[Phys-L] Re: pulse on a vertical rope

[jsd at AV8N.COM (John Denker)]

Carl Mungan wrote:
> > > ps: It is not hard to find STANDING wave solutions to the above
> > > modified wave equation. If you substitute the trial form y =
> > > f(x)*cos(wt-d), you will get a solution where f(x) are Bessel
> > > functions of zero order.

I suggested:

> >  Write it as
> >   f(kx)*cos(wt-d)
> > and solve for the dispersion relation, i.e. w as a function
> > of k.  The general behavior will be a superposition of waves
> > with different (w,k) values.

> Yes, good, a Fourier series.
>
> Can I make my work easier and construct my wave pulse shape as a
> superposition of Bessel functions instead of sinusoids?

That's what we've been talking about all along.  Each component varies
sinusoidally in time and Besseloidally in space, in accordance with
the equation
   f(kx)*cos(wt-d)

You can think of Bessel functions as being just slightly wonky
sine waves.  The main difference in this case comes when you have
to decide what values of k satisfy the boundary conditions.
  -- In the constant-tension case, the spatial variation is sinusoudal
   and therefore the k-values are evenly spaced.
  -- In contrast, in the present case, the Nth k-value is determined
   by the position of the Nth zero of the Bessel function, which are
   uneven but well known;  see e.g.
    http://mathworld.wolfram.com/BesselFunctionZeros.html
   For a nice picture, see
    http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html


To be precise, such an expansion is called a _generalized_ Fourier series,
in as much as the un-generalized Fourier series is restricted to sines
and cosines.

You can find tons of information about this in books on "Mathematical
Methods of Physics" under the heading of "expansion in orthogonal
functions".  You can also look under "Sturm-Liouville theory".  There's
a sketchy discussion plus a grunch of examples in Jackson _Classical
Electrodynamics_.