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

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

with the end conditions $u(0)=u_0$ and $u(L)=u_L$.