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.