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It's not a sine wave. It's the result of the viscous fluid (the atmosphere) lagging behind the spinning container (the planet)*. The shape is a ... mathematical coincidence. Rather, our perception of some meaningful pattern is the coincidence. The shape is just the math and physics.
* Or just lagging compared to the parts of the atmosphere that are spinning at different rates. Similar phenomena have been observed in the eyes of terrestrial hurricanes. I provided a good link above.
edit: No, I haven't recently returned from a vacation on Saturn, Uranus (or anyone else's for that matter... ha ha)
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Be Just and Fear Not.
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