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A triangular parabolic resonator
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- Опубліковано 15 сер 2024
- This video is analogous to the video • A hexagonal parabolic ... but with 3 instead of 6 parabolas. The domain looks a little bit like a Reuleaux triangle, but it is different, because its boundary is composed of parabolas sharing the center as a focal point, instead of circular arcs centered in the opposite vertex of an equilateral triangle.
Parabolas have the property of turning a circular wave coming from their focus into a linear wave, and vice versa. The initial circular wave is thus transformed into a triangular arrangement of wave fronts. However, the sides of this triangle do not hit an opposing parabolic side head on, but instead hit two sides at an angle, which prevents the triangle from being transformed back into a circle. Still, this produces some interesting wave patterns, including hexagons.
This simulation has two parts:
Wave energy: 0:00
Wave height: 3:14
The colors in the first half show the wave's energy, with blue indicating low energy and red indicating high energy. The colors in the second half show the wave height: blue means height zero, green-yellow-red hues indicate positive height, while purple-red-orange hues indicate negative height.
Render time: 19 minutes
Music: "The Stoic and the Sailor" by the Unicorn Heads@UnicornHeads
See also images.math.cn... 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....
C code: github.com/nil...
www.idpoisson....
Many thanks to my colleague Marco Mancini for helping me to accelerate my code!
It'd be interesting to see these wave simulations overlayed with particle based simulations using the same surfaces for reflection.
Thanks for the idea!
Cool!!!
@@NilsBerglund And also the actual heights of the wave plotted as actual heights so basically the physical realization PLEASE Nils
Except that it isnt particle based whatsoever, so why a visual for an incorrect observation?
@@liberatedcollective2124 Whoever said it was? Astoundingly incorrect observation there.
What do you know! The people who suggested the triangular resonator were right. It is quite interesting.
Thank you for making it!
You're very welcome! :)
@@NilsBerglund you're also welcome
In my almost 35 years on this planet, I don't believe I've ever seen a triangular, parabolic resonator before. But I'm glad that I have now!
The story of inventing shapes like cogwheels, reuleaux triangle, the biohazard symbol and many more by dropping a stone in a triangular parabolic pond.
May i use this idea for some fictional worldbuilding?
@@orbismworldbuilding8428 no
@@mono4171 aight, also are you the same person?
@@orbismworldbuilding8428 Yes he is
@@orbismworldbuilding8428 yes
Watching this makes me understand those kooks who obsess about "sacred geometry." This fills me with something I don't understand, a sense of awe or something like it. Watching waves in a triangle shouldn't do this to me.
I've gotta wonder how two of those overlaid ("star of David" style) would behave.
Love these sims - fascinating examples of wave mechanics
am i the only one watching these on 120FPS/120+htz by playing them in 2x speed? LOOKS SOOO GOOOOOOOOD
Imagine having a 3D tetrahedral analog to this simulation
Hexagons appear, woo!
I have a sudden, intense urge to hit Craigslist and look for RX-7's
Nooo000OOO 🤣
The spicy dorito engine
These videos are the natural evolution of the ole LSD posters.
And that, Nils, is why scientific visualisation is so awesome and this drives more inquiring minds towards research.
If it's the case that these simulations have motivated even one single person to start a career in research, it would already make me very happy!
Now I'm curious to see this with a Reuleaux triangle :D
I haven't done it for waves yet, but there is a particle version here: ua-cam.com/video/ynCK4D7h4M0/v-deo.html
@@NilsBerglund Ooh nice! It's fun to watch it _almost_ mimic the 3-parabola enclosure for a bit, then begin to quickly deviate!
Keeps thinking that this thing is going Nuclear. Especially around 1:05 .
Perfection, I can look at it for hours ...
Glad you like it!
Utterly Fascinating! I think this is your best render yet. Full of surprises!
Wow, thanks!
This needs to be a real life swimming pool.
