Re: [Phys-L] definition of "wave"

[John Denker <jsd@av8n.com>]

On 04/20/2016 08:13 AM, Donald Polvani wrote:
> Take the one dimensional wave equation to be:
>
>  d^2 f(x,t)/dx^2 = (1/v^2)*d^2 f(x,t)/dt^2   (1)
>
> Where "d ()" is the partial differentiation operator.
>
> Let f(x,t) =a*( x^2 + v^2*t^2) where a is a constant which provides the
> correct dimensions for the disturbance of interest. 
>
> f(x,t) satisfies (1).  Is this a counterexample to using the wave equation
> to define waves?

That's an interesting example.  I consider it a legitimate
wave.  It's the superposition of two traveling waves:

	g = (1/2)*a*(x - v*t)^2
	h = (1/2)*a*(x + v*t)^2

	f = g + h

Now g looks a bit weird, and h looks a bit weird, but
they are clearly traveling waves.  The superposition f
looks even more weird.

If we reject f because it looks weird, then we break
the idea that the superposition of two waves is a wave.
I would rather sacrifice the notion of "looks like a wave"
than sacrifice superposition.

This f is a challenging example because it might take
students a while to realize that it is a superposition.
By the same token, it's a good exercise, in that it
teaches you to trust the standard solution to the
wave equation:  you KNOW any solution has to be a
superposition of two traveling waves.  Sturm-Liouville
theory and all that.

Note that with more complicated equations you can get
some mighty peculiar wavefunctions.  Example:  electron
wavefunctions in atomic physics.  So "looks like a
wave" is probably not a good criterion.