What exactly is wrong with Huygens' principle in two dimensions?by frank_k_sheldon@yahoo.co.uk Tags: dimensions, huygens, principle 
#1
Dec2006, 05:00 AM

P: n/a

[This issue is still unlear in all discussions
of the topic, even after a thorough search, and despite a similar positing some weeks ago in another group.] It is stressed over and over again that Huygens principle is not valid in two dimensions. See for example, the web page http://www.mathpages.com/home/kmath242/kmath242.htm or the explanation by John Baez. But even those pages have pictures where the principle is illustrated in two dimensions. (As do almost all books and websites.) The enevelope of waves behind a ship is also often deduced in this way, and that is a purely 2dimensional effect. This leads to 2 issues: (1) What exactly does not work in two dimensions, given that all drawings to explain the principle are 2dimensional? (2) Is there a twodimensional wave effect that one CANNOT understand with Huygens' principle? (This should be possible, as it is "wrong" in two dimensions) Are there several such observations? Frank 


#2
Dec2006, 05:00 AM

P: n/a

Thus spake frank_k_sheldon@yahoo.co.uk
>[This issue is still unlear in all discussions >of the topic, even after a thorough search, >and despite a similar positing some weeks ago >in another group.] > >It is stressed over and over again that Huygens principle is >not valid in two dimensions. See for example, the web page >http://www.mathpages.com/home/kmath242/kmath242.htm >or the explanation by John Baez. > >But even those pages have pictures where the principle is >illustrated in two dimensions. (As do almost all books and websites.) >The enevelope of waves behind a ship is also often deduced in >this way, and that is a purely 2dimensional effect. > >This leads to 2 issues: > >(1) What exactly does not work in two dimensions, given that >all drawings to explain the principle are 2dimensional? I can't explain it any better than the website you cite. It is one of those explanations you can only understand by following the equations, and all the equation are given there. As for a verbal explanation as to how the principle breaks down, I refer you to this paragraph. The critical point is that waves of different wavelength propagate with different speeds. "It's worth noting that in the above derivation we were able to reduce the polar wave equation to a simple onedimensional equation by taking advantage of the fact that an unwanted term vanished when the number of space dimensions is n = 3 (or n = 1). For the case of two dimensional space this doesn't work (nor would it work with four space dimensions). We can still solve the wave equation, but the solution is not just a simple spherical wave propagating with unit velocity. Instead, we find that there are effectively infinitely many velocities, in the sense that a single pulse disturbance at the origin will propagate outward on infinitely many "light cones" (and subcones) with speeds ranging from the maximum down to zero. Hence if we lived in a universe with two spatial dimensions (instead of three), an observer at a fixed location from the origin of a single pulse would "see" an initial flash but then the disturbance "afterglow" would persist, becoming less and less intense, but continuing forever, as slower and slower subsidiary branches arrive. (It's interesting to compare and contrast this "afterglow" with the cosmic microwave background radiation that we actually do observe in our 3+1 dimensional universe. Could this glow be interpreted as evidence of an additional, perhaps compactified, spatial dimension? What would be the spectrum of the glow in a nonHuygensian universe? Does curvature of the spatial manifold affect Huygens's principle?)" > >(2) Is there a twodimensional wave effect that one CANNOT understand >with Huygens' principle? (This should be possible, as it is "wrong" >in two dimensions) Are there several such observations? > Yes. As an undergrad I had to write a computer programme to model the wave effect of a moving and pulsating source (e.g. that produced by a duck, where the paddling feet is quadrupole). The exercise actually suggested modelling using Huyghens principle, but the result was nonsense, so I solved the equations directly and got a completely different, and sensible, wave pattern. Regards  Charles Francis substitute charles for NotI to email 


