# Hidden patterns in sin(ax2)\sin(a x^2)

I discovered unexpected patterns in the plot of the function

$$f(x)=sin(a x2)f(x) = \sin(a\ x^2)$$

with $$a=π/ba = \pi/b$$, $$b=50000b=50000$$ and integer arguments $$xx$$ ranging from $$00$$ to $$100000100000$$. It’s easy to understand that there is some sort of local symmetry in the plot but the existence of intricate global patterns like these

astonished me.

Is there a somehow simple explanation of these regular patterns that emerge when combining such “incommensurate” functions like $$sine\text {sine}$$ and squaring? Especially of their specific shapes, their increasing distinctness and the distances between them?

Added: This pattern I found only today somewhere in the middle of the plot:

Do you see the “corridors”?

They long for explanation.

This is the phenomenon known as aliasing, caused to the fact that you are sampling a fast-varying signal with a too low frequency. It tends to create replicas of the original signal, but dilated in time.

When $a$ is close to $2m\pi$, we have at integers

which is a replica of the original function with the smaller coefficient $a'=a-2m\pi$, and this is similar to a time dilation with

so that

As you can check on the plot, the blue and green curve coincide at integers ($a=6, a'=6-2\pi$).

The other patterns are similarly obtained with a phase shift (such as the values at half-integers, corresponding to $\cos a'n^2$).