Faster than the wind that drives it

An ice yacht in a fifteen-knot breeze will do forty knots. A foiling catamaran does two and a half times the wind speed as a matter of routine. Neither has a motor, neither is going downhill, and both are being driven entirely by a wind slower than they are.

The usual objection — that the wind cannot push something faster than itself — is correct about pushing and wrong about sailing. A sail is not a sheet the wind pushes against. It is an aerofoil, the keel or the runner is a second aerofoil in a second fluid, and the boat’s speed is set by a velocity triangle rather than by the wind’s speed.

Two aerofoils, in two fluids, pulling against each other #

The whole model is a force balance between two objects.

The rig meets the apparent wind and produces a force. Like every aerofoil it produces lift perpendicular to the flow and drag along it, so its total force lies at an angle $\varepsilon_a = \arctan(D/L)$ from the perpendicular, leaning downwind. That angle is the rig’s drag angle, and it is the reciprocal of the lift-to-drag ratio in disguise — the same quantity that decides a glider’s glide ratio.

The hull, keel and rudder meet the water flowing past at the boat’s own speed and produce their own force, at their own drag angle $\varepsilon_h$ from the perpendicular to the track, leaning backwards.

In steady motion the two must be equal and opposite. That is one vector equation, and it has exactly one consequence:

$$\lambda = \varepsilon_a + \varepsilon_h$$

where $\lambda$ is the angle between the boat’s heading and the apparent wind.

No area appears. No lift coefficient, no displacement, no sail plan, no rig height. Doubling the sail area doubles both forces and changes nothing about the equilibrium; what changes it is making either surface better, which means a higher lift-to-drag ratio and a smaller drag angle.

That is the first surprising thing about this model, and it is worth pausing on. A larger sail does not make a boat faster in this account. It makes it heel more.

The absence has the same character as several results already on this site. Betz’s limit contains no turbine, the propulsive efficiency contains no engine, and the sailing polar contains no sail — in each case because the question being asked is about a ratio rather than a magnitude, and every magnitude cancels. What is left is the quality of the surfaces, expressed as an angle, and geometry.

It also means the model is unusually easy to falsify. There is exactly one measurement that fixes a boat’s whole polar in this account — its combined drag angle — and if two boats with the same combined drag angle have different polars, the model is wrong. Real boats depart from it chiefly through the three effects listed at the end of this essay, and the departures are largest exactly where those effects bite.

The triangle, which is where the speed comes from #

Now the kinematics. The boat sails at some course $\beta$ off the true wind. The wind it feels — the apparent wind — is the true wind minus its own velocity, as vectors, and the triangle closes.

Applying the sine rule to that triangle, with $\lambda$ already fixed by the force balance:

$$\frac{V_b}{V_t} = \frac{\sin(\beta - \lambda)}{\sin\lambda}$$

and that is the entire performance model. Two angles in, a speed out.

The loop is what makes it possible to exceed the wind. Going faster brings the apparent wind forward and makes it stronger; a stronger apparent wind makes more force; more force makes the boat faster. It is positive feedback, and the only thing that stops it is that the apparent wind angle cannot change — the force balance pins it at $\lambda$ — so the geometry eventually refuses to close. The equilibrium is where it refuses.

What was computed, and how the triangle was checked #

The speed is not obtained by evaluating the formula and drawing it. The solver computes the true wind vector, the boat vector and the apparent wind vector componentwise, and then checks two things.

The triangle closes. The apparent wind magnitude from the vectors is compared with the sine-rule value: they agree to $10^{-16}$.

The two forces really do cancel, built from the angle the triangle delivered rather than from the $\lambda$ the speed was computed with. Written both ways with $\lambda$ it would be an identity and would prove nothing; written with the triangle’s own answer it is a real test — if the speed formula were wrong, the apparent wind would arrive at some other angle and the two forces would not be opposite. The residual is $4\times10^{-17}$.

The maximum is then found by golden-section search over the course, without knowing where to look, and compared with the closed form:

• fastest course: 110.0000° off the true wind, against $90° + \lambda = 110°$;
• speed there: 2.923804, against $1/\sin\lambda = 2.923804$.

Both to six figures, both from a search that knew neither answer.

