Energy flux of waves of two different frequencies crossing a diffraction grating @NilsBerglund

Nils Berglund | Energy flux of waves of two different frequencies crossing a diffraction grating @NilsBerglund | Uploaded May 2024 | Updated October 2024, 4 minutes ago.
This is a variant of the simulation youtu.be/kPsCquKggeU , showing the energy flux instead of the wave height and energy, with sources slightly further apart.
The waves emitted at two different frequencies encounter a diffraction grating, made of regularly spaced circular reflectors. 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 flux intensity plot.
This video has two parts, showing the same evolution with two different color gradients:
Direction of energy flux: 0:00
Intensity of energy flux: 1:04
In the first part, the color hue depends on the direction of the wave's energy flux, which is proportional to the product of the spatial gradient of the wave height and its time derivative. The luminosity depends on the norm of the energy flux. In the second part, the color hue depends on the norm of the energy flux.
There are absorbing boundary conditions on the borders of the simulated rectangle. The display at the bottom shows the squared of the signal along the horizontal symmetry axis of the simulated rectangular area, averaged over a time window.

Render time: 40 minutes 31 seconds
Compression: crf 25
Color scheme: Part 1 - Twilight by Bastian Bechtold
github.com/bastibe/twilight
Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt
github.com/BIDS/colormap

Music: "Jaguar" by Quincas Moreira@QuincasMoreira

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 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 #grating

$Energy flux of waves of two different frequencies crossing a diffraction grating$

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$Waves of two different frequencies crossing a hexagonal lattice Several recent videos on this channel showed waves from two different sources, with different frequencies, crossing a diffraction grating. This simulation is similar, except that there are now 24 columns of obstacles, so maybe this should not be called a grating anymore. Nevertheless, it is interesting to see what kind of interference and diffraction patterns are created. One thing that stands out here is how more energy is able to cross certain channels, in particular from the lower source. 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: 1:07 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: 29 minutes 52 seconds Compression: crf 23 Color scheme: Part 1 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing Part 2 - Plasma by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: March of the Hares by Nathan Moore@NathanMooreSongs 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$

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$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$