# Shape of very long wire between two very tall posts (many km tall) which are attached to the earth

Consider for a moment a length of uniform wire or chain which goes between two posts of equal height. If we assume the earth to be flat then we can predict the shape of the curve using

$$y = a \cosh \left(\frac{x}{a}\right)$$

But if we had a very long wire between two very tall posts which were angle theta apart on a planet which is a perfect sphere then what would the equation be which gives the height of the wire as a function of the angle between the two posts.

Note that in the "flat earth" wire example the gravitational field strength which the wire is in will be constant, while in the case of a wire which goes between two very tall towers the gravitational field strength is not constant. This is likely to make the problem more complex.

• Interesting problem, for sure ! Jul 10 at 7:14
• What have you tried? Jul 10 at 8:23
• Are you familiar with the variational method? You need to find the $r(\phi)$ to minimize the potential energy $U=-\int d\phi \ r^{-1}\sqrt{r^2+r'^2}$ subject to constraint $\text{length}=L=\int d\phi \ \sqrt{r^2+r'^2}$. In principle, you then solve the Euler Lagrange equations to find $r(\phi)$
– Sal
Jul 10 at 17:11
• I had thought of some iteration method, I was considering trying to devise a brute force attack on the problem by breaking the wire into lots of short lengths. Jul 10 at 20:33
• Wikipedia describes a generalization for general force. en.wikipedia.org/wiki/Catenary#Other_generalizations Jul 11 at 10:13