Cecil Moore wrote:
Jim Kelley wrote:
So I'm happy to
leave it to you to explain to Cecil how waves cancel but without
anhiliating the energy "in" them.

But that's just the point, Jim. You seem to believe the
pre-existing energy in those waves has been destroyed.
They obviously possessed energy before cancellation and
you say they possess zero energy after cancellation. If
that pre-existing energy is not destroyed, where did it go?


Cecil,

Now that you have access to a copy of Born and Wolf, you might dig
inside to see if you can improve your understanding of conservation of
energy. It is not quite as simple as you seem to believe.

B&W discuss the Poynting vector and its use in an overview in the first
chapter. I don't have the 4th edition. I have a couple of later editions
that contain identical language, so perhaps the same thing is in the 4th
edition.

In any case, here is the relevant quote. My explanations are enclosed in
[...]. Otherwise the paragraph is completely intact.

"It should be noted that the interpretation of S [Poynting vector] as
energy flow (more precisely as the density of energy flow) is an
abstraction which introduces a certain degree of arbitrariness. For the
quantity which is physically significant is, according to (41), not S
itself, but the integral of S [dot] n taken over a closed surface.
Clearly, from the value of the integral, no unambiguous conclusion can
be drawn about the detailed distribution of S, and alternative
definitions of the energy flux density are therefore possible. One can
always add to S the curl of an arbitrary vector, since such a term will
not contribute to the surface integral as can be seen from Gauss'
theorem and the identity div curl = 0. However, when the definition has
been applied cautiously, in particular for averages of small but finite
regions of space or time, no contradictions with experiments have been
found. We shall therefore accept the above definition in terms of the
Poynting vector of the density of the energy flow."

[ S and n are vectors, shown in bold type in the original. ]

Now for my comments.

Two important concepts are contained in the B&W quote. First, the math
involved with Poynting vectors is not quite as simple as many amateur
radio operators seem to believe. It does not make any sense to simply
add and subtract Poynting vectors in elementary fashion and expect to
get correct results. This is true even for your favorite case of a
one-dimensional problem such as a transmission line.

Second, the Poynting vector by itself means little. It is only the
integral over a closed surface that has physical reality. In your
favorite case of reflections and re-reflections the only useful
non-trivial application of the Poynting vector would be the integration
of the Poynting vector over a small region that includes the line
discontinuity inside. And even then, only the total energy balance can
be determined. Put in direct terms, there is no available information,
and no need for any information about what happens to the energy
contained in the various component waves you like to consider. It simply
does not matter. The only energy balance that counts is the net energy
flowing through the surface of the integration volume. Anything else is
merely in your imagination. B&W allow you to add anything you like, as
long as it is the curl of a vector. But there is no physical reality in
doing so.

It has been pointed out numerous times that modern physical theory is
correct by design. Ian again pointed out that fact earlier today. If the
wave equations, the field equations, force equations, or whatever are
analyzed correctly the energy balance will automatically work out
correctly as well. A check of energy balance is sometimes useful to
highlight any errors that might have been made in the math, but no new
physical information should be expected.

Finally, it is well known by all physicists, and I believe most
engineers, that energy considerations by themselves can be very useful
for analyzing physical problems. Much of higher level classical
mechanics and essentially all of quantum mechanics techniques are energy
based. The so-called Hamiltonian formulation is well-known and widely
used. It is no more or less correct than techniques based on forces and
other fields, but the Hamiltonian technique is often much more
computationally convenient.

73,
Gene
W4SZ