Suppose a uniform flexible cable is suspended between two points (±L, H) at equal heights...

Suppose a uniform flexible cable is suspended between two points (±L, H) at equal heights located symmetrically on either side of the x − axis. Principles of physics can be used to show that the shape y = y(x) of Lhe hanging cable satisfies the differential equation

where the constant a = T/ρ; is the ratio of the cable’s tension T at its lowest point x = 0 (where y′(0) = 0) and its (constant) linear density ρ. If we substitute u = dymyslashdx, dv/dx = d2y/dx2 in this second-order differential equation, we get the first-order equation

Solve this differential equation for y′(x) = v(x) = sinh(x/a). Then integrate to get the shape function

of the hanging cable. This curve is called a catenaiy, from the Latin word for chain.

FIGURE. The catenary.