I have a strange wave and I am trying to make it an equation. These are the points I have plotted:
| x | y |
|---|---|
| $0$ | $8$ |
| $\arcsin\left(\frac{\sqrt{40}}{20}\right)$ | $\sqrt{40}$ |
| $\arcsin\left(\frac{\sqrt{80}}{20}\right)$ | $\sqrt{80}$ |
| $\frac{\pi}{4}$ | $\sqrt{128}$ |
| $\arcsin\left(\frac{\sqrt{80}}{10}\right)$ | $\sqrt{80}$ |
| $\arcsin\left(\frac{3\sqrt{40}}{20}\right)$ | $\sqrt{40}$ |
| $\frac{\pi}{2}$ | $8$ |
| $\frac{\pi}{2}+\arcsin\left(\frac{\sqrt{40}}{20}\right)$ | $\sqrt{40}$ |
| $\frac{\pi}{2}+\arcsin\left(\frac{\sqrt{80}}{20}\right)$ | $\sqrt{80}$ |
| $\frac{\pi}{2}+\frac{\pi}{4}$ | $\sqrt{128}$ |
| $\frac{\pi}{2}+\arcsin\left(\frac{\sqrt{80}}{10}\right)$ | $\sqrt{80}$ |
| $\frac{\pi}{2}+\arcsin\left(\frac{3\sqrt{40}}{20}\right)$ | $\sqrt{40}$ |
| $\pi$ | $8$ |
| $\pi+\arcsin\left(\frac{\sqrt{40}}{20}\right)$ | $\sqrt{40}$ |
| $\pi+\arcsin\left(\frac{\sqrt{80}}{20}\right)$ | $\sqrt{80}$ |
| $\pi+\frac{\pi}{4}$ | $\sqrt{128}$ |
| $\pi+\arcsin\left(\frac{\sqrt{80}}{10}\right)$ | $\sqrt{80}$ |
| $\pi+\arcsin\left(\frac{3\sqrt{40}}{20}\right)$ | $\sqrt{40}$ |
| $\frac{3\pi}{2}$ | $8$ |
| $\frac{3\pi}{2}+\arcsin\left(\frac{\sqrt{40}}{20}\right)$ | $\sqrt{40}$ |
| $\frac{3\pi}{2}+\arcsin\left(\frac{\sqrt{80}}{20}\right)$ | $\sqrt{80}$ |
| $\frac{3\pi}{2}+\frac{\pi}{4}$ | $\sqrt{128}$ |
| $\frac{3\pi}{2}+\arcsin\left(\frac{\sqrt{80}}{10}\right)$ | $\sqrt{80}$ |
| $\frac{3\pi}{2}+\arcsin\left(\frac{3\sqrt{40}}{20}\right)$ | $\sqrt{40}$ |
| $2\pi$ | $8$ |
My only problem with this is that I haven't gotten to this kind of math yet, as I am only in 11th grade. I tried using a Lagrange Interpolation, but it was too chaotic. All I need is a wave that doesn't exceeds the height of $y = \sqrt{128}$ and doesn't go below $y = \sqrt{40}$ . The one reason I couldn't make this one is that the middle of the wave $y = 8$ isn't actually the midline and the wave has a higher amplitude on the top of and lower below the wave. Hopefully, you all can possibly come up with an equation. Thanks! :)
The problem is that you have multiple waves, and a constant term. By looking at the data, and playing a little with the numbers, I've got afunction that approximates your data quite well: $$y=A[\cos(k(x+\varphi))+\cos(2k(x+\varphi))]+B$$ with $A=\frac{\sqrt {10}}2$, $B=8$, and $\varphi=\frac{\pi}4$. The blue lines (and circles) are your initial data, the red is the function.
It is also possible that you have a few more frequencies.