Waves escaping a sunflower spiral of obstacles #art #waveequation #diffraction #fibonacci @NilsBerglund

Nils Berglund | Waves escaping a sunflower spiral of obstacles #art #waveequation #diffraction #fibonacci @NilsBerglund | Uploaded July 2024 | Updated October 2024, 2 hours ago.
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 youtu.be/fQaVL3mE6VY or youtu.be/c8sKkM8ZDTs .
The color hue depends on the energy of the wave, averaged over a sliding time window. 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.

Render time: 14 minutes 51 seconds
Compression: crf 23
Color scheme: Plasma by Nathaniel J. Smith and Stefan van der Walt
github.com/BIDS/colormap

Music: "Instant Crush" by Corbyn Kites@corbynkites8006

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

$Waves escaping a sunflower spiral of obstacles #art #waveequation #diffraction #fibonacci$

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$Filming a moving source with a gradient index lens Like the video https://youtu.be/KB_Roqs38F8 , this one shows a simulation of light from a nearby source crossing a gradient index lens. The difference is that here, the source is moving on a path perpendicular to the axis of the lens. Note that the speed of the source is very large, about 33% of the speed of light, but the simulation does not take any relativistic effects into account. The refractive index along the central axis of the lens is quite large, with a value of 1.82, in order to obtain the small focal distance required by the simulation. Im not quite sure what causes the echo of the transmitted waves, which was not present in other simulations with a static source. Is might be due to light reflecting on the horizontal boundaries of the lens. If someone has a better explanation, please feel free to leave a comment. Lenses focus incoming rays of light by delaying them more near the center of the lens than at its periphery. This is often done with a material of constant index of refraction, by making the lens thicker near the center, as shown for instance in the simulation https://youtu.be/rrJJBh9ubUE . However, one can also build lenses of constant thickness, by making the index of refraction of their material depend on the location in the lens. In this simulation, the index decreases like sqrt(n0² - a*r²), where r is the distance to the axis of symmetry. This results in the incoming waves being focused on the (estimated) focal plane, marked by a vertical line. The plot to the right shows a time-averaged value of the field along that plane. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:39 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 right shows the signal along the focal plane, which is indicated by a vertical line. Render time: 51 minutes 32 seconds Compression: crf 23 Color scheme: Part 1 - Twilight by Bastian Bechtold https://github.com/bastibe/twilight Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: Runner by Silent Partner 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 #lens #gradient_index$ $A gradient index lens with quadratic refractive index exposed to two light sources$

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