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⬐ ermir

I assume these waves are not continuous but made out of many points and then just simulated over time, like a game engine. But since fractals have infinite detail, does that mean that for every number of points we choose for the wave we'll get a radically different wave pattern?

⬐ cycomanic

This depends on the wavelength. Essentially a wave hitting an object that has features that are smaller than roughly the wavelength will see those features as an average. So we therefore don't need to simulate the infinite resolution fractal. This is somewhat more complicated in the video because the waves are pulses, i.e. they contain many frequencies.

The video actually illustrates this effect when simulating the "narrow" ring waves and the "wide" ones. Those two contain different frequency components and thus see more or less of the fractals.

⬐ ermir

My initial thought was that was two particles that are infinitesimally close at the start of the simulation will have radically different reflections and paths when they hit the fractal, since the fractal has no local similarity, therefore the choice of the number of particles to use in the simulation will greatly affect the shape of the reflected wave. However by the same logic that the fractal has no local similarity we can also conclude that because the fractal has no derivative, and therefore no tangent, and therefore no way to calculate the reflection angle for any point at all.

⬐ maqflp

Would that have then something in common with butterfly effect?

⬐ cycomanic

Generally it is better to think of waves as waves, not as being composed of particles as they behave quite differently, for example particles can't interfere (the dark bright patterns we see). I assume the simulation to create the video simulates the wave equations and the are not composed of "particles".

⬐ maqflp

precisely, this is solution to wave equation (I found this thread as suggested by youtube - you people here gave me quite a lot of traffic recently ;)

⬐ maqflp

these waves are produced by finite difference (spatially finite resolution) approximation to contunuous wave equation so in some sense you are right but on the other hand - these are NOT particles at all

⬐ ermir

I gave this some thought and came to the conclusion that a continuous wave hitting a fractal would not have a defined reflection, since the fractal does not have a traditional derivative it would also not have a tangent at a point.

⬐ cycomanic

No, below a certain resolution (depending on the wavelength) the waves just see the average shape (which is smooth has a derivative). Think about this in real live imperfections in the atomic structure of a mirror don't influence the reflected light rays, or the shape of waterwaves reflected of a concrete wall is not influenced by the roughness of that concrete.

⬐ maqflp

you are right, I've recently done some Points Billiard in Koch Fractals video which tries to do classical billiard in Koch Fractal which is exactly what you write about averaging etc. (I had to average the normals around hitting point as the normal vector was not defined here), see the video here: https://youtu.be/v9KS03hiSU8

⬐ _Microft

If someone wants to reason about fractals and waves, I would maybe start with something (that I would assume to be) simpler than that and see if I could deduce some things from there before moving on to complicated combinations like the one shown in the video.

Maybe a plane wave [0] hitting a Koch curve / single side of a Koch snowflake [1] perpendicularily and then checking how different wavelengths and iterations of the fractal change the reflections or so. Diagram:

  plane wave moving downwards

  __________    |  
  __________    |
  __________    V



      /\ 
  ___/  \___

  Koch curve / side of a Koch snowflake

This is the first (or is it second?) iteration of the side of a Koch snowflake. Then reason about one more iteration (i.e. each side being replaced by exactly this shape) and so on. See what influence the shape has depending on the wavelength...

[0] https://en.wikipedia.org/wiki/Plane_wave

[1] https://en.wikipedia.org/wiki/Koch_snowflake