Re: [Phys-l] phase relationship

[John Denker <jsd@av8n.com>]

On 03/20/2010 04:08 AM, ludwik kowalski posted what 
appeared to be the answer to a question, without quoting
the question.  Until a moment ago, I had no idea as to
what the question might have been.

On 03/21/2010 10:21 AM, Bernard Cleyet wrote:

> > Now I'm puzzled -- I for some time, erroniously?, thought E and M
> > were in phase in "free" space**.

There is no doubt that for any running plane wave in
free space, E and cB are in phase.  The wavefunction 
for E is identical to the wavefunction for cB, except
possibly for an uninteresting additive constant.
Reference:
  http://www.av8n.com/physics/maxwell-ga.htm#sec-plane-waves

I now begin to see an interesting question:  The question
is, how to reconcile this well-known and inescapable
result with situations where the cB field appears to be
out of phase with the E field.

> > X band freq. is convenient.

My suggestion is to *not* use any particular frequency
band, but rather to consider a wavefunction that is
not sinusoidal and not even periodic.  Consider an
isolated blip such as
  http://www.av8n.com/physics/maxwell-ga.htm#fig-blip

I suppose you "could" represent this blip as a sum of
infinitely many Fourier components ... but I don't
recommend it.  That would be more work and would 
result in less understanding.  The Fourier technique
is not the only technique for finding solutions to
the Maxwell equations.

In free space, for a blip (or anything else) running 
left to right, the wavefunction must obey the constraint
  E' = -cB'  pointwise everywhere in space and time.

Reference:
  http://www.av8n.com/physics/maxwell-ga.htm#sec-plane-waves

Now things get interesting when we realize that for a
blip (or anything else) running in the opposite direction, 
i.e. right to left, the wavefunction must obey a different 
constraint:
  E' = +cB'   pointwise everywhere in space and time

If you consider the superposition of a left-running
blip and a right-running blip, the whole notion of
"phase relationship" goes out the window.  You can have 
places where E is zero but cB is not, or vice versa, or 
anything you like, and the local relationship between E 
and cB will be wildly changing as a function of space 
and time.

A standing wave can be constructed as the superposition
of running waves.  Depending on boundary conditions and
initial conditions including polarization etc., you can
construct a standing wave where E and cB are out of phase.

Somewhere on the list of suggestions, not at the top of
the list, here's another possibly-constructive suggestion:

Given two conflicting ideas (e.g. in-phase and out-of-phase)
someone who has mastered the subject should be able to
see both ideas at the same time, and interpolate between
them.  For some students, this may be more trouble than
it is worth, but for some students, especially advanced
students, it may be just the ticket.  In this spirit,
especially if you insist on thinking in terms of sine
waves rather than blips, consider a wave propagating in 
a waveguide.  It can be considered a standing wave in 
two dimensions and a running wave in the third.  Depending 
on the frequency relative to the cutoff frequency, you 
can get any "phase relationship" you want, to the extent 
that "phase relationship" makes sense at all.  At cutoff,
the wave as a whole is a standing wave.  At ω ≫ cutoff,
the wave as a whole is essentially the same as a wave in 
free space.

This is more-or-less equivalent to having two running
waves meet not quite head-on, so that the superposition
is almost a standing wave but still propagates a little
bit.