For a transverse wave on a string: $v = \sqrt{T/\mu}$ where $T$ is tension and $\mu = m/L$ is linear mass density. If the string hangs vertically, tension varies with position: $T(x) = \mu x g$ at distance $x$ from the free end. Speed varies: $v(x) = \sqrt{xg}$. The wave is NOT uniform on a hanging string. A pulse takes time $t = 2\sqrt{L/g}$ to travel the full length (found by integration).