Many appearances by the bestagon in this episode
The patterns this makes is beautiful
That’s really interesting that the second demonstration shows the wave can bring almost the entire system to a higher wave height at the same time…
So, even-sided resonators regenerate the circular wave every two reflections, while odd-sided resonators don't...
I guess because in the even shapes, each parabola has another parabola lined up exactly opposite from it.
Yes, exactly.
Somehow it reminds me of a caleidoscope :-)
Just at the start I though "after 2 reflections it can't possibly, can it?". If it had converged perfectly I might have had a stroke...
Thank goodness you're all right...
@@NilsBerglund It was excitingly and beautifully close though!
Is there possible tricks to play in the corners to reduce the diffraction-effects? Something like smoothing the corners to circular sectors?
@@NilsBerglund - if you really want to nerd-snipe you should make a short version of only the first 30-35 seconds. That would get me agitated even though I know the ending...
This one is my favorite so far.
This is soo relaxing and at the same time interesting
Glad you enjoyed
Is the amplitude-time graph of the center point available?
i have a feeling that if this was a shape in real life it would roll in an unexpected way
The shape is a little similar to a Reuleaux triangle (a shape of constant width), but not quite the same, as the boundaries are pieces of parabolas and not circular arcs.
@@NilsBerglund okay thanks
Now I'd like to see a triangular laser resonator under rotational motion (laser ring gyro) 😃
is it possible to have a shape that "spins" the wave?
I'm imagining rotating each parabolic side a little, but then the focii are moved and a central circular wave doesn't exactly become flat wavefronts
maybe with enough sides and only a slight rotation it'll be close enough to look cool?
I can imagine that some kind of ratchet geometry may have such a spinning effect. It seems indeed more difficult to keep the circle-line property, but maybe there exists some clever geometric design doing that...
@@NilsBerglund the Zeiss Jena guys should know about that😉
@@NilsBerglund i thought about it a bit more. inducing spinning would violate conservation of angular momentum, but it might be possible to have two "streams" in opposite directions... but that might get really complicated fast
I think now I'm just curious about the following ratchet-wheel like shape:
take a regular polygon and replace each flat side with two lines bent inwards, but make sure they're skewed slightly
so replace the bottom flat line of the shape with "go up 30 degrees for a while then down 60 degrees" (does that make sense?)
So when a circular wave expands outwards, some energy will be directed sharply outwards and the rest will be closer to reflecting back at the source. This might look a little "spinny"
the first 3 minutes look like an awesome alien scope for a super weapon.
Now I'm very curious to see what wave patterns a pentagonal resonator would create
Yeah, that's definitely one I want to try as well!
It'd be fun to have a shorter version in played back in reverse.
Very intriguing 🤨 analogous to fusion with a star. Really quite awesome.
Thanks!
@@NilsBerglund your welcome! I’m just wondering if by changing the chambers shape you could increase the potential high pressure zones? I’ve been thinking of a star shape chamber with the arches pointing inward to take the impact( arches being the strongest thing known to man. Much like how beach groynes work to break the waves up. I’d imagined the cove section to be were the most interaction would take place. pending how many point of a star were used and how long the points were. I love to see something like this. I just haven’t quite got there myself. seeing your animation reminded me of something o had been thinking about. I like how you alway set it wave off from the central point In the chamber. Much how I thought it would work is just uncanny. The reflection and refraction mimic the force found in stars. Which you captured quite well indeed.
Thanks. Here is an example of a star shape: ua-cam.com/video/uI36H9w4D-Y/v-deo.html
You may be right about the concentration properties. Another factor one should probably keep in mind is how the resonant frequencies are distributed. They tend to be more random in shapes in which the particle dynamics is chaotic, which is the case for many star shapes.
@@NilsBerglund thank you for the link. I did notice the timing of the resonance in the star shape chamber helped in the cycles of high pressure and low. It definitely seam like it was self cycling aiding in the reaction. There seam also to be a very distinct pattern emerging especially from the star shape one. I noticed it could be broken down in to phases by the pattern and shape it makes.which I might say is really cool. These simulation might aid in designing a reactor.
