Does a glass of water sing because of the SO(2) symmetry?

real-analysis soft-question functional-analysis representation-theory lie-groups

  1. Well, we can certainly have singing/ringing from non-axisymmetric objects, cf. e.g. this, this Phys.SE posts and links therein. So $SO(2)$ symmetry is not necessary in that sense.

  2. However, according to e.g. Ref. 1, it is apparently a good mathematical model to consider a wineglass as having topology $S^1\times I$, where the interval $I$ has 1 free and 1 fixed endpoint. The circle $S^1$ has $SO(2)$ symmetry and can be analyzed via Fourier series. The interval $I$ typically has no symmetry. The lowest mode has 2 nodes along $S^1$ and 0 nodes along $I$. There are also higher harmonics/overtones.

References:

  1. Jundt, Radu, Fort, Duda, Vach & Fletcher, Vibrational modes of partly filled wine glasses, J. Acoust. Soc. Am. 119 (2006) 3793.

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