From: "A. Bossavit"
Subject: Re: Gaussian pulse in 2D FDTD mesh
Date: Fri, 11 Aug 2000 15:32:43 +0000
Newsgroups: sci.physics.electromag,sci.math.num-analysis
Summary: [missing]
Gordon D. Pusch wrote:
>Jos [Bergervoet] is referring to an old result by (IIRC)
>Sommerfeld that non-dispersive
>wave-propagation only occurs in spaces with an _ODD_ number
>of dimensions,
This is treated in
H. Soodak, M.S. Tiersten: "Wakes and waves in N dimensions",
Am. J. Phys., 61, 5 (1993), pp. 395-401.
(See also p. 133 of
A.K. Dewdney: The Planiverse, Computer contact with a two-dimensional
world, Pan Books (Londres), 1984.
Other pages are worth reading, too).
What S&T say, in short, is this: If an antenna emits during
a finite interval of time, the perceived signal some distance away will be
nonzero for a finite time only (hence, no "wake"). This, according to S&T,
holds in all odd dimensions. In even dimensions, and thus is 2D,
there *is* a wake, of vanishing intensity but infinite duration--which
I hope vindicates JB's explanation of thunder rumble; it's too nice
not to be true.) Moreover, in 3D, and only in 3D, emitted and received
signal have the same shape.
In view of this, the expression " non-dispersive wave-propagation"
is somewhat ambiguous: one may have, in other dimensions than 3, same
propagation speed for all frequencies (no "dispersion", in common
parlance), and yet alteration of the shape of the signal.
S&T don't mention Sommerfeld. They cite Morse & Feshbach and Courant &
Hilbert as two of the precious few textbooks which do address the subject.
*Numerical* dispersion, of course, comes on top of all that.
==============================================================================
From: Jos Bergervoet
Subject: Re: Gaussian pulse in 2D FDTD mesh
Date: 13 Aug 2000 08:59:39 GMT
Newsgroups: sci.physics.electromag,sci.math.num-analysis
In sci.physics.electromag A. Bossavit wrote:
> ... Moreover, in 3D, and only in 3D, emitted and received
> signal have the same shape.
One should add 1D as well (which is the more or less trivial case of
the transmission line). In both these cases the retarded potentials
can be written as Dirac_delta(t-x/c), with c the signal speed.
But as far as I know it is restricted to those two cases. It is not
true for any (positive) even number of dimensions and also not for 5D,
7D, etc.
-- Jos
==============================================================================
From: Jos R Bergervoet
Subject: Re: Gaussian pulse in 2D FDTD mesh
Date: Tue, 15 Aug 2000 11:41:59 +0200
Newsgroups: sci.physics.electromag,sci.math.num-analysis
Gordon D. Pusch wrote:
>
> Jos Bergervoet writes:
>> ...
>> But as far as I know it is restricted to those two cases. It is not
>> true for any (positive) even number of dimensions and also not for
>> 5D, 7D, etc.
>
> Do you have a reference for this?
I believe it was discussed before in one of the sci.physics
newsgroups, but I don't know the subject line of that thread.
> ... It was my understanding from the proof
> I earlier attributed to Sommerfeld that one obtained a sharp
> wavefront with no ``tail'' in any uniform odd-dimensional space...
I think that is only possible if you can scale the solution
with some r-dependent factor, such that the resulting function
satisfies the simple 1D wave equation, and hence has sin(r)
and cos(r) as solution. For 3D, the scaling factor is just one
extra factor of r. For 5D, energy conservation would require
r^2 as scaling. But with that factor, the Laplace operator
does not reduce to a second derivative of the scaled function
(an extra 2/r^2 pops in)
-- jos
-----------------------------------------------------------------------
Dr. Jozef R. Bergervoet Electromagnetism and EMC
Philips Research Laboratories, Eindhoven, The Netherlands
Building WS01 FAX: +31-40-2742224
E-mail: bergervo@natlab.research.philips.com Phone: +31-40-2742403