patterns on swimming pool floor
Narasimham
top undulating surface on sunny afternoons. These patterns appear to
form due to focusing of wave water lenses of rapidly changing power.
Can the patterns be predicted when pool dimensions are given?
Maarten van Reeuwijk
In theory, yes. However, one would also have to know the complete
trajectories of everyone in the pool, the kids splashing in etc. This would
make computation of the complete turbulent free-surface problem of the
pool, and after that the light refraction process, computationally
prohibitive.
However, if you would be willing to leave the people out of the water, and
consider the non-hydrostatic wave problem, you could probably even solve it
with analytical theory. Some random distribution could be used for wave
amplitudes and phases to model for the people swimming in the water.
HTH, Maarten
--
===================================================================
Maarten van Reeuwijk dept. of Multiscale Physics
Phd student Faculty of Applied Sciences
maarten.ws.tn.tudelft.nl Delft University of Technology
mme...@cars3.uchicago.edu
>Narasimham wrote:
>
>> In a swimming pool floor bright lines are seen after refraction from
>> top undulating surface on sunny afternoons. These patterns appear to
>> form due to focusing of wave water lenses of rapidly changing power.
>> Can the patterns be predicted when pool dimensions are given?
>
>In theory, yes. However, one would also have to know the complete
>trajectories of everyone in the pool, the kids splashing in etc. This would
>make computation of the complete turbulent free-surface problem of the
>pool, and after that the light refraction process, computationally
>prohibitive.
>
>However, if you would be willing to leave the people out of the water, and
>consider the non-hydrostatic wave problem, you could probably even solve it
>with analytical theory. Some random distribution could be used for wave
>amplitudes and phases to model for the people swimming in the water.
>
building the pool in the first place?
Mati Meron | "When you argue with a fool,
me...@cars.uchicago.edu | chances are he is doing just the same"
Herman Trivilino
>>However, if you would be willing to leave the people out of the water, and
>>consider the non-hydrostatic wave problem, you could probably even solve
>>it
>>with analytical theory. Some random distribution could be used for wave
>>amplitudes and phases to model for the people swimming in the water.
>>
> If you leave the people out of the water, what was the point in
> building the pool in the first place?
To sustain a theorist's employment? ;-)
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Jim Klein
If the people urinate in the pool, how fast does the color change to
green from blue?
Jim Klein
Narasimham
> Narasimham wrote:
> > In a swimming pool floor bright lines are seen after refraction from
> > top undulating surface on sunny afternoons. These patterns appear to
> > form due to focusing of wave water lenses of rapidly changing power.
> > Can the patterns be predicted when pool dimensions are given?
> However, if you .. consider the non-hydrostatic wave problem, you could
> probably even solve it with analytical theory.
> Some random distribution could be used for wave
> amplitudes and phases to model for the people swimming in the water.
We consider a situation when a lot of people were swimming before and
left the agitated pool at time t= 0, and the water surface becomes
still later on at large t.
Which equations are to be solved? Navier-Stokes, Laplace etc.? Whether
the analytical theory (may be solving PDEs with boundary conditions,
assuming internal damping ) automatically supplies wave frequencies,
amplitudes and relative phases?
What intrigued me is that at any instant of time, there is a transient
lens locus whose focal length equals the still pool depth giving rise
to observable focussed bright lines seen on pool floor...and these
lines form a curved net whose lines never seem to cut at sharp angles.
mme...@cars3.uchicago.edu
><mme...@cars3.uchicago.edu> wrote ...
>
>>>However, if you would be willing to leave the people out of the water, and
>>>consider the non-hydrostatic wave problem, you could probably even solve
>>>it
>>>with analytical theory. Some random distribution could be used for wave
>>>amplitudes and phases to model for the people swimming in the water.
>>>
>> If you leave the people out of the water, what was the point in
>> building the pool in the first place?
>
>To sustain a theorist's employment? ;-)
>
Gregory L. Hansen
Maarten van Reeuwijk <maarten@remove_this_ws.tn.tudelft.nl> wrote:
>Narasimham wrote:
>
>> In a swimming pool floor bright lines are seen after refraction from
>> top undulating surface on sunny afternoons. These patterns appear to
>> form due to focusing of wave water lenses of rapidly changing power.
>> Can the patterns be predicted when pool dimensions are given?