#3
Dec2406, 05:00 AM

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frank_k_sheldon@yahoo.co.uk schrieb:
> [This issue is still unlear in all discussions > of the topic, even after a thorough search, > and despite a similar positing some weeks ago > in another group.] > > It is stressed over and over again that Huygens principle is > not valid in two dimensions. See for example, the web page > http://www.mathpages.com/home/kmath242/kmath242.htm > or the explanation by John Baez. > > But even those pages have pictures where the principle is > illustrated in two dimensions. (As do almost all books and websites.) > The enevelope of waves behind a ship is also often deduced in > this way, and that is a purely 2dimensional effect. > > This leads to 2 issues: > > (1) What exactly does not work in two dimensions, given that > all drawings to explain the principle are 2dimensional? That' a simple issue: In three space dimensions a point excitation travels in space time on the surface of the light cone (ct)^2 (x^2+y^2+z^2) >=0. This is independent of the way you stimulate the amplitude or the velocity or both. So you can simply add all point excitations and this is the simple background of Huyghens principle. In 2 space dimensions you get the wave of a point excitation by adding all excitations along a one dimensional in 3space line simultaneously, so the wave is independent of this line direction. This excitations cannot be independently superposed in 2space. > > (2) Is there a twodimensional wave effect that one CANNOT understand > with Huygens' principle? (This should be possible, as it is "wrong" > in two dimensions) Are there several such observations? This effect in even dimensional wave problems is called "wave diffusion" in the interior of the light cone.  Roland Franzius 


#4
Dec2406, 05:00 AM

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What exactly is wrong with Huygens' principle in two dimensions?
In the below metioned url I read the following:
"and if Huygens' Principle was valid in two dimensions," This sentence seems to imply that Huygens Principle does not apply in two dimensions. (The same with the subject of this thread) In the book "Physics" by Prof Kronig 1958 at page 354 I read: (Free translation from Dutch) With the aid of waterwaves we can explain (make clear) Huygens Principle. We make make waves at a certain point A. We have a wall at a certain distance with a narrow slit S. When you do that behind point S a new set of waves will appear centred around point S. S will be a new centre of vibrations. (End of free translation) That means you can explain the 3 dimensional Huygens Principle with a 2 dimensional set up. If that is the case why is the huygens principle than also not valid in "two" dimensions ? or am I missing something. Nicolaas Vroom http://users.pandora.be/nicvroom/ <frank_k_sheldon@yahoo.co.uk> schreef in bericht news:1166344055.124563.258610@73g2000cwn.googlegroups.com... > [This issue is still unlear in all discussions > of the topic, even after a thorough search, > and despite a similar positing some weeks ago > in another group.] > > It is stressed over and over again that Huygens principle is > not valid in two dimensions. See for example, the web page > http://www.mathpages.com/home/kmath242/kmath242.htm > or the explanation by John Baez. > > But even those pages have pictures where the principle is > illustrated in two dimensions. (As do almost all books and websites.) > The enevelope of waves behind a ship is also often deduced in > this way, and that is a purely 2dimensional effect. > > This leads to 2 issues: > > (1) What exactly does not work in two dimensions, given that > all drawings to explain the principle are 2dimensional? > > (2) Is there a twodimensional wave effect that one CANNOT understand > with Huygens' principle? (This should be possible, as it is "wrong" > in two dimensions) Are there several such observations? > > > Frank > 


#5
Dec2406, 05:00 AM

P: n/a

On Tue, 19 Dec 2006 frank_k_sheldon@yahoo.co.uk wrote:
>It is stressed over and over again that Huygens principle is >not valid in two dimensions... >(1) What exactly does not work in two dimensions, given that >all drawings to explain the principle are 2dimensional? Those drawings are intended to depict the sharp propagation of a wave, i.e., you are to understand that the leading and trailing edges of a single wave propagate at the same speed, so the wave doesn't get "thicker". Thus if you turn on a light bulb for one second, someone viewing the light from a mile away will see it "on" for only one second, no longer. You can depict this in two dimensions, but waves don't really propagate that way in two dimensions. For example, if you drop a pebble in a calm pond, a circular wave will emanate outward. If Huygens' Principle applied in two dimensions, the surface of the pond would be perfectly quiet both outside the expanding spherical wave AND INSIDE the spherical wave. But Huygens' Principle does not apply in two dimensions, so the surface of the pond inside the expanding wave will not be perfectly calm. It will still be undulating slightly, and this will persist indefinitely, although the magnitude becomes extremely small. The point is, the leading edge of the wave always propagates at the characteristic speed c, regardless of whether Huyghens' Principle is true. Huygens' Principle is more about the speed of the trailing edge, i.e., it is about what happens BEHIND the leading edge of the disturbance. >(2) Is there a twodimensional wave effect that one CANNOT understand >with Huygens' principle? (This should be possible, as it is "wrong" >in two dimensions) Are there several such observations? Sure, just look at how the surface of the pond undulates AFTER the wave front has passed through. This wouldn't happen if Huygens' Principle was true for two dimensions. Conversely, look at the sharp coherent images of stars and distant galaxies. Those would be somewhat smeared out if Huygens' Principle was not true for a space of three dimensions. 


