Wave Propagation in a Hanging Rope: Time for Reflection and Return

[wayfarer]

Homework Statement

A uniform rope of length L and negligible stiffness hangs from a solid fixture in the ceiling
The free lower end of the rope is struck sharply at time t=0. What is the time t it takes the resulting wave on the rope to travel to the ceiling, be reflected, and return to the lower end of the rope?

Homework Equations

Equation for wave motion - is it this:
y(t ) = A cos (kt - w),
for some constants k,w.

The Attempt at a Solution

v = sqrt (F/u) = sqrt (mg/ (m/L) ) = sqrt (gL)
this implies that velocity is constant for the wave. I was wondering if this was correct, since it looks suspicious.
From here, I'm not exactly sure where to go - which wave equation should I use to go further (to solve and find out what I want to find out?). Would plugging into the equation I had before, y( t) = A cos(kt-w), be the way to go?

 

[michalll]

F is not a constant in this case, each part of the rope is beeing stretched differently depending on how close it is to the ceiling

 

[wayfarer]

In that case, how would I deal with a situation where F is not constant? I have only learned so far how to deal with cases where it is constant.

 

[michalll]

the equation for v is still valid you just have to write F in terms of x (where x is distance from the bottom part of the rope)

 

[Doc Al]

You'll need to integrate. Find the speed of the wave as a function of position along the rope. (What's the tension as a function of position?)