>
>In theory, yes. However, one would also have to know the complete
>trajectories of everyone in the pool, the kids splashing in etc. This would
>make computation of the complete turbulent free-surface problem of the
>pool, and after that the light refraction process, computationally
>prohibitive.
>
>However, if you would be willing to leave the people out of the water, and
>consider the non-hydrostatic wave problem, you could probably even solve it
>with analytical theory. Some random distribution could be used for wave
>amplitudes and phases to model for the people swimming in the water.
On the other hand, with sufficiently detailed observations of the waves in
part of the pool, and sufficient computing power, you should be able to
determine reasonably likely locations and activities of people in the rest
of the pool. Same idea as passive sonar, but surely more difficult.
Engine noises in the ocean are relatively uncluttered compared with a
dozen kids jumping in and splashing around.
--
"No other major companies were working on [computer-controlled homes], and
that was exactly the problem. Microsoft does best when it has a
successful competitor it can copy and then crush." -- Marlin Eller,
"Barbarians Led by Bill Gates", 1998
mme...@cars3.uchicago.edu
biology. And since the color of the urine depends on diet, there is a
social and cultural angle to consider, as well.
Brian Blandford
"Gregory L. Hansen" <glha...@steel.ucs.indiana.edu> wrote in message
news:dko907$dfu$5...@rainier.uits.indiana.edu...
> In article <a3cb8$436f2d1d$82a1bcd4$23...@news2.tudelft.nl>,
> Maarten van Reeuwijk <maarten@remove_this_ws.tn.tudelft.nl> wrote:
>>Narasimham wrote:
>>
>>> In a swimming pool floor bright lines are seen after refraction from
>>> top undulating surface on sunny afternoons. These patterns appear to
>>> form due to focusing of wave water lenses of rapidly changing power.
>>> Can the patterns be predicted when pool dimensions are given?
There is a long paper ca. 1960 by Longuet Higgins, published when he was
working for the Admiralty Research Labs in Teddington, and before he went to
Edinburgh as Professor of Artificial Intelligence, giving a detailed
mathematical treatment of the nature of waves in the sea. It is a classic in
that it illustrates how far you can go without numerical simulation. There
is also a later paper by Professor Mike Berry of Bristol on the calculation
of the caustics reflected onto the ceiling by the surface of a swimming
pool, which is related to "catastrophe theory". Again very difficult to
follow, but fascinating to see what sort of problems a brilliant physicist
will tackle.
Brian Blandford
Charles Bernard
"Brian Blandford" wrote
>
> There is a long paper ca. 1960 by Longuet Higgins, published when he was
> giving a detailed
> mathematical treatment of the nature of waves in the sea. It is a classic
in
> that it illustrates how far you can go without numerical simulation. There
> is also a later paper by Professor Mike Berry of Bristol on the
calculation
> of the caustics reflected onto the ceiling by the surface of a swimming
> pool, which is related to "catastrophe theory". Again very difficult to
> follow, but fascinating to see what sort of problems a brilliant physicist
> will tackle.
Brilliant indeed, now ***Sir*** Michael, but hard to follow. ;-)
I have seen nice and very realistic looking simulations
of these (caustic) patterns done by people using the free
(but quite remarkable) sofware POVray.
They have their own site and also newsgroup.
Maybe some info could be gleaned from them.
Skywise
@corp.supernews.com:
I've dabbled with POVRay for years, from before it was even called
POVRay (used to be DKBTrace). I'm not exceptional at it.
Generating realistic caustics is a relatively recent addition to
the program. It can be done but is computationally expensive.
POVRay is a raytracer, but it traces backwards; from camera to
scene. To add the caustics it has to add forward tracing from the
light source to the object in question in a pre-render sequence.
POVRay can also _simulate_ refractive dispersion.
I've toyed around with caustics in the program using simple shapes
like spheres and such. There is one animation sequence I made of a
rotating crystal with multiple spotlights showing both caustings
and dispersion.
http://www.skywise711.com/misc/crystal/crystal2.mpg
It's over 9 megs so be patient.
Brian
--
http://www.skywise711.com - Lasers, Seismology, Astronomy, Skepticism
Seismic FAQ: http://www.skywise711.com/SeismicFAQ/SeismicFAQ.html
Quake "predictions": http://www.skywise711.com/quakes/EQDB/index.html
Sed quis custodiet ipsos Custodes?
