Proving Transverse Wave Travel Time in Suspended Rope: A Case Study in Physics

[Wen]

A rope of total mass M and length L is suspended vertically.Show that a transverse wave pulse travel in the length of the rope in a time t=2(L/g)^0.5

If i take the distance of any point on the rope ,A, from the lower end to be X.
linear density is M/L
Tension due to the segment of rope below A is MgX/L
velocity =(T/linear mass density)^0.5
=(L^2/Xg)^0.5
if X=L
V= (L/g)^0.5

which is not what is needed to be shown. So where am i wrong?

 

[mukundpa]

velocity =(T/linear mass density)^0.5
=(L^2/Xg)^0.5

Check your calculations ! and write $$v = dx / dt$$

 

[quasar987]

The speed of the wave is not constant. It diminishes as the wave gets closer to the lower end. So to find the time, you have to integrate dx/v.

(And yes, check your calculations )