Strange pattern in $x\sin(\left \lceil{x}\right \rceil )$.

Question

Below is a graph of $y=x \sin(\left \lceil{x}\right \rceil )$ from Desmos, $-140 < x <140$, in radians:

Nothing too strange, some symmetry about $y=0$, but not much else. Here is a graph of the same function, $-2000 < x <2000$:

A similar pattern appears for $y=x \sin( \left \lfloor{x}\right \rfloor )$, and for a slew of other functions of the form $f(x) \sin (\left \lfloor{x}\right \rfloor)$. I cannot quite wrap my head around this. Why does this pattern, which seems to be more or less symmetric about the $x$-axis, arise? What equations describe the lines which appear?

It should be noted that these aren't individual points, just very short (linear) line segments.

Answer 1

What you have are the values of $n\sin n$ for integer $n$. The point ordinates for different $n$ are close to each other when the abscissas differ by about the period ($2\pi\approx6.28$) or multiples ($14\pi\approx43.98$).

On the plot below, we overlay $n\sin n$ and $n\sin\dfrac n{14}$. The intersections will roughly correspond to your dotted curves. You have similar ones for $n\sin\dfrac{n+k}{14}$, and for other denominators.

This is called an aliasing effect.

Answer 2

Let's compare your "sophisticated" function to a slightly simpler "cousin," $x\sin(x)$. It has approximately the same shape, and it is symmetric about the y-axis—i.e. it is an even function—since it is the product of two odd functions.

If we graph the two functions together, we notice that the right side of each of the line segments appears to touch the graph of $x\sin(x)$. This is no illusion; they touch exactly where $\lceil x \rceil=x$, since then $x\sin(x)=x\sin(\lceil x\rceil)$.

We notice that the horizontal length of each line segment is 1 unit, which happens because on that interval, $\lceil x\rceil$ and hence $\sin(\lceil x\rceil)$ are constant, making $x\sin(\lceil x\rceil)$ a linear function on that interval.

Edit: Here is a screenshot of the two functions graphed together: