[Phys-L] definition of "wave"

[John Denker <jsd@av8n.com>]

On 04/10/2016 11:41 AM, Donald Polvani wrote:

> It made me search through my introductory physics books to review the basic
> properties of waves.  The general consensus was that waves have three
> characteristics:
>
> 1) Some type of disturbance is propagated at a FINITE speed.
> 2) The disturbance carries energy with it which, therefore, also propagates
> at a finite speed.
> 3) If the wave disturbance requires a medium to propagate in, "particles" of
> the medium only undergo short range motion as the wave disturbance passes.
> The same particles do not travel with the wave over long distances.

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.