Nils Berglund | When waves crossing a square lattice have a larger phase than group velocity @NilsBerglund | Uploaded July 2024 | Updated October 2024, 6 minutes ago.
The simulation youtu.be/15fpsopRyn8 , that showed waves from two different sources having different frequencies cross a regular square grid of obstacles, revealed another interesting feature: After a while, the waves from the source with a lower frequency seemed to propagate through the lattice at a larger speed than the wave speed. This has to be an instance of the situation where the phase velocity (measuring how fast wave fronts move) is larger than the group velocity (measuring the speed of energy transfer).
This video focuses on this phenomenon, by using only one source, having the lower frequency, and an additional plot of the signal in the x-direction.
This video has two parts, showing the same evolution with two different color gradients:
Wave height: 0:00
Averaged wave energy: 1:16
In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged over a sliding time window. The contrast has been enhanced by a shading procedure, similar to the one I have used on videos of reaction-diffusion equations. The process is to compute the normal vector to a surface in 3D that would be obtained by using the third dimension to represent the field, and then to make the luminosity depend on the angle between the normal vector and a fixed direction.
There are absorbing boundary conditions on the borders of the simulated rectangle. The displays at the right and bottom show a time-averaged version of the signal near the right boundary of the simulated rectangular area. More precisely, it shows the square root of an average of squares of the respective field value (wave height or energy).
Render time: 30 minutes 28 seconds
Compression: crf 23
Color scheme: Part 1 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt
github.com/BIDS/colormap
Music: "Future Girl" by RKVC@rkvc
See also https://images.math.cnrs.fr/des-ondes-dans-mon-billard-partie-i/ 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 hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf
C code: 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!
#wave #diffraction #phase_velocity
The simulation youtu.be/15fpsopRyn8 , that showed waves from two different sources having different frequencies cross a regular square grid of obstacles, revealed another interesting feature: After a while, the waves from the source with a lower frequency seemed to propagate through the lattice at a larger speed than the wave speed. This has to be an instance of the situation where the phase velocity (measuring how fast wave fronts move) is larger than the group velocity (measuring the speed of energy transfer).
This video focuses on this phenomenon, by using only one source, having the lower frequency, and an additional plot of the signal in the x-direction.
This video has two parts, showing the same evolution with two different color gradients:
Wave height: 0:00
Averaged wave energy: 1:16
In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged over a sliding time window. The contrast has been enhanced by a shading procedure, similar to the one I have used on videos of reaction-diffusion equations. The process is to compute the normal vector to a surface in 3D that would be obtained by using the third dimension to represent the field, and then to make the luminosity depend on the angle between the normal vector and a fixed direction.
There are absorbing boundary conditions on the borders of the simulated rectangle. The displays at the right and bottom show a time-averaged version of the signal near the right boundary of the simulated rectangular area. More precisely, it shows the square root of an average of squares of the respective field value (wave height or energy).
Render time: 30 minutes 28 seconds
Compression: crf 23
Color scheme: Part 1 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt
github.com/BIDS/colormap
Music: "Future Girl" by RKVC@rkvc
See also https://images.math.cnrs.fr/des-ondes-dans-mon-billard-partie-i/ 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 hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf
C code: 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!
#wave #diffraction #phase_velocity