#6
Dec2406, 05:00 AM

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The mathematics is clear but it does seem funny physically since I have
seen demonstrations of Huygen's principle that use waves in a pan of water. Also, I don't recall seeing the effect of afterglow and multiple speeds when throwing a rock on a pond. I wonder it there exists a modified wave equations that satisfies Huygen's principle. A related question is this. If an electromagnetic disturbance (light) in a medium really satisfies the ordinary wave equation in three dimensions then it should always travel at the same speed which is a constant k (less than or equal c) appearing in the supposed wave equation valid for that medium. Just sticking to a fixed wave equation, it is hard to see how dispersion is possible. So it appears that if there is dispersion then the light in a medium is a superposition of waves that satisfy different wave equations depending on frequency while the total disturbance does not satisfy any single wave equation. But if there is more than one wave equation involved (with various k's) why would we assume the superposition principle (which was all about a single PDE being linear). On Dec 20, 3:04 am, Oh No <N...@charlesfrancis.wanadoo.co.uk> wrote: > Thus spake frank_k_shel...@yahoo.co.uk > > > > > > >[This issue is still unlear in all discussions > >of the topic, even after a thorough search, > >and despite a similar positing some weeks ago > >in another group.] > > >It is stressed over and over again that Huygens principle is > >not valid in two dimensions. See for example, the web page > >http://www.mathpages.com/home/kmath242/kmath242.htm > >or the explanation by John Baez. > > >But even those pages have pictures where the principle is > >illustrated in two dimensions. (As do almost all books and websites.) > >The enevelope of waves behind a ship is also often deduced in > >this way, and that is a purely 2dimensional effect. > > >This leads to 2 issues: > > >(1) What exactly does not work in two dimensions, given that > >all drawings to explain the principle are 2dimensional?I can't explain it any better than the website you cite. It is one of > those explanations you can only understand by following the equations, > and all the equation are given there. As for a verbal explanation as to > how the principle breaks down, I refer you to this paragraph. The > critical point is that waves of different wavelength propagate with > different speeds. > > "It's worth noting that in the above derivation we were able to reduce > the polar wave equation to a simple onedimensional equation by taking > advantage of the fact that an unwanted term vanished when the number of > space dimensions is n = 3 (or n = 1). For the case of two dimensional > space this doesn't work (nor would it work with four space dimensions). > We can still solve the wave equation, but the solution is not just a > simple spherical wave propagating with unit velocity. Instead, we find > that there are effectively infinitely many velocities, in the sense that > a single pulse disturbance at the origin will propagate outward on > infinitely many "light cones" (and subcones) with speeds ranging from > the maximum down to zero. Hence if we lived in a universe with two > spatial dimensions (instead of three), an observer at a fixed location > from the origin of a single pulse would "see" an initial flash but then > the disturbance "afterglow" would persist, becoming less and less > intense, but continuing forever, as slower and slower subsidiary > branches arrive. (It's interesting to compare and contrast this > "afterglow" with the cosmic microwave background radiation that we > actually do observe in our 3+1 dimensional universe. Could this glow be > interpreted as evidence of an additional, perhaps compactified, spatial > dimension? What would be the spectrum of the glow in a nonHuygensian > universe? Does curvature of the spatial manifold affect Huygens's > principle?)" > > > > >(2) Is there a twodimensional wave effect that one CANNOT understand > >with Huygens' principle? (This should be possible, as it is "wrong" > >in two dimensions) Are there several such observations?Yes. As an undergrad I had to write a computer programme to model the > wave effect of a moving and pulsating source (e.g. that produced by a > duck, where the paddling feet is quadrupole). The exercise actually > suggested modelling using Huyghens principle, but the result was > nonsense, so I solved the equations directly and got a completely > different, and sensible, wave pattern. > > Regards > >  > Charles Francis > substitute charles for NotI to email Hide quoted text  Show quoted text  


