I apologize if this is elementary.
I'm trying to figure out how often the nodes of different sine waves line up with each other. As in, when $y = 0$ for different functions.
I've been using Desmos to try to figure this out inductively. So I saw, in the case of two waves: \begin{align} y &= \sin x \\ y &= \sin .5x \end{align} The nodes will line up every $2 \pi$.
I also saw that, in the case of two waves: \begin{align} y &= \sin x \\ y &= \sin .7x \end{align} The waves line up every $10 \pi$.
How do I figure out when these two waves line up? \begin{align} y &= \sin x \\ y &= \sin 0.71x \end{align}
And, ideally, how does one generalize to knowing when 3 or more waves' nodes synchronize?
Thank you! I'm trying to evaluate a theory of neural synchrony.
We have $\sin x = 0$ if $x = k\pi$ for some integer $k$, so as a consequence we have $\sin 0.71x = 0$ if $0.71x = k\pi$ for some integer $k$.
If we want both of these things to happen for the same value of $x$, then we want integers $k$ and $\ell$ such that $x = k\pi$ and $0.71x = \ell\pi$. But then, we have $0.71k \pi = \ell\pi$, or $71k = 100\ell$. Since $\gcd(71, 100) = 1$, this happens when $k = 100m$ and $\ell = 71m$, for some integer $m$.
This means that the nodes of $\sin 0.71x$ and $\sin x$ will line up when $x = 100m \pi$: the common nodes will be $100\pi$ apart.