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

In article <1167000546.363556.224750@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
andrew.higgins@xxxxxxxxx <andrew.higgins@xxxxxxxxx> 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.

.