I've often seen the DeBroglie wave illustrated by a two dimensional surface as a standing wave, but then the 'electron cloud' surrounding an atom is hardly two dimensional and furthermore held to the uncertainty principle. So I'm having trouble visualizing how electron 'shells' or 'clouds' might appear from a DeBroglie perspective. Do DeBroglie waves actually propagate around the nucleus of an atom as a spherical standing wave?

More generally I'm having trouble visualizing or conceiving standing waves with dimensionality any higher than two for any system. Do standing waves exist in any physical system along a spherical, closed surface?

As a hypothetical example I can imagine a perfectly spherical, gravitating body (a planet) covered in an ocean with no land mass, and no atmosphere. On such a planet imagine a comet striking the ocean. Waves would radiate outward in circular rings and eventually interfere with themselves, but I can't see that they would ever be able to reinforce as standing waves by any means. Doesn't the geometry of a sphere forbid standing waves?

I'm not even sure how I would approach this from a mathematical analysis.

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    $\begingroup$ The wave equation can be separated into a product of angular and radial components. The angular functions are the so-called spherical surface harmonics and are related to the (associated) Legendre polynomials (see Stratton: Electromagnetic Theory, page 403). These will give you the surface waves you are asking about. $\endgroup$ – hyportnex Apr 8 '17 at 21:49
  • $\begingroup$ Here is a paper about the standing waves on the Moon. digitalcommons.unl.edu/cgi/… $\endgroup$ – Farcher Apr 8 '17 at 22:31
  • $\begingroup$ It is worth noting that the same mathematical techniques used to find the associated Legendre polynomials that @hyportnex refers to for the angular partition of a sphere are also used to find the standing wave shapes of drumheads (which have the advantage of being relatively easy to exhibit in a practical demonstration). $\endgroup$ – dmckee --- ex-moderator kitten Apr 9 '17 at 0:11

In my opinion (for the hypothetical example), as the waves radiate outward from the comet strike point, they will eventually reach the opposite end of the globe opposite the strike point and intersect there synchronically, then radiate again outward, reaching the first point source, then back again, so what would be needed to create a standing wave is for half of the circumference of the globe to be an integer multiple of the wavelength.

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