When a boat beats the wind, and by how much #

From $V_b/V_t = \sin(\beta-\lambda)/\sin\lambda$, the boat exceeds the true wind speed whenever $\sin(\beta - \lambda) > \sin\lambda$, which is whenever

$$\beta > 2\lambda$$

For $\lambda = 20°$ that is any course more than forty degrees off the true wind. Most of the compass, in other words — and the maximum, at $\beta = 90° + \lambda$, is a broad reach rather than anything exotic.

The dependence on $\lambda$ is brutal, which is why this shows up on some craft and not others:

A cruising yacht with a soft mainsail, a fat hull and appendages designed for berthing rather than speed has combined drag angles nearer fifty degrees, sits at 1.3 times the wind at best, and rarely gets there. An ice yacht replaces the water with a steel runner on ice — a hydrodynamic drag angle of perhaps two degrees — and a rigid wing sail, and the sum falls to something like eight degrees, at which the model gives seven times the wind speed. Ice yachts do reach five or six.

The reason nobody expects this from a dinghy is that a dinghy’s drag angles are terrible, not that boats cannot do it.

The table also explains a piece of racing lore that sounds like superstition. In light airs, a boat whose rig has a poor drag angle is at a large $\lambda$ and its polar is small in every direction; in a breeze the rig is trimmed harder, the section works nearer its best lift-to-drag ratio, $\lambda$ falls, and the polar inflates disproportionately — because $1/\sin\lambda$ is a steep function at small angles. Going from 32° to 20° buys 55 per cent more speed at the optimum. That non-linearity is why small improvements to a rig are worth so much more at the top of the fleet than the bottom, and it is the same steepness that makes a lift-to-drag ratio the thing a glider designer cares about above everything else.

What the sail’s own performance has to be #

The drag angle is a familiar quantity in an unfamiliar place. $\varepsilon_a = \arctan(D/L)$ means a rig at 14° has a lift-to-drag ratio of 4; at 22°, of 2.5.

Those are poor ratios for an aerofoil. A glider wing reaches 40 or better; the site computes 20 or so for a modest section. A sail is worse for reasons that are all structural rather than aerodynamic: it is a thin cambered membrane with no leading-edge radius, its trailing edge flogs, its lower aspect ratio raises the induced drag, and the mast in front of it is a bluff body shedding a wake straight into the flow the sail depends on. A wing sail — rigid, with a proper section — is the single largest change available, and it is why every recent record-holder has one.

The keel’s drag angle is smaller for a good reason: water is dense. A keel needs very little area to produce the side force required, so its induced drag is low, and most of what remains is wetted-surface friction — which is why a racing hull’s underwater finish is worth arguing about and a cruiser’s is not.

The same device, going round in a circle #

There is a machine already on this site that does exactly what an ice yacht does, and nobody thinks of it as sailing: a wind turbine blade.

The tip of a large rotor at a tip-speed ratio of seven is travelling crosswind at seven times the wind speed. It is held on its course by a shaft instead of a keel, and the shaft’s drag angle — the bearing friction and the generator’s reaction, expressed the same way — is very small. Its section meets an apparent wind almost entirely made by its own motion, arriving from a few degrees off the plane of rotation, exactly as an ice yacht’s sail meets a wind almost entirely of its own making.

Read the other way, an ice yacht is a wind turbine with one blade, an infinite radius, and the load taken off as motion rather than as torque. The reason its speed is $1/\sin\lambda$ and a turbine’s tip-speed ratio is not obviously the same number is that a turbine is being asked to extract power while a yacht is not — a turbine that produced no power at all would run away to exactly this speed, and the induction factors that describe a loaded rotor are the account of how much it has been slowed down by being made to do work.

That is a satisfying place for the two anchors to meet. Crosswind motion is how a lift device outruns the wind, whether the constraint holding it on course is ice, water, or a bearing.

Where the model stops, and dead downwind is where #

At $\beta = 180°$ — dead downwind — the formula returns exactly $V_b = V_t$, and that number is an artefact. Feeding $\beta = 180°$ into the sine rule gives $\sin(180° - \lambda)/\sin\lambda = 1$ by trigonometric coincidence, while the apparent wind speed $V_t \sin\beta/\sin\lambda$ has gone to zero. A boat with no wind on it produces no force, and the equilibrium is nothing divided by nothing.

The solver refuses that course rather than reporting the number, and the refusal says why:

The apparent wind goes to zero there, the model has no force to balance, and the boat that appears to sail at exactly wind speed is an artefact of dividing zero by zero. Downwind performance is a question about the best angle to sail, not about the direct one.

