Faraday waves with heptagonal symmetry
Channel: Nils Berglund
Date: 2023-07-29

This is a variant of the simulation https://youtu.be/GBzvPrFKTMw attempting to show Faraday waves. The difference is that the pattern has a seven-fold symmetry, instead of a six-fold one as in the previous simulation. 
 Faraday waves can appear when you gently shake a cup containing a beverage. The tweet https://twitter.com/Rainmaker1973/status/1607348915959123968?s=20 provides an example of such a situation. The set-up used here is similar to the one in the video https://youtu.be/r8JFpuo4gSg, where a circular force field is used to confine the waves. However, the initial state is different. It consists in a wave-like oscillation, concentrated near the rim of the circle, and with a velocity alternating between being directed inward and outward. The oscillation has an angular period of 360/7 degrees, which is why the resulting pattern has a heptagonal symmetry. 
The shallow water equations are nonlinear equations, which give a better description of the motion of water than the linear wave equation. In particular, unlike the linear wave equation, they conserve the total volume of water. The linear equation gives an approximation of the solutions, when the wave height remains close to its average over space. 
The equations used here include viscosity and dissipation, as described for instance in 
https://en.wikipedia.org/wiki/Shallow_water_equations#Non-conservative_form , without the Coriolis force. The confinement to the circle is obtained by combining a smooth but steep cut-off of the velocity field, and a repulsive force from the boundary of the circle.
The video has four parts, showing the same simulation with two different color schemes and two different playback speeds:
Wave height: 0:00
Velocity (norm and direction): 0:58
Wave height (time lapse): 1:55
Velocity (time lapse): 2:15
In the first and third part, the color hue and z coordinate show the height of the water. In the second and fourth part, the color hue shows the direction of the flow, and the z-coordinates shows its speed (the norm of the velocity). The playback speed has been multiplied by 3 in parts 3 and 4. 

Reference: W. S. Edwards and S. Fauve, "Patterns and quasi-patterns in the Faraday experiment", Journal of Fluid Mechanics 278, 123-148 (1994)
https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/abs/patterns-and-quasipatterns-in-the-faraday-experiment/8FF2B22DE046BC52D144128237B9C194 
 
Render time: 42 minutes 36 seconds
Color scheme: Parts 1 and 3 -   Cividis by Jamie R. Nuñez, Christopher R. Anderton, Ryan S. Renslow
https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0199239
 Parts 2 and 4 - Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
 
Music: "First Class" by DJ Williams@djwmusic

See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html for more explanations (in French) on  a few previous simulations of wave equations. 

The simulation solves the 2D shallow water equation by discretization (finite differences).

C code: https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code!

#shallowwater #waves #wave