[FRIAM] Diffracton: minding the gap

Sat Aug 14 17:28:57 EDT 2021

Stephen,

"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."
Its not clear to me what your question is, so I'm going to say some stuff
and hopefully it either clarifies your understanding or helps me understand
your question better. Do you see your issue as different from understanding
how a micro-level of patch participates in a wave in the first place?
Whether the gap acts as a point source depends on the collective response
of the fluid surrounding it, so you can't understand whether a given gap
will act as a point source on a purely micro level.

As a thought experiment, imagine the situation where I have planar shallow
water waves incident on a gap, attached picture from your students'
presentation relevant. Assume that the incident waves have fixed temporal
frequency, e.g. I'm generating them by moving a plate up or down or
something like that and I keep the frequency of oscillation fixed. I then
keep the gap width fixed, but vary the strength of gravity. Changing the
strength of gravity, in turn, will vary the wavelength of the waves
incident on the gap. Since the ratio of the wave's wavelength to the gap
width determines the diffraction behavior, whether the micro-patch behaves
like a point source is determined by the strength of gravity in this
context. If I answered your question about why the gap acts like a point
source with "the strength of gravity is high or low enough", I imagine you
would find it deeply unsatisfying. I could say that when it acts like a
point source, it's doing so because the particles which make up the fluid
interact via gravity, this will probably be even less satisfying even
though it is explaining the point source like behavior at a micro-level.

More generally, this kind of dispersion relation trickery can be thought of
as varying the strength of the restoring force which allows the waves to
exist. The restoring force acts at a micro level (it determines how
asymmetries between neighboring micro patches get resolved) and it informs
how waves act at a macro level, here by determining the relation between
wavelength and frequency of the waves. It seems like your question should
be resolved by considering why the waves you care about are relevant to the
system they exist in, and why the thing we're thinking of as a wall is a
wall for the purposes of these waves. Does any of this help?

[image: Capture.PNG]
Best,
Alex

On Sat, Aug 14, 2021 at 10:41 AM Stephen Guerin <stephen.guerin at simtable.com>
wrote:

> The images in my initial email may have not come through. Attached is a
> PDF of the message
> _______________________________________________________________________
> Stephen.Guerin at Simtable.com <stephen.guerin at 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 10:17 AM Stephen Guerin <
> stephen.guerin at 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: 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: 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: 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: 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: 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: image.png]
>>
>>
>>
>> _______________________________________________________________________
>> Stephen.Guerin at Simtable.com <stephen.guerin at 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
>>
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