Coffee and regular polygons

Question

To save some money, I decided to brew my own morning-fix coffee and skip buying it from the coffee shop. BTW, I drive to work and put my coffee cup in between the two front seats. While driving on the freeway (especially while driving through the less smooth parts of the road), I have observed regular polygons forming on the coffee surface: For example, here is a heptagon that I observed today:

And here is an octagon:

I also have actually observed what I believe $15$, $16$ or $17$-gons although I do not have a pictorial proof for them. I was wondering about a mathematical theory behind these things (I remember reading somewhere that the water, when it is drained, can take these regular polygon shapes, but forgot where). What is surprising to me is that these shapes appear to occur out of pure "noise," i.e. the random bumps of the road. I would like to see an explanation, such as, "Well, if these conditions are satisfied, you get a differential equation whose solution has some $\frac{1}{1+x^7}$ term in it, and those bumps are the singularities of the solution of this equation." I am also interested in the case where the cup is not circular, e.g. a triangle, I will try it if I can find such a cup :)

$\textbf{Edit:}$ DO NOT try this yourself. My girlfriend (who took these pictures) is a professional photographer!

Answer 1

These most likely correspond (loosely!) to the vibrational modes of the (roughly circular) surface making up the top of the coffee. Since the first couple of harmonics are (roughly) planar, what you observe visually is more likely to correspond to higher-order (and thus higher-energy) harmonic modes. These can be 'separated' into a product of a functional term whose only parameter is the radius (generally a Bessel Function of the radius) and another term harmonic in the angle; it's the sixth- and seventh-order harmonics that you're possibly seeing here. For more details on the specific mathematics, I'd recommend Wikipedia's page on Vibrations of a circular membrane; for some excellent illustrations of lower-order modes in motion, check out Daniel Russell's animations of vibrational modes.

Answer 2

The system is similar to waves on a drum, but the boundary is not fixed.

Is the car's suspension not properly damped?

Coffee waves in a cylindrical cup of radius a are approximated by $\displaystyle{c^2\nabla^2 h=\frac{\partial^2 h}{\partial t^2}}$ with $\displaystyle{\frac{\partial h}{\partial t}}=0$ at $r=a$, where $h$ is the vertical displacement.

In polar coordinates, $\displaystyle {c^2 \left(\frac{\partial^2h}{\partial r^2}+\frac{1}{r}\frac{\partial h}{\partial r}+\frac{1}{r^2}\frac{\partial^2h}{\partial \theta^2}\right) =\frac{\partial^2 h}{\partial t^2}}$.

Let $h$ be a linear combination of terms of the form $R(r)\Theta(\theta) T(t)$. Following the methods in the link in the answer above,$\Theta=C\cos m\theta + D\sin m\theta)$ for $m \in \mathbb{Z}$ and the other terms are not worth bothering with. If the value of R(r) is largest with $m=n$, terms with $m=kn$ will constructively interfere with the corresponding peaks to make narrower, taller peaks marking out the vertices of an n-gon, and terms with other values of $m$ will be more likely to interfere destructively.