I have come across three sinusoidal functions f1, f2, and f3 which, up to scaling and translation, are very close to each other. When normalized and plotted together, they are hard to tell apart. For example, if I subtract f2 from f1, the difference is between $0$ and $2\times10^{-2}$. This is illustrated in the first graphic directly below.
I do not know the actual expression of these three functions, nor if such a form exists. That is my question. I am trying to figure out if these data points conform to a nice expression--that is, not just an ugly polynomial fitted with meaningless coefficients, but something meaningful in terms of trigonometric functions.
Or maybe the three functions are not particularly interesting in themselves, but are converging to some meaningful expression.
In case it is of help, when I take the discrete Fourier transform of f1 minus f2, I get the result plotted in the second graphic below.
Expanding on my comment, it appears that you have a sinusoidal wave whose frequency increases as $x$ increases. If you take any even or odd function $f$ with $f(0)=0$ and $f$ concave up on $\mathbb{R}_+$, then $\cos\left(f(x)\right)$ will give you a graph like this. Using $f(x)=kx^2$ is one option, but using $f(x)=kx^{1+a}$ with $a>0$ will be similar. And there are many more options, like $f(x)=\sinh(kx)$ or $f(x)=\cosh(kx)-1$. See if you can find something that fits your curves.
You could also try to plot $f$ straight from your curves, by applying different branches of $\arccos$ at the turning points. Since your curve moves up and down from $0$ to $1$, you would first apply $y\mapsto2y-1$ to obtain values to apply branches of $\arccos$ to.