#7
Dec2506, 05:00 AM

P: n/a

In article <1166344055.124563.258610@73g2000cwn.googlegroups.com>,
<frank_k_sheldon@yahoo.co.uk> wrote: >But even those pages have pictures where the principle is >illustrated in two dimensions. (As do almost all books and websites.) >The enevelope of waves behind a ship is also often deduced in >this way, and that is a purely 2dimensional effect. In any dimension, the *envelope* a pattern of waves satisfying the wave equation can be deduced by drawing a circle of radius t/v about each point that emitted a wave at a time t ago, if the waves move at velocity v. This "weak" version of Huyghens' principle works in any dimension. The fullfledged Huyghens principle says that if a point emits a wave at a time t ago, there will be no wave anywhere *except* at the circle of radius t/v about that point. This works only when the dimension of space is odd and greater than 1. >This leads to 2 issues: >(1) What exactly does not work in two dimensions, given that >all drawings to explain the principle are 2dimensional? > >(2) Is there a twodimensional wave effect that one CANNOT understand >with Huygens' principle? We can give the same answer to both questions. First consider 3 dimensions. If a point source of light blinks on for an instant at time 0, at some later time t there'll be no light visible except right on the surface of the sphere of radius t/c centered at this point. This is the strong version of Huyghens' principle. Next consider the same situation in 2 dimensions. We can figure out what happens using 3dimensional reasoning, since a point source of light in 2 dimensions acts exactly like a *line* source of light in 3 dimensions! Using the 3d Huyghens principle together with the superposition principle, we see that a point at a distance t/c from the line source will *first* see light at time t. But, it will continue to see light at later times, emitted from points further away along the line. So, it will see a decaying "afterglow" after the initial burst of light. In short: since the strong version of Huyghens' principle holds in 3 dimensions, it cannot hold in 2 dimensions. And, there's nothing special about the numbers 2 and 3 here. The same argument shows this: if the strong version of Huyghens' principle holds in ndimensional space, it cannot hold in (n1)dimensional space. So: since the strong version of Huyghens' principle holds in 3dimensional space, it cannot hold in 4dimensional space. We can't draw any further conclusions about Huyghens' principle in various dimensions from what I've said so far. But, explicitly solving the wave equation for a point source of light shows that the strong version of Huyghens' principle holds in ODD dimensions greater than 1, but not EVEN dimensions or  the curious exception  1 dimension. This is consistent with the fact that if the principle holds in ndimensional space, it cannot hold in (n1)dimensional space. All this stuff was worked out in ancient discussions on sci.physics.research... I forget by whom. By the way  the Dutch write "Huyghens", but NASA has taken to writing "Huygens". Happy Holidays! ...................................................................... Puzzle 33: Who owns all the unmarked mute swans on the River Thames? If you get stuck, see http://math.ucr.edu/home/baez/puzzles/33.html 


