# Hanging cable

A cable is suspended between two towers like a power line between two towers. Assume that the towers are on a level parcel of ground but the heights of the towers maybe different. Using the following notation

${\displaystyle u(x)}$ - height of cable center

${\displaystyle L}$ - distance between two towers

${\displaystyle u_{0}}$ - height of cable at tower 1

${\displaystyle u_{L}}$ - height of cable at tower 2.

${\displaystyle T}$ - horizontal component of tension.

${\displaystyle w}$ units of weight per unit length of cable.

and the forces acting on the cable, it can be shown that the cable height ${\displaystyle u(x)}$ is the solution of the differential equation

${\displaystyle {d^{2}u(x) \over {dx^{2}}}={w \over T}{\sqrt {1+({du \over {dx}})^{2}}}}$

with the end conditions ${\displaystyle u(0)=u_{0}}$ and ${\displaystyle u(L)=u_{L}}$.