Catenary Arc Length

Here the catenary repays its difficulty. The parabola's arc length has no elementary closed form and has to be expanded in a truncated series; the catenary's is simply $s = 2a\sinh(L/2a)$, exact, no dropped terms, no domain limit. It is one of the small handful of curves whose arc length integral collapses to something clean, and the reason is that $\sqrt{1 + \sinh^{2}} = \cosh$ exactly — the same identity that makes the whole catenary family tractable.

The number worth looking at is not $s$ but $s - L$, the slack. That excess is what a temperature swing, an ice load and years of creep all have to be taken out of, and it is what a reel has to hold. It grows fast: at a sag ratio of 1 in 40 the cable is about 0.17% longer than its span, at 1 in 10 about 2.7%, and at the near-1-in-4 of this shard's deep reference chain, 17.5%. On a 200 m span that is thirty-five metres of extra chain, which is not a rounding on anybody's purchase order.

Set the parabolic series alongside it on a taut span and the agreement is almost comic. The shard's shallow reference cable — 800 ft between supports, sagging 20 ft — comes out at 801.3340 ft by the exact catenary and 801.3313 ft by the truncated parabolic series. Under a millimetre apart, over 244 metres. That is the practical justification for a century of engineers using the series and never once regretting it.

The parameter $a$ does not come back out of this relation any more than it does out of the sag relation, and for exactly the same reason: it multiplies the expression and also sits inside the hyperbolic sine. A measured length and a known span do not yield the tension in closed form. The span, on the other hand, inverts cleanly, because it appears only inside the sinh — $L = 2a\operatorname{arsinh}(s/2a)$, exact, one step, no iteration.

Two things this length is not. It is not the length to cut, because the cable as it hangs has already stretched elastically under its own tension and will creep further over years; the unstressed length needs $EA$, a creep model and a construction sequence, and none of those is modelled here. And it is not the length at any other temperature — a steel-cored aluminium conductor strung on a cool morning is genuinely longer on a hot afternoon, and that difference is the whole reason stringing charts are drawn as families of curves against wire temperature rather than as single numbers.