Waves starting off center in a four-sided parabolic resonator
Channel: Nils Berglund
Date: 19/11/2021

This is another video for the viewers that wondered what happens when you start a wave slightly off center in a parabolic resonator. It is similar to the particle version shown in the video https://youtu.be/JTqCVk60pTY
Parabolas have the property of turning a circular wave coming from their focus into a linear wave, and vice versa. Since the wave starts here a small distance away from the common focus of the four parabolic arcs, it is not transformed in an exactly linear wave, and deviations from the perfectly symmetric situation seen in the video https://youtu.be/AUPhTGrukHY accumulate over time. Nevertheless, the wave keeps concentrating in its point of origin, and the symmetric images of that point.
This simulation has two parts: 
Wave height: 0:00
Wave energy: 2:15
The meaning of the colors is indicated by the color bar on the right. The slow dispersion as time goes on is most likely due to round-off errors near the parabolic boundaries, and is more visible owing to the color scheme used here. 

Render time: 16 minutes
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "Lost and Found" by Jeremy Blake@RedMeansRecording

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 wave equation by discretization. The algorithm is adapted from the paper https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf 
C code: https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html 
Many thanks to my colleague Marco Mancini for helping me to accelerate my code!