That refusal is checked in the site’s gate, alongside the one for courses inside the no-go zone, because both are places where a formula returns a plausible number for a situation that has no solution.

Three further absences, each of which a real boat has:

• No hull speed. A displacement hull generates a wave system it eventually has to climb, and that is a ceiling this model contains no trace of. It is precisely what foiling and planing exist to escape, and it is why the model’s predictions come true on ice and on foils and not on a heavy keelboat.
• No heeling. The side force the rig makes is enormous and acts high up; the righting moment is finite; a boat sailed to this model’s optimum would capsize. Reefing is the management of that, and it costs performance the model does not see.
• Nothing unsteady. Gusts, waves, and the fact that a boat accelerating into its own apparent wind takes time to get there.

The craft that goes straight downwind faster than the wind #

The model gives out dead downwind, and the reason it gives out is instructive: the apparent wind goes to zero, so a sail has nothing to work with. That is a limitation of the sail, not of the physics, and there is a machine that removes it.

A wheeled cart with a propeller geared directly to its wheels — no motor, no battery, no stored energy — can travel directly downwind faster than the wind that is driving it. It has been built, raced and certified: the vehicle Blackbird was measured at very nearly three times wind speed on a straight downwind course in 2010, and the result caused an argument that ran for years because it sounds like a perpetual-motion claim.

It is not, and the resolution is this essay’s own argument with one substitution. A sailing boat is a machine coupling two media in relative motion — air and water — with an aerofoil in each. Its speed is not bounded by either medium’s speed because neither medium is pushing it; what it is extracting from is the difference between them. A downwind cart couples the same kind of pair: the air and the ground.

Once that is seen, the bookkeeping is straightforward. The wheels, rolling on ground that is moving relative to the air, act as a turbine and take power out of that relative motion. The propeller, turning in air, acts as a propeller and puts thrust in. Which of the two devices is extracting and which is delivering depends on which frame the observer stands in, exactly as the sail and the keel swap roles between the air’s frame and the water’s — and the energy source in every frame is the same one: the wind and the earth are in relative motion, and a machine that touches both can take from it.

There is no free lunch anywhere in that, and the arithmetic says where the limit is. The cart is taking power from the ground and returning it to the air, and each transfer has an efficiency; the speed ratio it can reach is bounded by the product of those efficiencies, and a cart with a poor propeller or a lossy transmission does not exceed the wind at all. That is the same structure as this essay’s $1/\sin\lambda$: a ceiling set entirely by how good the two surfaces are, with no size, no area and no absolute speed in it.

And it makes the sailing result look less exotic rather than more. An ice yacht outruns the wind by going across it, using the velocity triangle to build its own apparent wind; a downwind cart outruns it going straight along it, using a gear ratio instead of a triangle. Both are extracting from the shear between two media, both are limited by the quality of two energy transfers, and neither is being pushed. The wind’s speed is not a speed limit for anything that is not being pushed by it, and the objection this essay opens with is an objection to a mechanism that sailing craft do not use.

Who found it, and when #

The course theorem — that a boat’s closest approach to the apparent wind is the sum of the two drag angles — is usually credited to Manfred Curry in the 1920s and was placed on a firm aerodynamic footing by Kenneth Davidson at Stevens Institute in the 1930s, whose towing-tank work founded modern yacht research.

The physics is older than the analysis by a great margin. Fore-and-aft rigs capable of working to windward existed in the Indian Ocean and the Mediterranean for a thousand years before anybody could say why they worked, and the lateen and the crab-claw sail were solving this equation empirically while the prevailing theory of sailing was still that the wind pushed. The gap between the practice and the explanation is the same one the dimpled golf ball had, and about the same length.

Where the ladder goes #

The polar in this essay says how fast a boat goes in each direction. It does not say which direction to go, and the answer to that is not the one anybody expects.

The mark is usually upwind or downwind, and the boat cannot sail straight at it — upwind because the no-go zone is in the way, downwind because the model has just given out there. So the course has to be a compromise between speed and direction, the quantity to maximise is the component of velocity towards the mark, and it has a maximum at an angle that comes out in closed form: $45° + \lambda/2$ going up and $135° + \lambda/2$ going down.

Both are found by search on the next rung and both agree to six figures. The consequence is that a boat sails seventy per cent further than the straight-line distance in order to arrive first.