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

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?

(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

Oh No

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

Roland Franzius

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

Nicolaas Vroom

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 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
>

Dale Mortenson

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.

Cyberkatru

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 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- Hide quoted text -- Show quoted text -

John Baez

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 2-dimensional 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 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.

>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?

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 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.

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 n-dimensional space, it cannot hold in
(n-1)-dimensional space.

So: since the strong version of Huyghens' principle holds in
3-dimensional space, it cannot hold in 4-dimensional 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 n-dimensional space, it cannot hold in (n-1)-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

Oh No

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(wt-k.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 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- Hide quoted text -- Show quoted text -
>

Regards

--
Charles Francis
substitute charles for NotI to email

John Baez

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

andrew.higgins@mcgill.ca

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 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.
>

I still don't get it.

I see what you mean if you are trying to create a 2-D system in 3-D by
using line sources of light. But in a *true* 2-D 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 2-D
system.

John Baez

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 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.

>I still don't get it.

Okay.

>I see what you mean if you are trying to create a 2-D system in 3-D 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* 2-D 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.

Maarten Bergvelt

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
Urbana-Champaign, IL 61801
Ph: 217-333-6326 email: bergv@math.uiuc.edu

John Baez

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 consonant-deprived 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

carlip-nospam@physics.ucdavis.edu

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 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.

> 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 n-dimensional space, it cannot hold in
> (n-1)-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

Gerry Quinn

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 3-dimensional 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

Hans de Vries

carlip-nospam@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://chip-architect.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

Nicolaas Vroom

"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 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/