#8
Dec2606, 05:00 AM

P: n/a

Thus spake Cyberkatru <cyberkatru@gmail.com>
>The mathematics is clear but it does seem funny physically since I have >seen demonstrations of Huygen's principle that use waves in a pan of >water. Also, I don't recall seeing the effect of afterglow and multiple >speeds when throwing a rock on a pond. It is difficult to carry out a scientific test on a pond, but next time you get the chance repeat the test a few times and examine the pattern carefully as it expands. In water low frequencies travel faster than high. In consequence you will see the leading wave altering shape and becoming broader. Behind that, you will see shorter wavelength ripples forming, just as Dale Mortenson has described. > >I wonder it there exists a modified wave equations that satisfies >Huygen's principle. > > >A related question is this. If an electromagnetic disturbance (light) >in a medium really satisfies the ordinary wave equation in three >dimensions then it should always travel at the same speed which is a >constant k (less than or equal c) appearing in the supposed wave >equation valid for that medium. Just sticking to a fixed wave equation, >it is hard to see how dispersion is possible. So it appears that if >there is dispersion then the light in a medium is a superposition of >waves that satisfy different wave equations depending on frequency >while the total disturbance does not satisfy any single wave equation. >But if there is more than one wave equation involved (with various k's) >why would we assume the superposition principle (which was all about a >single PDE being linear). A dispersive wave still satisfies a pde, just not the simple wave equation (or it may be the simple wave equation in a wave guide). It's solutions have the form exp(i(wtk.x) where w=w(k), and the general solution still obeys the superposition principle. > >On Dec 20, 3:04 am, Oh No <N...@charlesfrancis.wanadoo.co.uk> wrote: >> Thus spake frank_k_shel...@yahoo.co.uk >> >> >> >> >> >> >[This issue is still unlear in all discussions >> >of the topic, even after a thorough search, >> >and despite a similar positing some weeks ago >> >in another group.] >> >> >It is stressed over and over again that Huygens principle is >> >not valid in two dimensions. See for example, the web page >> >http://www.mathpages.com/home/kmath242/kmath242.htm >> >or the explanation by John Baez. >> >> >But even those pages have pictures where the principle is >> >illustrated in two dimensions. (As do almost all books and websites.) >> >The enevelope of waves behind a ship is also often deduced in >> >this way, and that is a purely 2dimensional effect. >> >> >This leads to 2 issues: >> >> >(1) What exactly does not work in two dimensions, given that >> >all drawings to explain the principle are 2dimensional?I can't >> >explain it any better than the website you cite. It is one of >> those explanations you can only understand by following the equations, >> and all the equation are given there. As for a verbal explanation as to >> how the principle breaks down, I refer you to this paragraph. The >> critical point is that waves of different wavelength propagate with >> different speeds. >> >> "It's worth noting that in the above derivation we were able to reduce >> the polar wave equation to a simple onedimensional equation by taking >> advantage of the fact that an unwanted term vanished when the number of >> space dimensions is n = 3 (or n = 1). For the case of two dimensional >> space this doesn't work (nor would it work with four space dimensions). >> We can still solve the wave equation, but the solution is not just a >> simple spherical wave propagating with unit velocity. Instead, we find >> that there are effectively infinitely many velocities, in the sense that >> a single pulse disturbance at the origin will propagate outward on >> infinitely many "light cones" (and subcones) with speeds ranging from >> the maximum down to zero. Hence if we lived in a universe with two >> spatial dimensions (instead of three), an observer at a fixed location >> from the origin of a single pulse would "see" an initial flash but then >> the disturbance "afterglow" would persist, becoming less and less >> intense, but continuing forever, as slower and slower subsidiary >> branches arrive. (It's interesting to compare and contrast this >> "afterglow" with the cosmic microwave background radiation that we >> actually do observe in our 3+1 dimensional universe. Could this glow be >> interpreted as evidence of an additional, perhaps compactified, spatial >> dimension? What would be the spectrum of the glow in a nonHuygensian >> universe? Does curvature of the spatial manifold affect Huygens's >> principle?)" >> >> >> >> >(2) Is there a twodimensional wave effect that one CANNOT understand >> >with Huygens' principle? (This should be possible, as it is "wrong" >> >in two dimensions) Are there several such observations?Yes. As an >> >undergrad I had to write a computer programme to model the >> wave effect of a moving and pulsating source (e.g. that produced by a >> duck, where the paddling feet is quadrupole). The exercise actually >> suggested modelling using Huyghens principle, but the result was >> nonsense, so I solved the equations directly and got a completely >> different, and sensible, wave pattern. >> >> Regards >> >>  >> Charles Francis >> substitute charles for NotI to email Hide quoted text  Show quoted text  > Regards  Charles Francis substitute charles for NotI to email 


#9
Dec2606, 05:00 AM

P: n/a

In article <1166633885.548868.206360@80g2000cwy.googlegroups.com>,
Cyberkatru <cyberkatru@gmail.com> wrote: >The mathematics is clear but it does seem funny physically since I have >seen demonstrations of Huygen's principle that use waves in a pan of >water. That's the *weak* Huyghens' principle, which we use to compute the *envelope* of some waves. > Also, I don't recall seeing the effect of afterglow and multiple >speeds when throwing a rock on a pond. Oh? It's actually a lot of fun to throw a rock into a pond and see the elaborate complicated ripples of different wavelengths following the main wavefront. But, when we last discussed this here on sci.physics.research, I was informed that water waves don't satisfy the wave equation very accurately  so some of these effects are due to "dispersion": the fact that different wavelengths move at different speeds. >I wonder it there exists a modified wave equations that satisfies >Huygen's principle. That's a fun question. One can start with a desired "Green's function" or "propagator" and try to figure out what wave equation it satisfies. But, I suspect that in 2d space a wave equation that satisfies Huyghens' principle would have to be nonlocal  i.e., not just a differential equation, but an integrodifferential equation. >A related question is this. If an electromagnetic disturbance (light) >in a medium really satisfies the ordinary wave equation in three >dimensions then it should always travel at the same speed which is a >constant k (less than or equal c) appearing in the supposed wave >equation valid for that medium. Just sticking to a fixed wave equation, >it is hard to see how dispersion is possible. So it appears that if >there is dispersion then the light in a medium is a superposition of >waves that satisfy different wave equations depending on frequency >while the total disturbance does not satisfy any single wave equation. >But if there is more than one wave equation involved (with various k's) >why would we assume the superposition principle (which was all about a >single PDE being linear). I don't understand this question very well, but: 1) It's easy to get dispersion from a fixed equation; just try something like Laplacian f + (Laplacian)^2 f = d^2 f / dt^2 giving the dispersion relation k^2 + k^4 = omega^2 so that k and omega are no longer proportional. 2) The failure of the strong Huyghens principle is not due to dispersion. .................................................................... Puzzle 32: which job are only blind people allowed to do in Korea? If you get stuck see: http://math.ucr.edu/home/baez/puzzles/32.html 


