The physics
A flick makes a pulse. The pulse moves along the string. Each bead only moves up and down.
v = √(T / μ) μ = m / L t = L / v
More tension makes a faster pulse. A heavier string, with more mass per metre μ, makes a slower one.
A rope hanging from a hook holds up its own weight. So its tension grows from zero at the foot. The pulse speeds up as it climbs.
v = √(g x) t = 2 √(L / g)
In a lift speeding up going up, use g + a in place of g. A block at the foot adds its weight Mg to the tension everywhere.
A loop of rope spinning at ω has tension T = μω²R². So a pulse moves along it at ωR, the rope's own speed.