What is the shape of the surface of the water in the animation below?

Clearly, the dots that compose the surface are following a sinusoidal path. The curve isn't a simple sine wave, since the peaks of the waves curve much more sharply than the troughs. Neither is it

$$y = \left |\sin (\theta) \right |$$

Since that has cusps at the zero crossings of the sine wave.

enter image description here

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    $\begingroup$ I believe that it is a trochoid. That is, the curve described by the motion of a fixed point on a moving wheel (not on the boundary). $\endgroup$ – lulu Jan 14 '17 at 14:56
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    $\begingroup$ Here is a reference. $\endgroup$ – lulu Jan 14 '17 at 14:57
  • $\begingroup$ You should make that an answer. $\endgroup$ – marty cohen Jan 14 '17 at 15:11
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    $\begingroup$ @lulu, to be precise, that's a curtate trochoid. $\endgroup$ – J. M. is a poor mathematician Jan 14 '17 at 15:20
  • $\begingroup$ It's not a common trochoid, since those have cusps. It does indeed appear to be a curtate trochoid. @J.M.isn'tamathematician, can you post that as an answer so I can accept it? $\endgroup$ – Duncan C Jan 14 '17 at 15:27

Experimental evidence suggests that the curve is a sort of trochoid. Here is a reference.

Specifically, it looks like the trajectory of a point in the interior of a disk which is rolling along a line, hence a "curtate trochoid" (N.B. personally, I'd have just called it a trochoid, but I think the crowd has it right here).

Perhaps there is a variational argument which would lead to this conclusion. That would be very interesting, but if there is such a line of reasoning, I am unaware of it.

Here is a derivation of the form, derived from fluid dynamics. I have not reviewed it, but it seems directly relevant.

  • $\begingroup$ In the illustration I posted, the curve is formed by points rotating on circles that don't move laterally at all. (see the orange circles in the animation). The common trochoid appears to be a shape formed when the circle rolls along a surface like a bicycle wheel (neither slipping nor moving faster than a wheel would while rolling along a surface.) $\endgroup$ – Duncan C Jan 14 '17 at 15:50
  • $\begingroup$ Quite so. And the trochoids relevant here are also made by looking at moving wheels. I just edited my post to include a reference to a derivation of the trochoidal form. The math looks straight forward enough, but I don't know enough about fluid dynamics to see where the starting equation comes from. $\endgroup$ – lulu Jan 14 '17 at 15:53
  • $\begingroup$ I just realized something. The surface of water waves forms a trochoid, but a trochoid is not a function - at least a prolate trochoid is not, since it loops back on itself. $\endgroup$ – Duncan C Jan 17 '17 at 14:16
  • $\begingroup$ But this one is curtate, so there is no problem with that. $\endgroup$ – lulu Jan 17 '17 at 14:33

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