#10
Dec2606, 05:00 AM

P: n/a

John Baez wrote:
> In article <1166344055.124563.258610@73g2000cwn.googlegroups.com>, > > Next consider the same situation in 2 dimensions. We can figure > out what happens using 3dimensional reasoning, since a point source > of light in 2 dimensions acts exactly like a *line* source of light > in 3 dimensions! > > Using the 3d Huyghens principle together with the superposition > principle, we see that a point at a distance t/c from the line > source will *first* see light at time t. But, it will continue > to see light at later times, emitted from points further away > along the line. So, it will see a decaying "afterglow" after the > initial burst of light. > I still don't get it. I see what you mean if you are trying to create a 2D system in 3D by using line sources of light. But in a *true* 2D system (Flatland), if you are a distance from t/c from the source, you see the source only at that instant. There is no "further away along the line" in a true 2D system. 


#11
Dec2706, 05:00 AM

P: n/a

In article <1167000546.363556.224750@73g2000cwn.googlegroups.com>,
andrew.higgins@mcgill.ca <andrew.higgins@mcgill.ca> wrote: >John Baez wrote: >> Next consider the same situation in 2 dimensions. We can figure >> out what happens using 3dimensional reasoning, since a point source >> of light in 2 dimensions acts exactly like a *line* source of light >> in 3 dimensions! >> Using the 3d Huyghens principle together with the superposition >> principle, we see that a point at a distance t/c from the line >> source will *first* see light at time t. But, it will continue >> to see light at later times, emitted from points further away >> along the line. So, it will see a decaying "afterglow" after the >> initial burst of light. >I still don't get it. Okay. >I see what you mean if you are trying to create a 2D system in 3D by >using line sources of light. It's a mathematical fact that this works. Any solution of the wave equation in 2d can be *perfectly* mimicked by a solution of the wave equation in 3d which is constant in one direction. So, we can use 3d reasoning to understand the wave equation in 2d. This is easy to check: If f(t,x,y) is a solution of the 2d wave equation (d^2/dt^2  d^2/dx^2  d^2/dy^2) f = 0 then we can define g(t,x,y,z) = f(t,y,z) and this satisfies the 3d wave equation (d^2/dt^2  d^2/dx^2  d^2/dy^2  d^2/dz^2) g = 0 since it's independent of z. Conversely, any solution of (d^2/dt^2  d^2/dx^2  d^2/dy^2  d^2/dz^2) g = 0 which is independent of z gives a function f(t,y,z) = g(t,x,y,z) which satisfies (d^2/dt^2  d^2/dx^2  d^2/dy^2) f = 0 >But in a *true* 2D system (Flatland), if you are a distance from >t/c from the source, you see the source only at that instant. No! Your claim here amounts to the strong Huyghens principle, which is precisely what's under contention. I haven't proved that the strong Huyghens principle holds in 3d  though that's true. But, I've shown that if it's true in 3d, it fails in 2d. So, if you (correctly) believe in this principle in 3d, you have to not believe it in 2d. And if you (wrongly) believe it in 2d, you have to not believe it in 3d. 


#12
Dec2706, 05:01 AM

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In article <emmglk$ob1$1@glue.ucr.edu>, John Baez wrote:
> By the way  the Dutch write "Huyghens", but NASA has taken to > writing "Huygens". Well, not quite. In the 17th century Dutch didn't have a fixed spelling, so you could have found both forms. But nowadays the Dutch use Huygens. For instance there is the Huygens institute of the Royal Dutch Acadamy of Sciences, honoring both Christiaan and Constantijn Huygens, see http://www.huygensinstituut.knaw.nl/...mid=56&lang=en There are also streets in Amsterdam named after Constantijn Huygens, the poet, http://nl.wikipedia.org/wiki/Eerste_...at_(Amsterdam).  Maarten Bergvelt Mathematics Department, University of Illinois UrbanaChampaign, IL 61801 Ph: 2173336326 email: bergv@math.uiuc.edu 


