light being reflected, or sound being echoed), and continue it in a straight line from there, it’ll pass through the other focus. What’s more, the total distance — from focus to ellipse to focus — is constant, regardless of which direction from the first focus was chosen. Ellipse Properties Showing Construction with string

What this means is that if you set off a starting pistol at one focus of a prolate spheroid, the sound will travel out from it in all directions, bounce of the walls, and arrive at the other focus from all directions at the same time, making the first echo there very nearly as loud as if your ear (or, more sensibly, your recording equipment, if it won’t be damaged) was right next to the starting pistol.

Then, of course, the sound will travel out from the second focus in all directions, bounce off the walls again, and arrive back at the starting pistol at the first focus, still nearly as loud as the original shot. With the echo bouncing back and forth between the foci like this, I reckon it’d be a contender for the most repetitive echo, assuming such a chamber exists somewhere.

Actually, a similar principle is demonstrated by a pair of whisper dishes outside a museum in Dunedin. (It’s been a while since I visited that museum, though, so I can’t recall its name, which has probably changed by now.) They look like a pair of satellite dishes pointing directly at each other horizontally, quite some distance apart. If you stand at one focus and speak in a normal voice, and your friend stands at the other focus and listens, they’ll be able to hear you much more clearly than you might expect, given the distance between you.

This example in Dunedin might actually be a pair of paraboloidal dishes. Such a dish would reflect sound from the focus along parallel lines heading towards the other dish, which would then reflect it back towards its own focus. This would mean that the dishes could be set up without worrying about getting them precisely the right distance apart (though they’d still need to be pointing fairly precisely towards each other).

Even if these Dunedin dishes are the ends of a prolate spheroid, most of the spheroid is missing, so I wouldn’t expect the repeated echoing back and forth to work as I described above.


One thought on “The most repetitive echo

  1. Still, not the place to choose for private confidences, eh?
    Like the tourist mother who whispered to her little son that he could probably have a surreptitious pee among the stones – at the base of an amphitheatre.

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