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.
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.