Roy Lewallen wrote:
[... very nice explanation]

Sine waves are another problem -- there, we can easily have
overlapping waves traveling in the same direction, so we'll run into
trouble if we're not careful. I haven't worked the problem yet, but
when I do, the energy will all be accounted for. Either the energy
ends up spread out beyond the overlap region, or the energy lost
during reflections will account for the apparent energy difference
between the sum of the energies and the energy of the sum. You can
count on it!

As always, I appreciate any corrections to either the methodology or
the calculations.

Roy Lewallen, W7EL

How about analyzing a vibrating string? If you play guitar, there's a very
nice note you can make by plucking a high string, then putting your finger
at exactly the correct spot and removing it quickly. The note will jump to
a much higher frequency and give a much purer sound. Clearly, the
mechanical energy has split into two waves that cancel at the node.

In principle, you could show the node is stationary, thus contains no
energy. But there is energy travelling on both sides of the null point -
you can hear it.

You can also create other notes by touching different spots on the
vibrating string. These create standing waves with energy travelling in
both directions, but cancelling at the null points. Very similar to
transmission lines.

Regards,

Mike Monett