Congratulations, Steve! These moments of insight are rare and wonderful. On Sat, Aug 14, 2021, 9:57 PM Stephen Guerin <stephen.gue...@simtable.com> wrote:
> Ed, > > Yes, that's how I'm seeing it. > > For others, Ed's Step function is what I was calling Rect pulse function > which is Fourier dual of the Sinc function. Attached is an .mp4 recording > of Ed description of the relationship of gap width, frequency, and the > amount of spread of the Sinc function which is the diffusion pattern > observed. I recorded the mp4 from the illustration/animation" I linked > to earlier > <https://www.olympus-lifescience.com/en/microscope-resource/primer/java/diffraction/> > . > > And I think it's pretty cool to think of the gap as a sampler. I suspect > this is a well-known idea in optics/physics and old hat to many of you but > it's exciting to come onto ideas like this for oneself :-) I can almost > hear John Zingale saying, "of course, it's just the convolution theorem > <https://en.wikipedia.org/wiki/Convolution_theorem> applied to a square > wave". In the past I would nod and frankly, my eyes glaze if I can't ground > it in a microscopic understanding that guides my intuition. > > Or if given this amazingly deep statement I came across as I'm searching > for connecting sampling and diffraction - "*the diffraction pattern of > an object is the Fourier Transform of the object*" from here > <http://www.sci.sdsu.edu/TFrey/Bio750/FourierTransforms.html>, it finally > makes sense to me. > > And I can practically hear Steve Smith and our dear and late Fred > Untersher calling out, "that's how we've been describing holograms to you > for 20 years and you always nodded like you understood". > > Or Ed who brought Pradeep Sen into our world saying what do you think I > was showing you with Dual Photography > <https://graphics.stanford.edu/papers/dual_photography/>, you idiot? > > And Alvy Ray Smith, again Ed bringing into our office, saying that's > what a Pixel is <https://youtu.be/dvHDXUV7hmQ>! it's not a little square > nor gaussian point sample, it's Kotelnikov Sampling (Nyquist-Shannon > Sampling Theorem), > > or potentially worse is Eric Smith and Roger Critchlow shaking their heads > saying "you're just confused and making connections that aren't there". :-) > > -------------------------- > > Now even after having said this, I *still* want to know how the > diffraction is happening using only the interaction rules in the model. > Obviously, there are no Sinc or Rect functions in the code, nor Fourier > transforms explicitly coded. All these wonderful explanations above are > emergent properties from the model I would call a macroscopic explanation > and description. If nothing else perhaps I learn a better phrase for the > level of explanation I'm asking for when you trace an algorithm and > understand where the emergent property comes from. (BTW, I think I have a > micro answer and will put it in my response to Alex). > > -S > > _______________________________________________________________________ > stephen.gue...@simtable.com <stephen.gue...@simtable.com> > CEO, Simtable http://www.simtable.com > 1600 Lena St #D1, Santa Fe, NM 87505 > office: (505)995-0206 mobile: (505)577-5828 > twitter: @simtable > z <http://zoom.com/j/5055775828>oom.simtable.com > > > On Sat, Aug 14, 2021 at 3:57 PM Angel Edward <edward.an...@gmail.com> > wrote: > >> I hope someone can check out the analysis below. >> >> If you look at the gap as a sampler, you can do the following analysis >> using Fourier methods: >> >> A gap is a window on a continuous function. A perfect gap is a step >> function multiplying the continuous function. >> >> In the Fourier domain, the Fourier transform of the continuous function >> on the input side of the gap is convolved with the Fourier transform of gap >> (the step function). >> >> The Fourier transform of a step function is a sinc (sin(ax)/(ax)) >> function. >> >> The width of the main lobe of the sinc is inversely proportional to the >> width of the gap. >> >> Consequently, the smaller the width of the gap, the more a given >> frequency is distorted because the sinc is wider. Convolution applies the >> sinc at each frequency of the input function. >> >> I think it gets more complicated when we add in sampling. If we take a >> number of samples that is proportional to the width of the gap, then as we >> make the gap smaller there are fewer samples, hence more reconstruction >> issues which is the second, often overlooked, part of the sampling theorem. >> >> In the limit as the gap goes to zero width, there is no distortion to the >> continuous function but in the digital world you could have only a single >> sample. >> >> Ed >> __________ >> >> Ed Angel >> >> Founding Director, Art, Research, Technology and Science Laboratory (ARTS >> Lab) >> Professor Emeritus of Computer Science, University of New Mexico >> >> 1017 Sierra Pinon >> Santa Fe, NM 87501 >> 505-984-0136 (home) edward.an...@gmail.com >> 505-453-4944 (cell) http://www.cs.unm.edu/~angel >> >> >> On Sat, Aug 14, 2021 at 10:17 AM Stephen Guerin < >> stephen.gue...