I know the main issue with toroidal chambers is they don’t focus the energy to where it will fuse hydrogen efficiently. Induction,compression, power and exhaust. Are the combustion cycle. The more high temperatures high, pressure zone that occurs, the more like fusion will take places. I think it might be possible to reach the temperatures required far easier with this method. than spending trillions On overly complicated designs. Essential you could have something as simple as super dense coils surrounding an iron core set with cooling. set up in a star shape. If the science is done right? then kids could be playing with a table top version 100 years from now? It seem far simpler to make a 2D resonance chamber for fusion than it does a 3D one. The best thing I find about the simulations is it can be fine tune before any experiment starts. Which mean it can be dialled in and cross reference with results. Great work! You basically have the means to check for an optimal frequency, the minimum wall thickness and have something to aim for. Fantastic keep it up! I don’t know if ITER have done anything like this? It would be certainly worth reaching out. As you never know? The future is only what we can make today, in this moment, in this now. For tomorrow always seam a lifetime away. I’d love to see what you come up with next. As this is fantastic work. Definitely like what I imagined. But with far more details. I realised when I thought about how it would work. that the pressure waves would compress those in front leading to increase moments where fusion could take place. With internal reflection also playing a part I knew it was something special. Practically could run it like a car engine and it would become self sustaining. I don’t think ITER are even vaguely close to threshold on returns. The resonance analysis is definitely going to help there.
I'm not a specialist of research into controlled fusion energy production, but have talked to some people who are - and you can believe me, they are making lots of far more sophisticated simulations. There is an additional problem in fusion reactors, which is that ions in a plasma repel each other. Magnetic bottles seem to be a promising design to control that, and while there may be other designs that toroidal ones as used in tokamaks, those seem to be doing quite well. In fact, it is my understanding that experimental fusion reactors currently do produce energy, it is mainly a matter of efficiency, because also a lot of energy is needed to keep the confining electromagnetic fields going.
Now this is how you triangle a circle
The beauty of mathematics
Would be cool to get audio of what it would sound like at different points if it were sound waves too
The rounded hexagon appears in nature often. The winds on Saturn and many craters, especially on small bodies such as moons and comets. This simulation may be a tell.
beautiful
Thank you! :)
Can you also fo noice cancellation walls? 😅
Can you do a simulation of a Marconi Anderson EM backscatter in free space?
This is really beautiful I expect this has applications in waveguide channelling amongst other things.
THIS IS AMAZING!!!!
Glad you like it!
So now I've started imagining some of your simulations as coloured wave attacks in Undertale or some other bullet hell...
Don't tell anyone
(if you're reading this and want to steal my ideas you better hit me up for a concept collab right away)
I'm waiting for a Brandon Sanderson magic system based on the properties of different resonnators/billards.
@@clementdenis4212 Hmm, I've heard about that. Don't know what you mean, though...
Cool! I wonder how it will sound if you make it metal and ring?
Wave height version just cool
could be interesting to see if you can make a sorta sound synthesis of this, making the waveform of various things (i was thinking car sounds) and seeing what kinda affect it has on stuff
I wonder if it's possible for the waves to reform the circle that started it. I guess that kind of violates entropy though.
This is possible in specific cases, for instance when the number of sides is even like in the simulation ua-cam.com/video/lVDwH5PfEGY/v-deo.html and the wavelength is small enough for dispersion to be negligible. Otherwise, the wave will rather be a superposition of standing waves oscillating at different frequencies.
This looks really "clean". Did you simulate the full shape, or did you make use of the 6 way symmetry?
No, I did it in the full domain. Computing this did not take so long, and I can use the same code for other, non-symmetric initial states.
Interesting. It's like. All the major brands and what not came here for their logo inspiration
Wooooow this is stunning!
Thank you so much!!
So mesmerizing
Надо купить такую чашку и попускать по ней круги
I need to find a program that’ll turn these resonator patterns into sound. I wanna hear these.
