# What mathematical shape is the surface of waves on water?

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.

• 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). – lulu Jan 14 '17 at 14:56
• Here is a reference. – lulu Jan 14 '17 at 14:57
• You should make that an answer. – marty cohen Jan 14 '17 at 15:11
• @lulu, to be precise, that's a curtate trochoid. – J. M. is a poor mathematician Jan 14 '17 at 15:20
• 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? – Duncan C Jan 14 '17 at 15:27