Question

A flexible steel cable of total length L and mass per unit length $\mu$ hangs vertically from a support at one end. Show that the speed of a transverse wave down the cable is $v=\sqrt{g(L-x)}$, where $x$ is measured from the support.

Answer 1

Tension at a point at a distance $x$ from the support is the due to the weight of the cable below it.

Let $m$ be the mass of the cable.

Thus $T=m\frac{(L-x)}{L}g$

Mass per unit length, $\mu =\frac{m}{L}$

Thus speed of the wave at a given point=$\sqrt{\frac{T}{\mu }}=\sqrt{\frac{m\frac{(L-x)}{L}g}{m/L}}=\sqrt{g(L-x)}$