Re: [Phys-l] physics lessons from wikipedia

[John Denker <jsd@av8n.com>]

Note:  I apologize for the typos in my previous note.
The typos were entirely my fault, and no reflection
on wikipedia.  I regret that the typos distracted from
the point I wanted to make.

I would never discuss superficial typos in this forum.
This is supposed to be a physics forum, and I would
like to keep it that way.  There is some interesting
physics to be found, in a perverse way, in this
wikipedia passage:
  http://en.wikipedia.org/wiki/Wave_equation#Spherical_waves

On 03/11/2010 08:46 AM, curtis osterhoudt wrote:
>  If you want to follow a dit with "sharp edges" by another
> such dit in an even-dimensioned space, you have to wait until the
> first dit has died out via dissipation, or until the wake formed by
> the first dit is of a significantly different shape. In an
> odd-dimensioned space (at least in 3D and higher), that first dit can
> have just as much of a sharp trailing edge as a rising edge, and so
> your dahs may be much shorter and don't have to rely on dissipative
> processes.

That's all fine, but what's true for a baseband dit isn't
necessarily true for a wave packet in the far field.

By definition, in the baseband there are important k=0 
contributions.  In contrast, by definition the far field is 
where kr >> 1, which necessarily is far outside the baseband.

The aforementioned wikipedia section ends with the statement:

> >  Fortunately, we live in a universe that has three space
> > dimensions, so that we can communicate clearly with acoustic and
> > electromagnetic waves.

This is nonsense.

*** I mention it in this forum because it makes an
*** amusing exercise in critical thinking.  Most
*** people would just read that sentence and accept
*** it without asking themselves whether it might
*** be complete nonsense.

Here's my analysis:

As for acoustic waves, the fact is that indoors, the
actual propagator is a mess due to reflections and
resonances.  I'm talking about the actual propagator,
not ideal plane waves or ideal spherical waves.  There
are poles and zeros all over the place, as you can
easily verify using a PC connected to a speaker and
a microphone.  Yet somehow people manage to communicate
anyway.  The situation would not be qualitatively 
better or worse in an even-dimensional space.

Even outdoors, the fact is that nearby things sound 
different from distant things.  You can easily verify
this by taking a portable music player to a big open
field.  Also it is well known that when people shout
across long distances, if they want to be understood,
they they speak slowly in a peculiar, stilted style.  
Wavelength-dependent attenuation (not dispersion) is
probably dominant in this situation.  Distance would 
be no more of a barrier (and usually less of a barrier) 
in a two-dimensional situation, such as a large empty 
showroom with a low ceiling.

As for ordinary visible light, communication is always
in the far field: wavelengths on the order of microns,
distances on the order of meters.  In the far field,
a spherical Bessel function doesn't look much different
from a cylindrical Bessel function ... and neither of
them looks much different from a plane wave, except
for some funny business with the phase, and the eye
is not sensitive to phase.

In particular, the spherical Bessel functions of order
n are conveniently expressed in terms of the cylindrical
Bessel functions of order n+1/2.  See the last two 
equations at
  http://mathworld.wolfram.com/SphericalBesselDifferentialEquation.html

In the far field, the phase of any Bessel function of
order n is shifted by nπ/2 relative to what it would 
have been in the near field.
  http://en.wikipedia.org/wiki/Bessel_function#Asymptotic_forms

This shift would not be there for plane waves.  It means 
that the wave carrying the nth multipole moment is 
phase-shifted relative to the other multipole moments.  
To say the same thing in more physical terms, it means 
that the wave in the far field looks very different from 
what it would have looked like in the near field ... but 
on the other hand, as we go from place to place *within* 
the far field, the shape of the wave does not undergo much 
in the way of further changes;  the phase shift is maxed
out.  This is equally true in odd and even dimensions.

Last but not least, the equation that is the subject of
the wikipedia section in question
  http://en.wikipedia.org/wiki/Wave_equation#Spherical_waves

does not apply to electromagnetic waves at all!  So the
conclusion about communicating with electromagnetic waves
is completely off the wall.

When we solve the wave equation by separation of variables,
the radial part is given by equation 1166 at
  http://farside.ph.utexas.edu/teaching/jk1/lectures/node97.html
It includes a term involving n(n+1) where n (aka l) is
the multipole order.  The wikipedia discussion considers
only the maximally-symmetric case i.e. n=0 and therefore
omits this term ... but alas there are *no* electromagetic
waves (and few acoustical waves) of zero order.  For EM
waves the dipole (n=1) is the lowest-order radiation
pattern;  the n=0 term is impossible because of conservation
of charge.  For ordinary sounds such as speech or the snapping 
of a twig, again the dipole is the lowest-order radiation 
pattern.  The vibration of an ordinary bell has a quadrupole 
(n=2) pattern.

The sound of popping a balloon has a large n=0 component, 
and also qualifies as a baseband signal, but this is not 
what people normally use for communication.  It is not even 
remotely representative of the general case.

Bottom line:  the wikipedia statement in question:

> >  Fortunately, we live in a universe that has three space
> > dimensions, so that we can communicate clearly with acoustic and
> > electromagnetic waves.

is wrong several times over.  
 -- Undistorted propagation is not necessary for communication.
 -- Odd dimensionality is neither necessary nor sufficient for 
  undistorted propagation.  
 -- In any case, the stated conclusion cannot possibly follow 
  from the stated starting point.

If you want to improve the wikipedia, delete this sentence.  
It probably won't stay deleted, but you can try.


... or you could just leave the passage as is, and use it as
an object lesson for your students.  See how many of them 
pick up on the errors.  I'll bet that even if you give them
huge hints, most of them will take the offending statement
at face value.