This is one of the best
Thank you! :)
Cool vids. Could you do some acoustic models of stethoscopes?
I'm not familiar with the principle of their design, is it some kind of wave guide?
What do you get if you rotate it as a cylinder? Do you think you could time pulses for a tractor beam tuned to the receiver element?
Violin body is two resonant chambers of different volumes with a coupling region.
How would that look, and could the placement of 'f-holes' be aligned with internal reflections?
I'll have to look into that. If there is some equation or data on the shape of these chambers, it may be possible to make a simulation.
it'd be super cool to see this rotating like the widdershine one you later posted... or maybe not just simple rotation but rather a more localized oscillation (like only +/-xx° so it has a "back and forth" "nature").
Coming up soon (with constant speed of rotation).
Nice fractals
Is that a reauleux triangle or however its spelt
nope! the difference is in the description.
I make coffee by first putting coffee and sugar in a round pan and then I add milk. most of the times a square forms, before everything starts boiling. (it starts boiling round the inside edge of the pan, but not round, square) one time however a triangle first formed. 🤷♂️ the first 12 seconds of this film reminded me of that.
can someone explain that?
Parabolic resonator
The triangle thing in the Wankel engine
Dorito stax
Just out of curiosity, how much have you learned in the process of making this series of videos? I'm guessing you've learned new coding tricks, but have you had any "a-ha" momements?
I'm also trying to think of what else you could do. Maybe something that has both a reflective surface and a high-IoR lens?
I have learned that quite a number of people seem to like these simulations... ;)
That being said, the efficiency of fractals as absorbing energy impressed me. I also learned new things, about Poisson disc sampling and translation surfaces, for instance.
Nils, could this methodology be used to simulate hitting a cymbal or gong? Perhaps the progression of the sound over time could be seen somehow.
The linear wave equation should be able to model the vibrations of such systems as long as they are small enough for non-linear effects to be negligible. Maybe the boundary conditions I use here (zero wave height outside the domain) are not quite correct for that.
I seem to remember something about drum heads resonating with non-harmonic frequencies, described by Bessel functions. It's been a long time since I read it, but I remember a story about Richard Feynman wanting a picture of the resonant waves of a drum head on the printed versions of his famous lectures, and someone mis-understood and ended up putting a picture of him beating a drum instead. It's also completely possible that this was apocryphal and or my neurons have gotten scrambled by time...
All hail the parabolic resonator uterus
It would be interesting to see tsunami on shoreline reflection.
Thank you!
Welcome!
Does it ever reach an equilibrium? Or ever go back to the origin all at once? This is so interesting to me
Such systems behave like a superposition of different standing waves, oscillating at different frequencies. In very special situations, the waves can all get back to the starting point at once, but in general they will keep oscillating.
Matter from Energy. All is just Nothingness Vibrating.
Would the starting pattern ever repeat itself? A perfect circle?
I don't think so. But it may eventually approach a pattern that loops.
Is there a non-circular shape that will eventuallty reflect the waves back into a perfect circle?
Yes, parabolic resonators with an even number of sides do that: ua-cam.com/video/AUPhTGrukHY/v-deo.html (wave optics), ua-cam.com/video/jRhrbb9Hhq0/v-deo.html (ray optics)
How is red both in positive and negative height? I mean, for example, Is red next to green different than red next to purple? Very beautiful btw.
Thanks. I used an HSL color wheel like here: thenewcode.com/61/An-Easy-Guide-To-HSL-Color
The hue for zero wave height is 220°, and it changes with an amplitude of 230°, so between -10° (a slightly purplish red), and 450° (or 90°, a greenish yellow).
@@NilsBerglund got it! Thanks
Make it symmetric. Why the offset? Got a triangle point it up.
can the same process through the resonance of a decahedron create all shapes? or do we need a hendecagon?
Cases with even number of sides will look more regular, because they transform circular waves into linear ones and back. For odd number of sides, the result will probably always be less regular.
