Catenary Tension at a Point

This is the most beautiful result in the subject and the one most worth remembering, because it replaces a hyperbolic function with a length you can measure. The tension at any point on a free-hanging cable equals $w$ times the height of that point above the directrix. The directrix is a horizontal line sitting a distance $a$ below the cable's lowest point, and the cable behaves as though it were hanging from it.

Written the other way, $T = H\cosh(x/a)$ with $x$ measured horizontally from the low point, it says the same thing in the form a calculator wants. Both are exact and both come from the same free-body diagram. At the low point $x$ is zero, $\cosh$ is one, and the tension is $H$ — because there the cable is horizontal and has no vertical component at all. That is the minimum tension anywhere on the span, and it grows monotonically outward in both directions.

Three consequences fall straight out. The maximum tension is always at the highest support, because that is the point furthest above the directrix — which is why the uphill dead-end governs on any inclined span, and why a line running down a hillside is checked at the top of the slope rather than the bottom. The tension is symmetric about the low point, so the same tension occurs at $+x$ and $-x$, and asking "where does the cable reach this tension?" always has two answers. And the vertical component at any point is $w$ times the arc length from the low point to it — the cable is quite literally holding up the weight of everything between there and the bottom, and nothing else.

The classic mistake this page exists to prevent is sizing hardware on $H$. A tension meter at the dead-end does not read the horizontal tension; it reads this. On a taut conductor at 1 in 40 the two differ by less than two thousandths of a percent and the error hides perfectly. On a deep sag it is several percent, and on a slack rope — a zip line with somebody on it, a chairlift haul rope, a temporary rigging line — it is a great deal more. The direction of the error is always the same, and it is always unconservative.

One quiet piece of good news lives on this page. It is the only place in the catenary set where the parameter $a$ can be recovered algebraically, and it is worth understanding why. The ratio $T/H$ fixes $\cosh(x/a)$ outright; $a$ appears only inside the hyperbolic cosine and nowhere outside it as a multiplier, so one inverse-hyperbolic and one division finish the job. In the sag and arc-length relations $a$ sits on both sides of the hyperbolic function at once and no rearrangement exists. So two tension readings on one span — one at the low point, one at a known offset — pin the cable's entire geometry down exactly, with no iteration anywhere.