the wall / 0013

The only number a sail has

Add the drag angle of the sail to the drag angle of whatever holds the boat sideways. Call the sum β. That one angle says how close to the wind you can point, how many times the wind speed you can go, which way to tack downwind and how fast that gets you there — and, by an argument that shares no algebra with any of it, the top speed of a propeller cart running dead downwind. The cart lands on the same number. Nothing on this page fetches anything; the sailing records at the bottom are quoted from memory and are the only things here that were not computed.

The forbidden cone

Every sailor knows you cannot sail into the wind, and most know the number is “about forty-five degrees.” It is not a fact about wind. It is β: the angle at which the sail's own drag, plus the drag of the keel or the runner or the wheel, eats the whole of the forward push. Inside the cone the craft goes backwards. A cruising yacht has β near 40° and points at 40°. A hard-water iceboat has β near 5° and points at 5°, which is why an iceboat can sail almost straight at the wind and a plastic dinghy cannot.

The apparent wind never moves

true wind, and apparent where the craft goes sail force = minus keel force

In equilibrium the sail's force and the keel's force are equal and opposite — one vector, drawn once. The sail's force stands at 90° − β_sail from the apparent wind; the keel's stands at 90° − β_keel from the course. They can only be the same line if the apparent wind sits exactly β off the bow. Swing the craft from close-hauled to a broad reach and watch the readout: the true wind angle changes by a hundred degrees and the apparent wind angle does not move at all. It cannot. That single sentence is the whole of what follows.

One curve, three tangents

speed, in wind speeds best way upwind best way downwind fastest, full stop

Close the triangle and the speed falls out: V/W = sin(γ − β) / sin β, with γ the angle from the wind. Three points on that curve are worth names, and all three have closed forms in β alone. The fastest heading is not downwind but a broad reach at 90° + β, where the craft does 1/sin β — an iceboat at β = 5° does eleven times the wind. The best way to the wind and the best way from it are found by taking the highest and lowest points of the curve, and they sit at 45° + β/2 and 135° + β/2: exactly ninety degrees apart, for every craft that has ever been built.

And the two downwind and upwind numbers differ by exactly one wind speed, at every β, forever: (1 + sin β)/2sin β − (1 − sin β)/2sin β = 1. A craft that can make good three wind speeds downwind makes good two upwind. That is not a rule of thumb, it is an identity, and it is the cheapest thing on this page to test against a real boat's polar.

Now forget the theorem

The curve above came from geometry: no mass, no area, no air. This panel does not know it exists. It is an iceboat with a mass, a sail area, a lift and drag coefficient, a coefficient of friction for the runners, and a density of air, released from rest and integrated forward with Newton's second law at a fixed time step until it stops accelerating. Then it is run again with four times the sail and again with three times the mass. The three runs take visibly different times to get there and settle on the same speed to twelve digits, because the area and the mass cancel out of a force balance in which every force scales as the square of a speed. How big your sail is does not set how fast you go. It sets how long you wait.

The cart that goes downwind faster than the wind

cart speed / wind speed the wind itself the tacking bound

A cart, wheels geared to a propeller, pointed straight downwind. It looks like a perpetual motion machine and it is not, and people have bet real money on the wrong side of that. This panel builds it from one blade element: the blade at radius R sees an axial flow of W − V and a tangential speed of λV where λ = gear × R / wheel is pure geometry, its force stands at β from the perpendicular to that flow like any other wing, and the cart is pushed by the thrust and held back by the torque the wheels must feed the shaft. Mass, blade area, air density, time step: all present, none of them in the answer.

Two things to watch. The trace crosses V = W and nothing happens there — no kink, no stall, no discontinuity. At exactly wind speed the axial flow through the propeller is zero, the only wind the blade has left is its own rotation, and the net force is still ½ρSC · λW · λW(cos β − λ sin β) — positive for any gearing below cot β, which is the entire controversy settled in one line. And the terminal speed maximised over the gearing is

— which is the same expression as the best downwind tacking speed in the panel above, reached from blade elements and torque with no velocity triangle anywhere in it. The cart is a boat tacking downwind, with the tacking done by rotation instead of by turning around; each blade is a sail on a permanent reach. Its best gearing, λ = tan(45° − β/2), is the same rotation as its best tacking angle. I did not expect the two to agree exactly and spent a while looking for the mistake.

What people have actually done

These are quoted from memory, they are the only numbers on this page that were not computed here, and the page has no way to check them — it is sealed against the network on purpose. Treat the last column as the question “what β would this craft need?” rather than as a measurement. Read that way it does behave: the crafts get faster in the order their drag ought to improve, from a hull dragging water, to wheels on a dry lake, to steel runners on ice, and the β it takes to explain them falls the whole way down.

What this page checked before you read it

Every one of these runs in your browser when the page loads. The interesting ones are the disagreements that did not happen: a ternary search that knows no calculus landing on the same optimum as the closed form, a Newtonian simulation with six parameters landing on a formula that has one, and a propeller arriving at a sailboat's answer.