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

[Carl Mungan <mungan@USNA.EDU>]

John Denker has continued to help me greatly. So let's review where I am now.

Our equation is:

d^2y/dt^2 = gx*d^2y/dx^2 + g*dy/dx             [1]

and we seek a traveling wave solution of the form:

y = sum_i {b_i f(k_i x) cos(w_i t)}              [2]

where b_i is a generalized Fourier coefficient
(i=1,2,3,...). I am assuming the string starts
displaced and at rest, so I have dropped the
phase constants in the cosine.

The solution is Bessel's functions of zeroth order:

f(k_i x) = J0(sqrt[k_i x])                         [3]

where k_i = 4*w_i*w_i/g. We fit to the boundary
values (node at the top end x=L), giving k_i =
N_i*N_i/L where N_i is the i-th zero of the
Bessel function (N_1 = 2.405, N_2 = 5.520, etc).

So now I can explicitly write out y(x,t) given
values for b_i. To be explicit, suppose I form a
small Gaussian pulse.

Now for the last step. While the pulse disperses
as it moves along the string, I can still
recognize a peak to the pulse, since I chose a
Gaussian. How do I figure out the speed v of the
pulse?

In principle, I could just plot the above
solution in Mathematica and look at where the
peak is located for different instants in time.
But is there an analytical way to get this? I'm
still assuming that in the end the answer will
come out to be approximately:

v = sqrt(gx).                                           [4]

But I'd like to see this starting from [2] and improve on this approximation.

> 1) The whole discussion of dispersion relations and group velocity is
> a bit fishy.  The usual derivation that gives dw/dk as the group
> velocity is predicated on the equation of motion being translationally
> invariant ... which is not the case with the equations we have been
> considering.
>
> Certainly there is no such thing as "the" group velocity for the whole
> rope.  The best we can do is to consider a short segment of the rope,
> not too near the free end, and approximate the tension as being
> locally approximately constant there.  That leads to a local notion
> of group velocity.

That's helpful. I completely missed this point!

> 2) That has been called the "naive" approximation, but a better name
> would be the WKB approximation.  None of those guys (Wenzel, Kramers,
> Brillouin) was particularly known for his naiveté.
>
> Some people first encounter WKB in connection with evanescent waves,
> but it works just fine with propagating waves such as we have here.

Also helpful. Can you help me fill in the details that lead from [1] to [4]?
--
Carl E. Mungan, Asst. Prof. of Physics  410-293-6680 (O) -3729 (F)
       U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5040
mailto:mungan@usna.edu       http://usna.edu/Users/physics/mungan/
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