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I collected some resources and some thoughts on the topic
of "What Is a Wave?"
1 Examples and Counterexamples
1.1 Waves
1.2 Non-Waves
2 What is a Wave?
3 What is a Wave Equation?
4 Reflection from an Impedance Mismatch
5 Nonlinearity and Dispersion
6 References
It's a first draft, very incomplete, but still you may
find some bits of it to be useful.
In particular: If anybody has a better definition of
"what is a wave", please let us know! I've seen lots
of definitions, but the simple ones are not good, and
the good ones are not simple.
Also: I would sure like to see a demonstration of a
wave hitting an impedance mismatch /without/ a speed
mismatch. This should not be particularly hard to do
using a Shive-style torsion wave machine. It might
make a nice science-fair project.