Spirals in the rock-paper-scissors equation on the sphere @NilsBerglund

Nils Berglund | Spirals in the rock-paper-scissors equation on the sphere @NilsBerglund | Updated October 2024, 3 hours ago.

Vortex dynamics on the sphere with weaker Coriolis force

$Waves of two different frequencies crossing a sunflower lattice Some recent simulations on this channel, such as https://youtu.be/Y9Zn8T8dYgg , showed waves of two different frequencies, produced by two different sources, crossing a regular lattice of scatterers. Here the regular lattice has been replaced by a sunflower pattern of discs. This pattern is obtained by staring with a disc at the center of the display, and then gradually adding shifted discs, each time turning by a golden angle of (phi - 1) times 360°, where phi = 1.618... is the golden ratio or golden mean. The distance to the original circle 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 . The frequency of the lower source is three times the frequency of the upper one. The resulting wavelengths are such that the open intervals in the grating are between both wavelengths. As a result, waves of the lower source pass the grating more easily, as can be best seen on the energy plot. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:19 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 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: 35 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 https://github.com/BIDS/colormap Music: Raw Deal by Gunnar Olsen 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 #diffraction #grating$

Using six sources to excite resonant modes in a circle

Vortices in the compressible Euler equation on the sphere with Coriolis force

The shallow water equation on the sphere

$Symmetric low-frequency pulses in a magnetron-shaped resonator This simulation of a magnetron-shaped resonator uses pulses separated by relatively long time intervals as a source, with the aim of creating long wavelengths, that are closer to what occurs for real magnetrons. The anode is slightly refracting, with a high damping coefficient. The reason for this choice is that I was afraid that perfectly reflecting boundary conditions would prevent the waves from entering the smaller cavities. Neumann boundary conditions would probably be more appropriate, but they are harder to simulate, and making the walls slightly refracting is an easier surrogate for that. Magnetrons were used in early radars, and are still used in microwave ovens, as sources of the microwaves. Cavity magnetrons use a resonating cavity with several chambers, together with a magnetic field. In this simulation, there is no magnetic field, and waves are produced as pulses in the center of the device, so it does not represent a real magnetron. However, I still found it interesting to look at the effect of the resonating cavities on the output. Since this simulation is in 2D, I added a channel at one side to act as an outlet for the waves. In 3D, my understanding is that the outlet would rather be along the axis of the device. This video has two parts, showing the same evolution with two different color gradients: Wave height: 0:00 Averaged wave energy: 1:52 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 graph at the right shows a slightly time-averaged version of the signal. Render time: 43 minutes 52 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: Blue Macaw by Quincas Moreira@/UCL1zFMJb0sthwdAlGjGbdyg 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 #resonator #magnetron$ $[Flash warning] Two planar waves crossing a gradient index lens The first part of this video shows some flashing due to the formation of standing waves. This is a variant of the simulation https://youtu.be/CdpTq3dSrXA , with a different wave form. Though it looks like several parallel beams, the incoming wave is actually obtained by superimposing two planar waves, one of them angled upwards, and the other one downwards. This results in two spots forming in the focal plane of the lens. The videos are inspired by Huygens Optics recent short https://youtube.com/shorts/VGd3Ajnp6e0 showing the principle of a gradient index lens. 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 planar waves being focused at two points in 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: 1:08 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 periodic boundary conditions between the top and bottom boundaries, absorbing boundary conditions on the right boundary, and a time-periodic signal is imposed on the left boundary. The display at the right shows the signal along the focal plane, which is indicated by a vertical line. Render time: 32 minutes 40 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: Surrender by Asher Fulero@AsherFulero 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$

$Waves from a point source crossing a percolation-style arrangement of obstacles The arrangement of obstacles in this video is obtained by randomly deleting circles from a regular square lattice (here a point is kept with a probability of 25%). It is thus more random than a square lattice, but not as random as a Poisson disc process. A point source emits waves at constant frequency, and the video shows how the waves interact with the obstacles. 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 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: 30 minutes 28 seconds Compression: crf 23 Color scheme: Part 1 - Twilight by Bastian Bechtold https://github.com/bastibe/twilight Part 2 - Magma by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: Verve by Benjamin Martins@soundtherapy-bestsleepandr8945 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 #phase_velocity$