The @mathematicsprof tweet above links to a paper, by Carolyn Gordon and David Webb and published in American Scientist in 1996, that’s about one of those questions that’s both mathematically interesting and of obvious everyday interest. The question was originally put, in nice compact and real-world-relevant form, in 1966 by Mark Kac: can one hear the shape of a drum?
At first glance the answer may seem, “of course” — you can hear the difference between musical instruments by listening to them. You might need experience, but, after all, you’re not going to confuse a bass drum from a bongo even if you haven’t been in the music store much. At second glance, why would Kac bother asking the question if the answer were obvious? He didn’t need the attention. He had, among other things, his work in ferromagnetism to be proud of (and I should write about that some.) And could you tell one bass drum from another?
The question ties into what’s known as “spectral theory”: given a complicated bundle of information what can you say about the source? One metaphorical inspiration here is studying the spectrum of a burning compound: the wavelengths of light emitted by it give you information about what elements go into the compound, and what their relative abundances are.
The sound of a drum is going to be a potentially complicated set of sound waves produced by the drum’s membrane itself oscillating. That membrane oscillation is going to depend, among other things, on the shape of the membrane, and that’s why we might suppose that we could tell what the shape of the drum is by the sound it makes when struck. But then it might also be that multiple different shapes could produce the exact same sound.
It took to about 1990 to get a definite answer; Gordon and Webb, along with Scott Wolpert, showed that you can get different-shaped drums that sound the same, and very nicely showed an example. In the linked article, Gordon and Webb describe some of the history of the problem and how they worked out a solution. It does require some technical terms that maybe even re-reading several times won’t help you parse, but if you’re willing to just move on past a paragraph that looks like jargon to the rest I believe you’ll find some interesting stuff out, for example, whether you could at least hear the area of a drum, even if you can’t tell what the shape is.
nebusresearch 
When I attended college in the early 70’s, somebody (probably Mark Kac) did a presentation at the University on “Can You Hear the Shape of a Drum?” I had one professor who was especially intrigued and shared tidbits of the presentation with the class afterwards. It is so nice to get this update. Thank you!
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Oh, I’m interested to hear about the history of the problem. Also I’m glad you were able to learn something about how the problem’s turned out.
It’s curious what presentations and seminars will last. One that I remember from about 1995 or 1996 was a guy who was modeling viscous fluid flows, and to give this study a nice punchy real-world application, brought up the Sherwin-Williams “Cover The Earth” logo. If you suppose the paint being dropped on the sphere to be about average, then, does the way it’s dripping tell you the scale of the logo — how big the imitation Earth there would be? And, yes: if I’m not remembering this incorrectly, the logo implies a model Earth that’s the size of a grapefruit.
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I read most of it. These guys are very good at explaining stuff. I learned quite a bit. When I was a student I avoided differential equations like the plague !
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They did explain it quite well, yes. I think it helps they were writing for an audience that could be supposed to know a fair bit of mathematics; the less background stuff you have to explain the easier it is to get to the interesting point.
I was a poor student the first time I took a differential equations course (that is, the kind for mathematics majors) and that hobbled me through a lot of my work, sad to say.
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