My first encounter with Spherical Harmonics must have been in the course on electrodynamics, when these intimidatingly looking functions showed up in series expansion of the Coulomb potential and in the multipole expansions of charge distributions. Half a year later, they appeared again, in the solution of the hydrogen problem in the quantum mechanics class. Here, at least, they were used to produce nice figures of electron orbitals.
But I did miss out on one of the most elementary occurrences of Spherical Harmonics.
Spherical Harmonics Yℓm(θ, φ) describe the angular part of the solution to Laplace's equation in spherical coordinates. As such, they are ubiquitous in physical problems with a spherical symmetry. Thus, they describe not only the behaviour of the electron in the hydrogen atom, but also the wobbling deformations of an oscillating, elastic sphere. What sine and cosine are for a one-dimensional, linear string, the Spherical Harmonics are for the surface of sphere.
Deformation modes of a bouncing oil droplet (radius R = 0.765 mm) described by the Spherical Harmonics Y20, Y30 and Y40, as observed with a high-speed camera. From S. Dorbolo et al.: Resonant and rolling droplet, New J. Phys. 10 (2008) 113021.
This has been demonstrated very nicely in a paper by S. Dorbolo, D. Terwagne, N. Vandewalle and T. Gilet just published in the New Journal of Physics, Resonant and rolling droplet. In the experiment, a tiny oil droplet is placed on an oil bath which is set into vertical vibrations to prevent coalescence of the droplet with the bath. The droplet, which at rest would have a spherical form due to surface tension, bounces periodically on the bath. At the right frequencies of the vibrating surface, the droplet oscillates in resonance – and deforms according to spherical harmonics!
A movie (Quicktime, 11.0 MB – someone should explain to the NJP how to upload these movies to YouTube ... ) shows the oscillations of the drop and the corresponding calculations using Spherical Harmonics Yℓm with ℓ = 2, 3, 4 and m = 0. "Magnetic quantum number" m = 0 means rotational symmetry of the wobbling around the vertical axis. For m ≠ 0, deformations are not symmetric with respect to the vertical, and in this case, the droplet starts to move around on the oil bath. This can be seen in a second movie (QuickTime, 5.6 MB).
That's a beautiful example of Spherical Harmonics in action I would like to have known when I was struggling for the first time with multipole expansions!
The first application of Spherical Harmonics to describe the wobbling of a droplet was by Lord Rayleigh, in the appendix of a paper On the Capillary Phenomena of Jets, Proceedings of the Royal Society of London 29 (1879) 71-97. You can check out his calculation - the PDF (2.5 MB) is available for free.