Is there a use for a wave made of alternating parabolas? Messing around in Desmos, I constructed (with a lot of copy+pasting) a wave-like object made of the heads of a parabola I'd shifted, flipped, and connected together. The result closely resembles a cosine wave with equation $$y = \cos(\frac{\pi x}{2})$$
Original Desmos graph: https://www.desmos.com/calculator/cmhoghswxo
Picture of the graph: https://i.stack.imgur.com/PEZTa.png
Is there any application for this type of wave? The parabola is a shockingly useful shape so I'd think there might be some hyperspecific use for a wave like this, but I have no idea what that might be when the cosine and sine functions are that much better for modeling wave mechanics.
Entirely possible this thing's pretty useless, but thanks for any information that can be provided.
 A: I arrived at this exact function myself just now while trying to imagine what a "one-dimensional orbit" might look like. I googled "alternating parabola wave" to see if anyone has talked about this, and found this question lol.
In a 3D world, gravity is governed by the inverse-square law. Here's Newton's equation for gravity:
$$|F|=\frac{Gm_1m_2}{r^2}$$
Notice this contains an inverse $r^2$. This is an effect of the inverse-square law, which states that in three dimensions, effects emanating from a point decrease in magnitude proportional to the inverse-square of the distance from the point. The reason for this is that the effect spreads out proportionally to the surface area of a sphere.
This means that in two dimensions, instead of gravity spreading out proportional to the surface area of a sphere, it would spread out proportional to the circumference of a circle, i.e. proportional to the inverse of $|r|$. One can imagine Newton's law of two-dimensional gravitation being the following:
$$|F|=\frac{km_1m_2}{|r|}$$
Likewise, in a one-dimensional world, the effect would not decrease at all and would remain constant. Thus:
$$|F|=km_1m_2$$
This would mean the equation governing the acceleration of an object orbiting a point would be:
$$a=-k*\text{sgn}(x)$$
And, of course, $a=x''$. The solution to this equation is precisely this alternating parabola wave (let's call it $P$), ignoring any constants:
$$x=P(t)$$
Using some trig hacking, I was able to create a single formula for this wave equation:
$$x=\left(1-\frac{2}{5}\left(\arcsin\left(\cos\left(\frac{\pi}{2}t\right)\right)\right)^2\right)*\text{sgn}\left(\sin\left(\frac{\pi}{2}t\right)\right)$$
Here it is in Desmos: https://www.desmos.com/calculator/egn9orerco
I got lost in the math and am still confused about where exactly the $\frac{2}{5}$ came from. But oh well.
PS: It might alternatively be reasonable to assume that the force exerted by one-dimensional gravity would actually not change direction, and would therefore be completely independent of the relative position of the object. This would make the "orbit" a regular parabola. But that's much less fun.
