Faraday waves (short version, velocity)
Channel: Nils Berglund
Date: 2023-08-09

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. 
This simulation of the shallow water equations uses a circular force field to confine the waves. The color hue shows the direction of the flow, and the z-coordinate and luminosity show its speed (the norm of the velocity). 
The initial state 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 30 degrees, which is why the resulting pattern has a 12-fold 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. 

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: 14 minutes 51 seconds
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a
 
Music: Hit the Lights by Twin Musicom is licensed under a Creative Commons Attribution 4.0 licence. https://creativecommons.org/licenses/by/4.0/
Artist: http://www.twinmusicom.org/

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 #Faraday_waves