Supernews sucks - blocking google, usenet.com & newsfeeds.com posts
Andy Resnick
Those patterns are known as 'caustics' and consist of an envelope of
rays. There's a great book called "Natural focusing and the fine
structure of Light" by J. F. Nye (Professor Sir Michael Berry's
advisor, IIRC). Caustics are singularities similar to wave
dislocations, only that the amplitude is formally infinite rather than
identically zero.
Nye and Berry have spent a lot of effort studying caustics and
catastropies, and because they are general properties of waves they
apply not only to electromagnetic waves but also matter waves (Berry's
phase, for example). Also, it is intrinsically interesting (to me,
anyway) to be able to produce a true singluarity in the lab (a wave
disloation has zero amplitude, and thus the phase is indeterminatie)
--
Andrew Resnick, Ph.D.
Department of Physiology and Biophysics
Case Western Reserve University
Narasimham
For the simplet case I can imagine when sunlight is incident overhead
( parallel along z-axis), pool depth = D, wave profile z = A sin ( 2
pi x / lambda) cos ( 2 pi y / lambda) what are the refracted caustics?
Do we find the local normal, apply Snell's Law to get refracted ray,
use C-disriminant to eliminate the slope? How exactly does one
calculate the caustics?
Louis Boyd
This is an interesting subject as a mathematical exercise, but I'm
curious if there are practical applications of being able to predict
the patterns? Is there much similarity to the patterns formed by a
single boundry layer compared to atmospheric scintillation?
Skywise
@onion.ccit.arizona.edu:
Adaptive optics for astronomical telescopes?
The atmosphere does to starlight what the waves in a swimming pool
does to sunlight. Although the telescope only cares about a small
area and deforms it's mirror to cancel out the scintillation effects.
I just had a thought, though. What if there was a very large array of
small sensors placed in a grid over a large field, all focused onto
the same star. Could the real time light levels be used to image the
caustics caused by the atmosphere projected on the ground the same
as those seen on the bottom of a swimming pool? There'd probably have
to be thousands of sensors over a wide area to get enough resolution.
Perhaps there might be an application in the atmospheric sciences, for
this idea?
Brian Blandford
"Narasimham" <math...@hotmail.com> wrote in message
news:1131667080.0...@g44g2000cwa.googlegroups.com...
<snip>
How exactly does one
> calculate the caustics?
As I understand it, a caustic is simply the location of the turning point in
the transverse ray aberration function. Even a rainbow can be thought of as
a caustic - it is simply the turning point in the
refraction-reflection-refraction ray deviation.
Brian Blandford
![Sbharris[atsign]ix.netcom.com's profile photo](http://lh3.googleusercontent.com/a-/ALV-UjURN6b50hrZp1VmOKY6vUoD4xke_qxfdpwlGhkd6mvzimVI=s40-c)
Sbharris[atsign]ix.netcom.com
Also, elementary Schroedinger color vector combination theory suggests
a change from blue to green, not the other way around.
SBH
Narasimham
For getting caustic in a spherical mirror we find envelope of
reflected rays using differential calculus C-discriminant method in 2
dimensions. For a ruled surface an edge of regression could be also
found using partial derivatives in 3 dimensions. Does that give the
caustic?
Charles Bernard
<mme...@cars3.uchicago.edu> wrote
> Hmm, that's an interdisciplinary project, physics, chemistry and
> biology. And since the color of the urine depends on diet, there is a
> social and cultural angle to consider, as well.
Diet is very important, for example, eat 1kg of red beets and at least
500g of asparagus, then if you misbehave you'll surely be noticed,
visually or olfactorily.
I was told that in pre WWII Germany a special chemical was added to
swimming pools water for detection and deterrence.
Truth or legend I don't know, an allegedly milky appearance of the water
around the culprit was the mark of shame.
mme...@cars3.uchicago.edu
about it. Still, may be true.
Charles Bernard
"Brian Blandford" wrote
> As I understand it, a caustic is simply the location of the turning
> point in the transverse ray aberration function. Even a rainbow
> can be thought of as a caustic - it is simply the turning point in
> the refraction-reflection-refraction ray deviation.
Could you explain what you mean by "turning point"?
In geometrical optics as you well know, a caustic is a surface,
i.e. the envelope of the rays that are normal to a given wavefront
(and therefore to an infinity of others as per Malus-Dupin).
When intersected by say a plane this surface becomes apparent
when the density of rays per unit of surface is high.
