What mathematical shape is the surface of waves on water?

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.