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

I discovered unexpected patterns in the plot of the function

f(x)=sin(a x2)

with a=π/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

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astonished me.

Is there a somehow simple explanation of these regular patterns that emerge when combining such “incommensurate” functions like 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:

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Do you see the “corridors”?

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They long for explanation.

Answer

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 2mπ, we have at integers

sinan2=sin(a2mπ)n2

which is a replica of the original function with the smaller coefficient a=a2mπ, and this is similar to a time dilation with

n=aan so that

sinan2=sinan2.

As you can check on the plot, the blue and green curve coincide at integers (a=6,a=62π).

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The other patterns are similarly obtained with a phase shift (such as the values at half-integers, corresponding to cosan2).

Attribution
Source : Link , Question Author : Hans-Peter Stricker , Answer Author : Community

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