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

$u(x)$ - height of cable center

$L$ - distance between two towers

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

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

$T$ - horizontal component of tension.

$w$ units of weight per unit length of cable.

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

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

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