On Fri, 4 Jan 2008 04:02:23 -0800 (PST)
Keith Dysart wrote:

On Jan 3, 2:14*pm, Jim Kelley wrote:
Keith Dysart wrote:
The example was carefully chosen to illustrate the
point, of course. But that is the value of particular
examples.
When the pulses are not identical, the energy that crosses
the point is exactly sufficient to turn one pulse
into the other.
The remainder of the energy must bounce
because it does not cross the mid-point.
...Keith

So it really is almost as though the pulses travel through one
another, rather than bounce off one another.

I have seen the concept that energy doesn't cross nodal points alluded
to in some texts. *However there are so many exceptions to it found in
physical systems as to render it a dubious notion at best. Useful
perhaps for illustration purposes.

In the discussion of standing waves on a string, Halliday and Resnick
says "It is clear that energy is not transported along the string to
the right or to the left, for energy cannot flow past the nodal points
in the string, which are permanently at rest. *Hence the energy
remains "standing" in the string, although it alternates between
vibrational kinetic energy and elastic potential energy."

So the idea is valid for a simple harmonic oscillator in which there
are no losses. *In such a case, once the system begins oscillating, no
further input of energy is required in order to maintain oscillation.
* Clearly there is no flow of energy into or out of such a system.
What is clear is that energy doesn't pass through the nodes. *It is
less clear that there exists an inherent mechanism which prevents the
movement of energy.

And so it appears in cases where there is no transfer of energy that
one might claim that waves bounce off of one another. *There are no
other examples, and no supporting mechanism for it of which I am
aware, and so one might be equally justified in claiming that waves
pass through each other in all cases.

I'd suggest that this is only if the concept of the
waves in question does not include energy. In the
limiting case of the two waves being identical no
energy crosses the nodes. In other cases, only a
portion of the energy crosses the nodes.

If the concept of the waves includes energy, some
explanation is required to account for the wave
crossing the node, but its energy does not.

Some readers like to superpose energy just as
they do voltage, but in general this is not a
valid operation so I am uncomfortable using
it as the explanation.

...Keith

Food for thought.

Consider an isolated transmission line charged to some DC voltage. Then initiate current by attaching a resistor. We can identify a wave moving back from the junction, beginning at the time of contact. We can also, by monitering the current or power through/into the resistor, plot a wave going through/into the resistor. The two waves would be mirror images of one another. The forward wave would clearly carry energy, the backmoving wave would be a "book keeping" wave that reported the energy removed from the transmission line.

The bookkeeping wave would really be the visible part/result of a power wave that is the negative equivalent of the wave passing through the resistor. Mathematically defining the energy component of the power wave, we should have If*Ef = 1 - Ib*Eb, where If and Ef are the instantaneous measured values of forward current and voltage, and Ib and Eb are the instantaneous measured values of bookkeeping current and voltage. The number 1 defines the beginning energy level as 1. We should observe that If = Ir. If so, then Ef = 1 - Er. Remember, these would be instantaneous values.

73, Roger, W7WKB