Why the straight line is wrong
Take the default rode: 150 ft of chain at 1.5 lb per foot in 30 ft from the roller to the seabed, with 400 lb of horizontal pull on the boat. The straight-line model says the anchor is 147 ft away. The catenary says 145.3 ft, of which 20 ft is chain still lying flat on the bottom and 130 ft is the hanging curve.
The reach barely changed, and that is the first surprise. What changed completely is the angle at the anchor. In the straight-line model the rode pulls up at 11.5 degrees. In the real one it pulls at zero, because the chain leaves the anchor lying along the seabed. That angle is the entire reason chain is worth its weight.
The catenary parameter is the whole model
Divide the horizontal pull by the chain weight per foot and you get a length — 267 ft in the default case. Everything else follows from it. The suspended length needed to reach a rise of h is the square root of h squared plus twice that length times h, which here is the square root of 900 plus 16,000, or 130 ft. Whatever is left over is lying down.
Notice what happens when the pull doubles: the parameter doubles, the suspended length grows, and the chain on the bottom disappears. Chain does not resist load by being strong. It resists it by having weight that has to be lifted before the angle at the anchor can change at all.
The number worth knowing is the lift-off pull
There is one load at which the last link comes off the bottom, and past it the rode behaves like a wire. For this rode it is 540 lb: chain weight times length squared minus rise squared, over twice the rise. Below 540 lb the anchor sees a flat pull and the curve absorbs surges. Above it the anchor starts being lifted, and the tension at the bow roller climbs with nothing left to soak it up.
The length term is squared, which is why letting out more chain helps so much more than it looks. Go from 150 ft to 200 ft in the same 30 ft of water and the lift-off pull goes from 540 lb to 977 lb — 81 percent more, from 33 percent more chain. That is the argument for scope, made in pounds instead of ratios.
Where it stops being true
All of the above is chain. A rope rode is nearly neutral in water and hangs close to straight, so a chain-and-rope combination follows this curve down to the splice and then does not. A snubber changes the picture again, deliberately, by putting elasticity where the chain has none.
And the model is a steady state. A boat at anchor sails from side to side, surges on the swell and yaws in gusts, so the actual load is a series of peaks around whatever number you typed. The catenary is what happens between the peaks. What happens at the peaks — whether the anchor moves, whether the snubber takes it, whether anything breaks — is not arithmetic and is not on this page.
Questions people ask
How much anchor chain actually lies on the bottom?
For the default rode — 150 ft of 1.5 lb per foot chain in 30 ft, with 400 lb of pull — 20 ft is still on the seabed and 130 ft is suspended. Raise the pull to 540 lb and the last link lifts. The share on the bottom drops fast as the load rises, because the suspended length needed grows with the square root of the catenary parameter.
What is the catenary parameter for anchor chain?
Horizontal pull divided by the submerged weight per foot of chain. With 400 lb of pull on chain weighing 1.5 lb per foot it is 267 ft. The suspended length that reaches a given rise is the square root of the rise squared plus twice the parameter times the rise, and everything else on the page comes out of that one length.
At what load does anchor chain go straight?
When the horizontal pull reaches the chain weight times the difference of the squares of length and rise, divided by twice the rise. For 150 ft of 1.5 lb chain in 30 ft that is 540 lb. Because the length is squared, letting out more chain raises that threshold sharply — 200 ft in the same depth takes 977 lb.
Does chain catenary really matter, or is it a myth?
It matters at moderate loads and stops mattering at high ones, which is why both claims get made. Below the lift-off pull the chain leaves the anchor flat along the seabed and the curve absorbs surges. Above it the rode is a straight wire with no give, and the catenary contributes nothing. The page prints that threshold so you can see which side of it you are on.
Does this work for a rope and chain rode?
Only down to the splice. Rope is close to neutrally buoyant in water, so it hangs nearly straight instead of forming a curve. A combination rode follows this model through the chain section and then behaves like a straight line, which means the flat pull at the anchor depends on the chain length being long enough to still be on the bottom.