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

[jsd at AV8N.COM (John Denker)]

Carl Mungan wrote:

> 1. Why does the naive approach give such a close value for the time?

We agree that the "naive" approach is an approximation, and that the
question is how good is the approximation.

So, consider the following even-more-approximate description.  Imagine
that the rope consists of N pieces joined end-to-end.  Each piece has
constant tension ... that is, we are making the approximation of locally
constant tension.  Now we can solve the conventional wave equation
exactly for each piece.  To get an overall solution, we need to patch
together the N piecewise solutions.  This comes down to the familiar
problem of finding the reflection off an impedance mismatch.  In the
limit of large N, the mismatch is small, and the reflection is
small-squared.  Necessary conditions for this to make sense include
the wave amplitude to be small compared to the wavelength, and the
wavelength to be small compared to the total length of the rope.
The approximation will be highly questionable very near the free
end, but it should be pretty good elsewhere.

> 2. Can one solve the modified wave equation above for traveling
> solutions exactly? If not, what can one say about it?
>
> 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 think you've answered your own question.  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.