# What exactly is wrong with Huygens' principle in two dimensions?

by frank_k_sheldon@yahoo.co.uk
Tags: dimensions, huygens, principle
 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 2-dimensional effect. This leads to 2 issues: (1) What exactly does not work in two dimensions, given that all drawings to explain the principle are 2-dimensional? (2) Is there a two-dimensional 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
 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 2-dimensional effect. > >This leads to 2 issues: > >(1) What exactly does not work in two dimensions, given that >all drawings to explain the principle are 2-dimensional? 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 one-dimensional 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 sub-cones) 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 non-Huygensian universe? Does curvature of the spatial manifold affect Huygens's principle?)" > >(2) Is there a two-dimensional 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
 P: n/a 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 2-dimensional effect. > > This leads to 2 issues: > > (1) What exactly does not work in two dimensions, given that > all drawings to explain the principle are 2-dimensional? 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 3-space line simultaneously, so the wave is independent of this line direction. This excitations cannot be independently superposed in 2-space. > > (2) Is there a two-dimensional 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
P: n/a

## 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
> [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 2-dimensional effect.
>
> This leads to 2 issues:
>
> (1) What exactly does not work in two dimensions, given that
> all drawings to explain the principle are 2-dimensional?
>
> (2) Is there a two-dimensional 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
>

 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 2-dimensional? 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 two-dimensional 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.
 P: n/a "John Baez" schreef in bericht news:emmglk$ob1$1@glue.ucr.edu... > In article <1166344055.124563.258610@73g2000cwn.googlegroups.com>, > 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 full-fledged 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 3-dimensional 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/