Tutorial: How to simulate the wave equation

This tutorial explains how the simulations of the wave equation on this channel have been coded. After giving a derivation and some properties of the wave equation, we discuss its simulation in 1D and in 2D, the choice of initial state and boundary conditions, and some tricks to speed up numerical simulations.

Intro: 0:00
Example - Parabolic reflectors: 0:27
Derivation of the wave equation: 4:26
Solutions in 1D: 9:30
Energy conservation: 17:38
Variants of the wave equation: 20:57
Numerical simulation in 1D: 23:40
Wave equation in 2D: 27:23
Examples - Variable wave speed: 29:00
Initial state: 30:43
Boundary conditions: 34:10
Optimizing computation time: 42:52

There is a typo around 25:55 - the Courant number is C = Delta x/Delta t, without the power 2.

Playlist of wave equation simulations:    • Wave equation  
Parabolic reflectors in 2D:    • Speak to me: Waves traveling between two p...  
Parabolic reflectors in 3D:    • A wave traveling between parabolic reflect...  
Total reflection:    • Total internal reflection (short version)  
Branched flow:    • A branched flow (super short version)  
Waves in an ellipse:    • Classics revisited: Wave in an ellipse sta...  

Discretizing the wave equation: https://hplgit.github.io/fdm-book/doc...
Absorbing boundary conditions: https://hal.archives-ouvertes.fr/hal-...

Code on GitHub: https://github.com/nilsberglund-orlea...

#wave #waves #simulation