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

["Donald Polvani" <dgpolvani@verizon.net>]

On Wednesday, April 20, 2016 11:48 AM, John Denker wrote:

> 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

Thanks, I find that both insightful and convincing.

Don
> -----Original Message-----
> From: Phys-l [mailto:phys-l-bounces@www.phys-l.org] On Behalf Of John
> Denker
> Sent: Wednesday, April 20, 2016 11:48 AM
> To: Phys-L@Phys-L.org
> Subject: Re: [Phys-L] definition of "wave"
>
> 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.
>
>
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