Catenary Parameter

Every catenary in the universe is the same curve. $y = a\cosh(x/a)$ has exactly one parameter, and changing $a$ does nothing but scale the picture — the way every circle is the same circle at a different radius. So a single length, $a = H/w$, fixes the entire shape of a hanging chain, and comparing it against the span tells you at a glance what sort of cable you are looking at.

The parameter is a real distance on the drawing, not just an algebraic convenience. It is the height of the lowest point of the cable above a horizontal line called the directrix, drawn a distance $a$ below that low point. That line is where the whole elegance of the catenary lives: the tension at any point on the cable equals $w$ times the height of that point above the directrix. The cable behaves as though it were hanging from a line that is not there.

Read $a = H/w$ as $H = wa$ and it says something you can feel: the horizontal tension is the weight of a length $a$ of the cable. A chain weighing 20 N/m hanging with a parameter of 50 m is pulling 1000 N at its lowest point — the weight of fifty metres of itself. That reading turns an abstract parameter into a number you can estimate by eye, and it makes the extremes obvious. A heavy chain pulled gently has a small $a$ and droops sharply. A light wire pulled hard has a large $a$ and looks almost straight. In the limit of infinite $a$ the cosh flattens into a straight line, which is exactly the weightless-cable case.

Compare $a$ against the span and you have your model-selection rule. If $L$ is much smaller than $a$, the cable is taut and shallow and a parabola describes it just as well — the two agree because both are dominated by their common leading term. If $L$ is comparable to $a$ or larger, the cable is deeply curved and the parabola is simply the wrong load case. Cable A in this shard's anchors has $L/a = 0.2$ and the two models agree to within a tenth of a percent; Cable B has $L/a = 2$ and they disagree by eight.

Be careful with what $w$ means here, because it is not the same $w$ as on the parabolic pages. The catenary's $w$ is weight per unit length of cable, running along the curve. The parabola's $w$ is load per unit length of span, along the ground. On a taut cable the two are near enough identical that nobody bothers to distinguish them; on a deep one they differ by exactly the ratio of arc length to span, which is the same 17% that separates Cable B's chain from its own span. Mixing them is a genuine modelling error, not a rounding.