#13
Dec2906, 05:00 AM

P: n/a

In article <slrnep2hvv.ete.bergv@u00.math.uiuc.edu>,
Maarten Bergvelt <bergv@math.uiuc.edu> wrote: >In article <emmglk$ob1$1@glue.ucr.edu>, John Baez wrote: >> By the way  the Dutch write "Huyghens", but NASA has taken to >> writing "Huygens". >Well, not quite. In the 17th century Dutch didn't have a fixed >spelling, so you could have found both forms. But nowadays the Dutch >use Huygens. Oh! Whoops. And here I thought NASA was just dumbing down the name for consonantdeprived Americans. Okay, henceforth I'll write "Huygens". Thanks! ................................................................... Puzzle 30: What happened when Fidel Castro's brother took a ride in a flying car? If you get stuck see: http://math.ucr.edu/home/baez/puzzles/30.html 


#14
Dec2906, 05:00 AM

P: n/a

John Baez <baez@math.removethis.ucr.andthis.edu> wrote:
[...] > We can give the same answer to both questions. First consider > 3 dimensions. If a point source of light blinks on for an > instant at time 0, at some later time t there'll be no light > visible except right on the surface of the sphere of radius t/c > centered at this point. This is the strong version of Huyghens' > principle. > Next consider the same situation in 2 dimensions. We can figure > out what happens using 3dimensional reasoning, since a point source > of light in 2 dimensions acts exactly like a *line* source of light > in 3 dimensions! > Using the 3d Huyghens principle together with the superposition > principle, we see that a point at a distance t/c from the line > source will *first* see light at time t. But, it will continue > to see light at later times, emitted from points further away > along the line. So, it will see a decaying "afterglow" after the > initial burst of light. > In short: since the strong version of Huyghens' principle holds > in 3 dimensions, it cannot hold in 2 dimensions. > And, there's nothing special about the numbers 2 and 3 here. > The same argument shows this: if the strong version of Huyghens' > principle holds in ndimensional space, it cannot hold in > (n1)dimensional space. But why can't I go down more than one dimension? For example, a point source in 1 dimension should look like a plane source in 3 dimensions  why doesn't the strong version of Huyghens' principle therefore fail in 1d? (My first guess is that you must have destructive interference from various points on the plane, but it's not obvious...) Steve Carlip 


#15
Dec3006, 05:00 AM

P: n/a

In article <emmglk$ob1$1@glue.ucr.edu>,
baez@math.removethis.ucr.andthis.edu says... > In article <1166344055.124563.258610@73g2000cwn.googlegroups.com>, > <frank_k_sheldon@yahoo.co.uk> wrote: > Next consider the same situation in 2 dimensions. We can figure > out what happens using 3dimensional reasoning, since a point source > of light in 2 dimensions acts exactly like a *line* source of light > in 3 dimensions! By the same argument, does a plane source in 5D act like a point source in 3D? That will require some impressive cancellation by superposition, because there will be large components at later times.  Gerry Quinn 


#16
Dec3006, 05:00 AM

P: n/a

carlipnospam@physics.ucdavis.edu wrote:
> > But why can't I go down more than one dimension? For example, a > point source in 1 dimension should look like a plane source in 3 > dimensions  why doesn't the strong version of Huyghens' principle > therefore fail in 1d? > > (My first guess is that you must have destructive interference from > various points on the plane, but it's not obvious...) > > Steve Carlip The photon propagator in 1d isn't "on the light cone" only. It is constant everywhere within the lightcone and zero outside the lightcone. If you take a look at a paper I wrote recently: http://chiparchitect.com/physics/Hi..._radiation.pdf Then, you'll see in fig.1 the propagators for the first 5 dimensions. Only the odd dimensions from 3d and higher are on the lightcone only. It's only in 3d that the propagator is a nice Dirac pulse. Higher odd dimensions have (higher) derivatives of the Dirac pulse as propagator. The paper contains a complete derivation of the propagators in any dimension. The 1d propagator is particular simple: see page 3, Sec. IV. at the top of the left column. On page 6, halfway the left column, it talks about the propagator from a plane in 3d which represents exactly the 1d situation you mentioned. Regards, Hans 