@simtable.com> wrote: >> >>> At yesterday's Virtual Friam I asked a question on diffraction and said >>> I would send more background. >>> >>> The gist of my question is: >>> >>> *Even though I completely understand the micro-level rules that generate >>> diffraction in the wave model described below, I still don't have an >>> intuition **how** the gaps in an obstacle have the emergent effect of >>> diffracting waves when wavelengths >= gap width. Can anyone help?* >>> >>> >>> Background: >>> The question arose from my mentoring NM School for the Arts high school >>> students in the NM Supercomputing Challenge >>> <http://nmsupercomputingchallenge.org/> where the students simulated >>> spatial acoustics by appropriating Saint-Venant equations used for shallow >>> water waves to instead model acoustic pressure waves. We wrote a >>> Netogo agent-based model with Python extension for reading / writing >>> the sound files and simulating spatial acoustics. >>> >>> <image.png> >>> >>> >>> The students explored the effects of different room configurations on >>> acoustics. >>> >>> One configuration of interest was a wall gap illustrated below in the >>> top right under Madelyn's video below. The wall gap is hard to see on right >>> side. >>> >>> <image.png> >>> >>> They simulated microphones in Netlogo by recording amplitudes at a patch >>> (red dot below in top-right visualization of room) and simulated speakers >>> (hard-to-see blue dot below red dot on other side of wall) by driving >>> amplitudes at a patch from the time series of amplitudes in .wav files >>> (recordings of a singer and viola performance). They could hear, and >>> through Fourier analysis, see the gap acting as a low-pass filter on the >>> acoustic signal. ie, only the low frequencies were "bending" around the >>> wall to reach the microphone. >>> >>> You can see and listen to this effect and the spectrogram visualization at >>> time 33:11 in their presentation <https://youtu.be/61p97NWJiQ8?t=2117>. >>> >>> <image.png> >>> >>> It took me a few weeks after their presentation in the NM Supercomputing >>> Challenge - they got second place - to connect the low pass filter behavior >>> to the concept of diffraction. Had this been a light model and I saw the >>> rainbow effects I would have clued in much faster. Their presentation was >>> a month after finals and they added this epilogue in the presentation >>> above to identify the effect as diffraction. >>> <https://youtu.be/61p97NWJiQ8?t=2761> >>> >>> Their presentation included this physical wavepool video demonstration >>> <https://youtu.be/BH0NfVUTWG4> which was helpful to me to begin to >>> understand the diffraction relationship with frequency and gap width. >>> >>> Note: my question is not about "describing" the behavior with >>> macroscopic equations or geometric models but fundamentally how does the >>> gap become a point source ala Huygens Principle at the micro-level of the >>> patches interacting with the emergent waves. To help with the distinction, >>> I consider this interactive model >>> <https://www.olympus-lifescience.com/en/microscope-resource/primer/java/diffraction/> >>> a >>> great macroscopic description of the phenomenon that nicely illustrates the >>> relationship of frequency and gap width but doesn't help me interpret the >>> micro-level interactions giving rise to the diffraction effect in our >>> simple shallow-water model. >>> >>> The students describe the details of the shallow water model at this >>> point in their presentation <https://youtu.be/61p97NWJiQ8?t=870>: >>> <image.png> >>> >>> >>> Here is my simplified Netlogo wave model >>> <https://anysurface.com/sguerin/models/shallowWaterDoubleSlit.html> of >>> the same shallow water equations without the acoustics. It's set up to >>> explore double slit but you can change it to single slit and mess with >>> frequency and gap and watch the wave propagations, diffractions and >>> interference patterns >>> https://anysurface.com/sguerin/models/shallowWaterDoubleSlit.html >>> <image.png> >>> >>> As a related aside, with some follow-up discussions with Ed Angel and >>> Steve Smith I am also trying to understand how the gap might be considered >>> a sampling function on the signal. My intuition is that the diffraction of >>> the wave creates a spreader Sinc function and the gap is Rect >>> function which are Fourier duals. In some way, i see Nyquist-Shannon >>> Sampling Theorem >>> <https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem> >>> related >>> to the gap. Note that diffraction creates a spreader function on the back >>> wall in single gap experiments and the gap may be considered a Rect pulse >>> when smaller than the wavelength. >>> >>> <image.png> >>> >>> >>> >>> _______________________________________________________________________ >>> stephen.gue...@simtable.com <stephen.gue...@simtable.com> >>> CEO, Simtable http://www.simtable.com >>> 1600 Lena St #D1, Santa Fe, NM 87505 >>> office: (505)995-0206 mobile: (505)577-5828 >>> twitter: @simtable >>> z <http://zoom.com/j/5055775828>oom.simtable.com >>> >>>> >> - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > FRIAM-COMIC http://friam-comic.blogspot.com/ > archives: http://friam.471366.n2.nabble.com/ >
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