The following problem appears in many texts: A heavy rope of linear
mass density u and length L hangs freely from a fixed attached point
at the ceiling. The free end of the rope is given a brief shake, to
send a pulse up to the ceiling and back. How long does it take the
pulse to make this return trip?
The naive approach is to say the speed of the pulse is
sqrt(T/u)=sqrt(gx) where T is the tension to support the hanging
weight of the rope below any point x along the rope measuring upward
from the free end. Since this matches the usual kinematic expression
sqrt(2ax), one concludes the acceleration of the pulse is the
constant a=g/2, so the time can be found from 2L=0.5at^2.
The error here however is that the derivation of the wave equation
ASSUMES constant tension, which clearly isn't the case.
Going back over the derivation of the wave equation, it is not hard
to show from Newton's second law that the wave equation is replaced by
d^2y/dt^2 = gx*d^2y/dx^2 + g*dy/dx
for small displacements in the transverse direction y.
If the last term were not present, we would indeed say the wave speed
is sqrt(gx).
However, the last term IS present. So I am quite baffled as to how to
proceed. Experimentally it IS possible to send a small pulse up and
down a rope and the solution DOES come out close to the naive
approach. (AJP 18:405, Oct. 1950 using a steel chain of small smooth
links weighing 1 pound per 12 feet with a length of 180 cm. Ten
return journeys were observed in 17 seconds.)
1. Why does the naive approach give such a close value for the time?
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. The restriction that there be a node at x=L
then determines the normal modes. You can see photos of the first 3
modes for example in TPT 36:108, Feb. 1998.
--
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.eduhttp://usna.edu/Users/physics/mungan/