# 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

## 1 Answer

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.

• 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.) – Duncan C Jan 14 '17 at 15:50
• 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. – lulu Jan 14 '17 at 15:53
• 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. – Duncan C Jan 17 '17 at 14:16
• But this one is curtate, so there is no problem with that. – lulu Jan 17 '17 at 14:33