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

In article <1166344055.124563.258610@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<frank_k_sheldon@xxxxxxxxxxx> 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


.

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