Hidden patterns in $\sin(a x^2)$

I discovered unexpected patterns in the plot of the function

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

with $a = \pi/b$, $b=50000$ and integer arguments $x$ ranging from $0$ to $100000$. 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 $\text {sine}$ 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.

Solution 1:

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

$$\sin an^2=\sin(a-2m\pi)n^2$$

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

$$n'=\sqrt{\frac{a'}a}n$$ so that

$$\sin an^2=\sin an'^2.$$

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 $\cos a'n^2$).

Solution 2:

^{[To check it all out, you may want to visit this interactive page.]}

There are several aspects of the plot of $f(x) = \sin(a x^2)$ for integer arguments $x$, that need explanation, especially

1. its periodicity

2. the symmetry of the period

3. its recurring patterns

All these can be explained straight forwardly in a similar way:

Periodicity

$$\sin(a x^2) = \sin(a (c + x)^2) = \sin(a (c^2 + 2cx + x^2)) = \sin(ac^2 + 2acx + a x^2) =\sin(a x^2)$$

if $ac^2 = 2\pi m$ and $ac = \pi$ when $x$ is an integer. With $a = \pi/b$ this is fulfilled for $c = b$, provided $b$ is even. Here, $b=500$.

Symmetry of the period

$$\sin(a x^2) = \sin(a (b - x)^2) = \sin(a (b^2 - 2bx + x^2)) = \sin(ab^2 - 2abx + a x^2) = \sin(a x^2)$$

Recurring patterns

The most prominent recurring pattern is the first peak of $f(x) = \sin(a x^2)$, e.g. for $b = 5000$:

Let's take as an example the third of these patterns which is found exactly in the middle of the period of $f(x) = \sin(a x^2)$, i.e. at $x_0 = b/2$.

Enlarged, it looks like this:

We find:

$$\sin(a (2x)^2) = \sin(a (x_0 + 2x)^2) = \sin(a (x_0^2 + 4x_0x + 4x^2)) = $$

$$\sin(ax_0^2 +4ax_0x + 4ax^2) = \sin(\pi b/4 + 2\pi bx + 4a x^2) = \sin(4a x^2)$$

which holds when $b$ is divisible by $8$. A similar calculation shows that in this case

$$\sin(a (2x+1)^2) = -\sin(a (x_0 + 2x+1)^2)$$

This is still not the whole story to be told, but a beginning.