If you phrase it that way “surface of revolution” describes bells much better than “solid of revolution”.
I think “solid of revolution” is the better term, though. That allows you to model varying wall thickness. You’d need more than a function u:ℝ->ℝ to do that.
Also, I don’t a priori see why the shape of a bell would be limited to such functions. If the edge of a bell bends inwards into the bell and goes up a bit, I think I would still call it a bell, especially if it somewhat sounds like one.
Finally, rotational symmetry isn’t required for a bell. Your typical cow bell, for example, is elliptical or even rectangular (https://en.wikipedia.org/wiki/Cowbell_(instrument))
⇒ the design space is a lot larger than you indicate.
> That allows you to model varying wall thickness
That's mentioned in the first chapter of this book [0].
[0] The Finite Element Analysis of Shells – Fundamentals, Chapelle & Bathe
EDIT: I'm aware it is not a surface, but the notion is closer than that of a solid of revolution.
But surface, by definition, has no concept of thickness, right? Surely there's a better term?
> 'solid' would mean its interior is completely filled.
I think the easiest way to define a bell would be the difference of two solids of revolution (front surface, back surface). What do you call the difference of two solids of revolution?
I disagree. Mathematically, I wouldn’t know how to define a solid other than “anything with a volume”.
Certainly, a bell, when defined as a surface of revolution, doesn’t have an interior in a mathematical sense (https://mathworld.wolfram.com/Interior.html), does it?
There’s the convex closure, but that would, for the prototypical bell, add volume ‘outside’ the bell, too.
Also, I think varying wall thickness in bells for centuries was the only known way to tune them. https://en.wikipedia.org/wiki/Pieter_and_François_Hemony#Lif...:
“When struck, a bell produces a number of partials which, if imprecisely tuned, can create an unpleasant sound and which prevents it from harmonizing in accordance with other bells. To address this problem, the Hemony brothers gave their bells a particular profile and thickened it in certain places. The bells were then tuned by hollowing ridges from specific parts of the inner wall until the first few partials were acceptably in tune.”
(Aside: if you want your work to be used for centuries, becoming a professional bell caster seems a good bet)
Actually, if you normalize the bell height to 1, you can take u from 0 to 1 as the inner surface (tracing down from the top), and u from 1 to 2 as the outer (tracing back up from the bottom). I'm not sure that's mathematically convenient, but it does work. (You could also use a periodic function on 0..pi for inner and pi..tau for outer, which might be nicer math-wise and also allows you to say that the height (pi) is half of whatever the function's period is.)
Now, to find the displacement given the eigenvalues (spectrum) is possible in a finite-dimensional setting, in infinite dimensions it's trickier. I'm sure there are approximation results of this kind, but not if there is an exact way.