Refraction and reflection of a shock wave @NilsBerglund

Nils Berglund | Refraction and reflection of a shock wave @NilsBerglund | Uploaded August 2024 | Updated October 2024, 1 minute ago.
This simulation has been suggested in a comment to a previous video. It shows the shock wave created by a source traveling faster than the wave speed in a medium. The wave encounters an interface to a different medium of relative refractive index 1.5 (the wave speed is 1.5 times slower in the lower part). As a result, part of the shock wave is refracted, changing its angle in the second medium, while part of it is reflected.
This video has six parts, showing the evolution with three different speeds of the source, and two different color gradients for each speed:
Speed 1, wave height: 0:00
Speed 1, averaged wave energy: 0:44
Speed 2, wave height: 1:36
Speed 2, averaged wave energy: 2:19
Speed 3, wave height: 3:12
Speed 3, averaged wave energy: 3:56
In parts 1, 3 and 5, the color hue depends on the height of the wave. In the other parts, 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.

Render time: Part 1 - 16 minutes 29 seconds
Part 2 - 16 minutes 13 secondsPart 3 - 16 minutes 15 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: "Eye Do" by Jeremy Blake@RedMeansRecording

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 #shock_wave #sonic_boom

$Refraction and reflection of a shock wave$ $Waves escaping a ring of obstacles: Sunflower grid This is a simulation of waves originating from a point source crossing a set of circular obstacles placed in an annular region based on the golden mean. This pattern is obtained by gradually adding discs, each time turning with respect to the center of the screen by a golden angle of (phi - 1) times 360°, where phi = 1.618... is the golden ratio or golden mean. The distance to the center is gradually increased, in such a way that the density of circles remains constant. This arrangement has appeared before on this channel, see for instance https://youtu.be/fQaVL3mE6VY or https://youtu.be/c8sKkM8ZDTs . This video has two parts, showing the same evolution with two different color gradients: Averaged wave energy: 0:00 Wave height: 1:11 In the first part, the color hue depends on the energy of the wave, averaged over a sliding time window. In the second part, it depends on the height of the wave. 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. The color in the central region has been brightened to white, because the waves tend to make large-amplitude oscillations there, which would not be very pleasant to watch. There are absorbing boundary conditions on the borders of the simulated rectangle. The display at the right shows 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: 29 minutes 22 seconds Compression: crf 23 Color scheme: Part 1 - Turbo, by Anton Mikhailov https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: Moorland by Underbelly@yousuckatproducing 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 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 Marco Mancini and Julian Kauth for helping me to accelerate my code! #wave #diffraction$

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$Cherenkov radiation of two particles moving in opposite directions This video was suggested in comments to a previous video on Cherenkov radiation. It features two particles moving in opposite directions in a medium with varying refractive index. Cherenkov radiation is produced when a fast moving particle enters a medium in which the speed of light is smaller than the speed of the particle. This does not contradict special relativity, because only the speed of light in a vacuum cannot be reached by a massive particle, while the speed of light in a medium is in general smaller than in a vacuum. In this simulation, two particles move at constant speed, one from left to right and the other one from right to left, emitting pulses at regular time intervals. The lower particle emits waves of smaller amplitude, to avoid that its radiation completely masks the radiation of the upper particle. The speed of light decreases linearly, dropping to 0 at the right boundary. When the speed of light in the medium is smaller than the speed of the particle, a shock waves appears, which is the origin of Cherenkov radiation. No shock wave is produced when the particle moves in a region where the speed of light is larger than its own speed. 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. There are absorbing boundary conditions on the borders of the simulated rectangle. The display at the bottom shows the signal along a horizontal line, located halfway between the paths of the two particles. Render time: 28 minutes 59 seconds Compression: crf 20 Color scheme: Part 1 - Twilight by Bastian Bechtold Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: The Coldest Shoulder by The 126ers@hutchinw 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 Marco Mancini and Julian Kauth for helping me to accelerate my code! #wave #Cherenkov #shock_wave$

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