= Wave travel time on a hanging uniform cable
{title2=$t=2\sqrt{L/g}$}
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 $\mu y_{tt}=\partial_x(Ty_x)$; its characteristic speed is still $\sqrt{T/\mu}$.
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