#17
Jan407, 05:00 AM

P: n/a

"John Baez" <baez@math.removethis.ucr.andthis.edu> schreef in bericht
news:emmglk$ob1$1@glue.ucr.edu... > In article <1166344055.124563.258610@73g2000cwn.googlegroups.com>, > <frank_k_sheldon@yahoo.co.uk> wrote: > > In any dimension, the *envelope* a pattern of waves satisfying > the wave equation can be deduced by drawing a circle of radius > t/v about each point that emitted a wave at a time t ago, if > the waves move at velocity v. This "weak" version of Huyghens' > principle works in any dimension. > > The fullfledged Huyghens principle says that if a point emits > a wave at a time t ago, there will be no wave anywhere *except* > at the circle of radius t/v about that point. This works only > when the dimension of space is odd and greater than 1. > Accordingly to the book Physics (See previous posting) page 352 the Huygens principle says: (Free Dutch Translation) "Each point of a wavefront can be considered a new vibration point, which transmits lightpulses. A new wavefront consists (comes across) by taking the envelop of these elementary wavefronts" In dutch we use the word wavefront which has a similar meaning as bowwave. Wavefront means something like the beginning of the wave which moves forward. We also have the word weatherfront. Together with this text there is a picture with the text: "Propagation (transmission) of sphereshaped wavefronts" This picture shows two circles one t with radius vt and one at t+t' with radius v*(t+t') > We can give the same answer to both questions. First consider > 3 dimensions. If a point source of light blinks on for an > instant at time 0, at some later time t there'll be no light > visible except right on the surface of the sphere of radius t/c > centered at this point. This is the strong version of Huyghens' > principle. > > Next consider the same situation in 2 dimensions. How do you perform this 2D experiment in reality ? Is this something similar as light transmitted by a lighthouse? Suppose the lighthouse does not turn around and the lighthouse emits one short flash and you are inside the beam. Is it not true that in that case you will only see one flash ? with rather sharp edges ? and no afterglow ? Or is this an example in 1D ? > We can figure > out what happens using 3dimensional reasoning, since a point source > of light in 2 dimensions acts exactly like a *line* source of light > in 3 dimensions! > > Using the 3d Huyghens principle together with the superposition > principle, we see that a point at a distance t/c from the line > source will *first* see light at time t. But, it will continue > to see light at later times, emitted from points further away > along the line. So, it will see a decaying "afterglow" after the > initial burst of light. > Nicolaas Vroom http://user.pandora.be/nicvroom/ 


#18
Jan407, 05:00 AM

P: n/a

The general problem I have is when people start discussions
in anything different than 3D. With 3D I mean the x,y and z coordinates. The next "dimension" is time. What is the physical meaning of one photon (or particle) in 1D, 2D, 3D, 4D, 5D etc ? What is the physical meaning of two photons (or particle) in 1D, 2D, 3D, 4D, 5D etc ? IMO when you speak of 4D, 5D etc this only makes sense in mathematical context. I expect they do not mean: Pressure and Temperature. The same with 1D and 2D. How do you perform experiments in 1D and 2D ? In the book "Not even Wrong" at page 147 Peter Woit writes: "this kind of string theory could be made sense of in 10 dimensions rather then the 26 dimensions of the original string" and at page 151 "E8 is the largest etc and it corresponds to a 248 dimensional space of possible symmetry transformations" At page 154 and 155 the concepts seven and eleven dimensions are frequently used. IMO the meaning of the word dimension requires clarification with a distinction between physics and mathematics. "Hans de Vries" <hansdevries@chiparchitect.com> schreef in bericht news:1167426816.533578.159970@a3g2000cwd.googlegroups.com... > carlipnospam@physics.ucdavis.edu wrote: >> >> But why can't I go down more than one dimension? For example, a >> point source in 1 dimension should look like a plane source in 3 >> dimensions  why doesn't the strong version of Huyghens' principle >> therefore fail in 1d? >> > > > The photon propagator in 1d isn't "on the light cone" only. > It is constant everywhere within the lightcone and zero outside > the lightcone. If you take a look at a paper I wrote recently: > > http://chiparchitect.com/physics/Hi..._radiation.pdf > > Then, you'll see in fig.1 the propagators for the first 5 dimensions. > Only the odd dimensions from 3d and higher are on the lightcone > only. It's only in 3d that the propagator is a nice Dirac pulse. > Higher odd dimensions have (higher) derivatives of the Dirac > pulse as propagator. > I expect everything what is written above is mathematical correct, but what does it mean physical (in 3D and 4D if time is included) ? Nicolaas Vroom http://users.pandora.be/nicvroom/ 


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