Wave rider function

Question

I'm looking for an example of a 'wave-like' function $f : \mathbb{R} \longrightarrow \mathbb{R}$ with the following properties:

• deterministic
• smooth (continously differentiable)
• periodic but with varying period lengths $p_i$, $p_i \neq p_j$
• mean period length $p=\lim_{n \rightarrow \infty}\frac{1}{n}\sum_{i=-n}^n p_i$ is finite but not $0$ (periods don't get constantly larger or smaller)

I tried various combinations of sine functions but that seems more tricky than I thought.

A non-trivial example might be the Riemann zeta function (normalized to have unit average spacing), i.e., $f(x) = \text{Re}\zeta(\frac{1}{2}+\mathsf{i} \delta(x))$.

Background: The question is related to how random the roots of a deterministic function can be?

Answer 1

I would try a function like $\sin(x\arctan(x))$ since the periods get shorter as you aproach infinity but never aproach 0.

Answer 2

I think $\displaystyle f(x) = \sin\big(x + 10\sin\big(x^{\frac{3}{5}}\big)\big)$ should work.

It has a vertical tangent at $x=0$ because $x^{\frac{3}{5}}$ is not differentiable there, but it satisfies all of your other conditions. In particular, the period varies but never goes to $0$. (I think it averages out to $2\pi$ because the second term's influence lessens over time, but I'm not sure.) You could fix the vertical tangent issue by using a differentiable-everywhere function in place of $x^{\frac{3}{5}}$, which would have to be sublinear so the period doesn't go to $0$.

On a sidenote, I found this function while playing around, which I think looks pretty cool.

$f(x) = \sin\big(x+\sin\big(x^{2}\big)\big)$