It gives me some new logo ideas
Using flamable gas and oxygen?
I like your funny colors magic man
:D
Maybe a stupid question but what field of study is this? Or is this just hobby?
I'd say this is part of wave mechanics in physics, and part of the analysis of partial differential equations in mathematics. It also has connections to spectral theory and other fields. These simulations are not directly related to my field of research, but there are some links.
Enchanting
Thank you!
What kinds of technique do you use for these visualization? FEM?
It's finite differences. The simulation grid is a simple square grid, which is not as elaborate as finite elements.
@@NilsBerglund nice! How coarse is the grid and im assuming it must still be fairly performant then?
I'm using a 1280 times 720 grid, same as the video's resolution. The important thing is to choose the time step small enough that the Courant number is below 1. Here it's value is 0.02.
Very interesting!
What are the parameters for initial pulse? Is a pure single frequency? And how wide is the initial pulse in time/wavelength domain? Also if it is not single frequency, is the refraction index of the environment set as 1?
The initial wave amplitude is given by exp(-r*r/0.002)*cos(r/0.1), where r is the distance to the center. So it is a wave packet containing a distribution of frequencies. Since there is only one medium with constant speed of propagation, there is no need to introduce an index of refraction here.
@@NilsBerglund Thanks, that was exaclty where i was a little bit confused. Have ever think of performing some simulations for wavefront passing throgh atmosphere/AO stuff?
Very nice, have you made a typical microwave resonator one
Not yet. That would be a magnetron, right?
It would be nice to see it in 3D instead of colored, too..
I'm thinking about it, thanks!
Is anyone writing down these new shapes?
In the Squares into circles and circles into squares, there was no dispersion, it would be nice to see triangles into circles and circles into triangles, is it even possible?
Not with this design anyway. The circle transforms into a triangle all right, but then things get more messy...
Is the color gradient keyed a variable of pressure?
In the second half of the video, yes (assuming it's sound waves). In the first half, colors represent the energy.
@@NilsBerglund thank you
Hexagons are the bestagons
Strangely, I'd like to see a "tesla valve" setup here. I wonder if it has any interesting properties.
Afaik, a Tesla valve would need a nonlinear equation to work.
@@NilsBerglund ah, that's right. I MAY already have the code for that, it was made to break a record for calculating large cosmological data.
It could be adapted, I'm sure.
@@NilsBerglund It's for large bodies, I'm sure that could be an issue.
nice!! I wonder what the normal modes of this shape would look like
I see you start to use primes ;)
Isn't that one of those objects with a constant height?
Not exactly. Those are Reuleaux triangles, and the sides are circular arcs, centered on the opposite vertex. Here the sides are parabolas.
What is this means?
For the end becomes at 4D wave...
Looks just like the trinity, and every other warning ⚠️ sign we label 🏷 for caution ☠️☠️☠️ like bio, nuclear, toxic, deadly.
Nobody:
Windows media player:
I think i saw many danger sings like radioaktive
In what application is this used??🤔
There are many lasers designed with mirrors aligned like this and light being generated in between them. They call it a confocal resonator cavity.
@@greenlungo3996 it took 2 years, but I got an answer 😂👌🏽thank you
When I made my comment, I was posting in the wrong Nils Berglund video. This is more like a ring resonator. The video depicting a confocal resonator cavity is here
ua-cam.com/video/n19XjuK_Dgs/v-deo.htmlsi=tG84IRsJmEcSPwrS
hexagon
Music bussin
What software is this magic 🧐
github.com/nilsberglund-orleans/UA-cam-simulations
@@NilsBerglund do you have this for 3D simulations?
Real 3d would be too costly computationally (about 1000 times longer). I may at some point consider 3d rendering of the 2d surface, but haven't done so yet.
Once again, hexagon is bestagon
Now... Is there a pentagonal derivative?
If you mean a five-sided equivalent, see here: ua-cam.com/video/6znsMo4000s/v-deo.html