As for example sometimes happens when you look wistfully
in your cup of coffee in the morning when certain conditions
are met.
Gregory L. Hansen
That seems like the sort of thing that modern society would judge damaging
to a child's self-esteem.
--
"There's nary an animal alive that can outrun a greased Scotsman!" --
Groundskeeper Willy
mme...@cars3.uchicago.edu
>In article <11naiu1...@corp.supernews.com>,
>Charles Bernard <inc...@nulle.part> wrote:
>>
>><mme...@cars3.uchicago.edu> wrote
>>
>>> Hmm, that's an interdisciplinary project, physics, chemistry and
>>> biology. And since the color of the urine depends on diet, there is a
>>> social and cultural angle to consider, as well.
>>
>>Diet is very important, for example, eat 1kg of red beets and at least
>>500g of asparagus, then if you misbehave you'll surely be noticed,
>>visually or olfactorily.
>>
>>I was told that in pre WWII Germany a special chemical was added to
>>swimming pools water for detection and deterrence.
>>
>>Truth or legend I don't know, an allegedly milky appearance of the water
>>around the culprit was the mark of shame.
>
>That seems like the sort of thing that modern society would judge damaging
>to a child's self-esteem.
>
Gregory L. Hansen
Which, I might have added, was the point-- maybe they'll learn to use the
restrooms.
--
"What are the possibilities of small but movable machines? They may or
may not be useful, but they surely would be fun to make."
-- Richard P. Feynman, 1959
mme...@cars3.uchicago.edu
>In article <HTtdf.18$25....@news.uchicago.edu>,
> <mme...@cars3.uchicago.edu> wrote:
>>In article <dl51hj$oea$4...@rainier.uits.indiana.edu>,
>>glha...@steel.ucs.indiana.edu (Gregory L. Hansen) writes:
>>>In article <11naiu1...@corp.supernews.com>,
>>>Charles Bernard <inc...@nulle.part> wrote:
>>>>
>>>><mme...@cars3.uchicago.edu> wrote
>>>>
>>>>> Hmm, that's an interdisciplinary project, physics, chemistry and
>>>>> biology. And since the color of the urine depends on diet, there is a
>>>>> social and cultural angle to consider, as well.
>>>>
>>>>Diet is very important, for example, eat 1kg of red beets and at least
>>>>500g of asparagus, then if you misbehave you'll surely be noticed,
>>>>visually or olfactorily.
>>>>
>>>>I was told that in pre WWII Germany a special chemical was added to
>>>>swimming pools water for detection and deterrence.
>>>>
>>>>Truth or legend I don't know, an allegedly milky appearance of the water
>>>>around the culprit was the mark of shame.
>>>
>>>That seems like the sort of thing that modern society would judge damaging
>>>to a child's self-esteem.
>>>
>>Yep :-)))
>
>Which, I might have added, was the point-- maybe they'll learn to use the
>restrooms.
>
important than slights to self esteem, yes:-)
Brian Blandford
"Charles Bernard" <inc...@nulle.part> wrote in message
news:11naml2...@corp.supernews.com...
>
> "Brian Blandford" wrote
>
>> As I understand it, a caustic is simply the location of the turning
>> point in the transverse ray aberration function. Even a rainbow
>> can be thought of as a caustic - it is simply the turning point in
>> the refraction-reflection-refraction ray deviation.
>
> Could you explain what you mean by "turning point"?
>
on a sphere of water decreases until the angle of deviation is 138 degrees
and then it increases again. It is only at the turning point, where the
gradient of
deviation angle versus incidence height is zero, that the light is visible.
Hence
the primary rainbow has a semi-angle of 42 degrees.
Brian Blandford
Narasimham
have from Mungan's article:
For refractive index n :
RainbowAngle = ( 4 f [ (4/n^2 -1)/3 ] - 2 f [ (4 -n^2)/3 ] ) * 180/Pi
where f [u] = ArcSin[ Sqrt[u]];
However, the wave geometry here is different, curvature varies point to
point.
Hero.van...@gmx.de
decoyed me to this thread and this group.
The word "caustic" was mentioned there, related to this article of
David R.Wilkins, celebrating Hamilton's 200 th birthday:
http://physicsweb.org/articles/world/18/8/7
And Wilkins published most of Hamilton's work on his website, so i hope
You will enjoy
"The Theory of Systems of Rays"
and the "Third Supplement..."
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Optics.html
Have fun
Hero