Wave along a vertical rope

[phil ess]

1. The problem statement, all variables and given/known data

A uniform rope of mass m and length L hangs vertically from the ceiling. The distance along the rope, as measured from the bottom of the rope is y (i.e., the bottom of the rope is y = 0 and the top is y = L).

2. Relevant equations

v = sqrt(T/u) ?

3. The attempt at a solution

ok so to find the speed of the rope i used the above equation, with:

u = m/L (linear density of string)
T = mg (tension in string)

So substituting gives: v1 = sqrt (gL)

now since gravity is accelerating downward, and we need v as a function of y:

v22 = v12 + 2ad

then substituting:

v22 = gL - 2gy

v(y) = sqrt (gL -2gy)

Does this look correct? The concern i have is that this equation says the wave cant go past half the length of the rope, which seems kinda wonky, though it may be the case. Can anyone clear this up please!

Thanks!

EDIT: ok the next question says how long does it take to get to the top of the string so I know this cant be right, since y =/= L in my equation.

[Carid]

As the wave goes up the rope it encounters greater tension. The velocity changes.