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I find it quite tricky to define "wave". The usual careless
definitions get into trouble because:
-- standing waves don't propagate at all
-- typical circular and spherical waves (unlike plane waves) do
not propagate without some change in shape.
-- some media are dispersive
-- some media are nonlinear
-- some media are dissipative
-- a wave is not necessarily repetitive
The best dodge I've found is this:
We define a wave to be something that exists as the solution
to a wave equation. It is best to focus attention on the wave
equation rather than the wave itself.
The wave equation must support some running-wave solutions,
even if not all solutions are of this form.
For examples and further discussion, see
https://www.av8n.com/physics/wave-intro.htm
If anybody has a better way of looking at this, i.e. simpler
and/or more correct, please let me know.