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

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...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
Thus spake frank_k_shel...@xxxxxxxxxxx





[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 -

.

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