Wave travel time on a hanging uniform cable ($t=2\sqrt{L/g}$)

At distance $x$ below the support of a flexible uniform cable of length $L$, the equilibrium tension is $\mu g(L-x)$, with linear mass density $\mu$. The local transverse speed is $\sqrt{g(L-x)}$. Integrating its reciprocal from zero to $L$ gives the displayed travel time. The speed vanishes at the free end, but the travel-time singularity is integrable. With variable tension, the small-amplitude wave equation is $\mu y_{tt}={\partial }_x(T{y}_x)$; its characteristic speed is still $\sqrt{T/\mu }$.