Define $$f_n(\mathbf{r})=\frac{1}{n}\sum_{k=1}^n\exp\left(2\pi i\binom{\cos\left(2\pi k/n\right)}{\sin\left(2\pi k/n\right)}\cdot\mathbf{r} \right)$$ as the sum of $n$ counterpropagating plane waves. Then, let $I_n^R(z)\geq0$ be the "density of states" image in $\mathbb{C}$ of the ball $\left\{\mathbf{r}\mid|\mathbf{r}|<R\right\}\subset\mathbb{R}^2$ under the map $f_n$, normalized in some "sane" way.
- Is $I_n^\infty$ known?
$I_1^\infty$ is just a uniform-density ring of radius 1, ie
$$I_1^\infty(re^{i\theta})\propto\delta(r-1)$$
where $\delta$ is the Dirac delta.
For $I_2$, since $f_n(x,y)=\cos(2\pi x)$ and so we get a "horizontal line" of length two in $\mathbb{C}$ with density somewhat like this:
$$I_2(x+iy)\propto\frac{\delta(y)}{\sqrt{1-x^2}}$$
where $-1\leq x \leq 1$ (for $x$ outside that it's clearly zero). In general, $I_n$ for even $n$ is just going to be a "horizontal line" of length two with some density function, since $f_n$ is purely real-valued and oscillates somewhere between -1 and 1.
For odd $n$ the situation gets more interesting, and I'm not sure how to approach it. In such situations I usually resort to brute-force computation and have fun (while making no symbolic progress at all!). For example, here is a plot of $f_5$ on the ball of radius 4 (magnitude is encoded by brightness, and phase encoded by hue):
Here are pictures of $I_5^1,I_5^2,I_5^3$ and $I_5^4$:
As we increase the computation radius, a pattern begins to emerge:
$I_5^{10}$:
$I_5^{20}$:
$I_5^{50}$:
$I_5^{300}$:
Does anyone recognize what function this is? And are there general formulas for $I_n^\infty$ for odd $n$?
At the very least, I hope people enjoy the images! Right-click and open in a new tab to see higher resolution versions.
