Yes, it's a vibration of an elastic axisymmetric thin shell (surface of revolution, as you meantioned). Going deeper into the theory, to know or characterize the sound it makes one should have a model of free or forced vibration and find the eigenvalues of the resulting linear system. That system is typically (K-omega^2 M) U = F, where K is the stiffness matrix of the material, M the mass matrix, U the displacement matrix, F the force matrix and omega the frequency.

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.