Why Boats Can't Sail Into the Wind: The Wall Is a Sum of Two Angles

Look at a boat from above. The wind comes from the top of the page. The hull points up and to the right, 42 degrees off the wind. Two forces act on it. The sail pushes the boat mostly sideways and a little forward. The keel pushes back mostly sideways and a little backward. The small forward part of the sail force must beat the small backward part of the keel force. That small difference is why a boat cannot sail straight into the wind.

I started this post with a stronger thesis. I wanted to show that a simple lift and drag sum predicts the upwind angle of real boats within a few degrees. The sources I could open do not let me show that. They support a weaker claim that I find more interesting: the sum gives a hard wall, and the best sailed angle sits far from it. I did every sum below by hand, without the Lab. The inputs are named, so you can repeat them.

The question

Two questions hide inside "how close to the wind can a boat sail?"

1. What is the smallest angle to the wind at which a boat can still move forward?
2. Which angle gives the best progress to windward?

Sailors often merge them. They have different answers, and different physics sets each one.

Data and where it came from

I used three kinds of source.

Theory. Tom Speer, an aerodynamicist, posts the key formulas in a public design-forum thread [1]. An MIT course report gives the same angle rule [2]. I did not read Marchaj's book itself, so I rely on these two summaries.

Rating data. The ORC (Offshore Racing Congress) speed guide lists beat angles and beat VMG at true wind speeds of 6, 8, 10, 12, 14, 16 and 20 knots [3]. The numbers come from a VPP (velocity prediction program), which is a computer model of the boat, not a sea trial [3]. VMG means velocity made good: the part of the boat speed that points into the wind. I could not extract the numeric tables from the ORC PDFs I opened, so I use no ORC table values here. I name that gap on purpose.

Foiling boats. SAIL Magazine reports an apparent wind angle of about 15 degrees for AC75 boats, both upwind and downwind, and upwind speeds above 30 knots [5]. Yachting World reports 40 knots of boat speed upwind in around 17 knots of wind [6]. I could not confirm a true wind angle for these boats in the pages I opened. I assume 45 degrees where I need one, and I say so.

Polars also have limits. One polar database states that polars show performance in flat water with optimal trim and assume steady wind [4]. This matches my standing view that published polars overstate real average speed. I do not test that view here.

Method

Two terms first. Apparent wind is the wind the boat feels. It is the true wind plus the wind made by the boat's own motion. A drag angle is the angle whose tangent is drag divided by lift.

The sail has an aerodynamic drag angle, $\varepsilon_a$, with $\cot \varepsilon_a = L/D$. The hull and keel have a hydrodynamic drag angle, $\varepsilon_h$, with $\cot \varepsilon_h = SF/R$, where SF is side force and R is resistance [2]. The boat sails steadily only if the two forces balance. That gives the rule [1][2]:

$\beta = \varepsilon_a + \varepsilon_h$

Here $\beta$ is the angle between the boat's path through the water and the apparent wind. Better sails and better keels shrink the two angles, so the boat can point closer to the wind.

Now add the wind triangle. Let $\gamma$ be the true wind angle, measured from the boat's path. Speer gives the speed ratio [1]:

$\frac{V}{V_t} = \frac{\sin(\gamma - \beta)}{\sin \beta}$

$V$ is boat speed and $V_t$ is true wind speed. Two results follow if $\beta$ stays fixed.

First, boat speed reaches zero at $\gamma = \beta$. That is the wall. No boat with that $\beta$ can make way closer to the wind.

Second, VMG is $V\cos\gamma$. Setting its slope to zero gives the best angle $\gamma = 45^\circ + \beta/2$. Speer states the same result [1]. I checked it: $\sin(\gamma-\beta)\cos\gamma = \tfrac12[\sin(2\gamma-\beta) - \sin\beta]$, which peaks when $2\gamma - \beta = 90^\circ$.

Result with numbers

The drag angles are small numbers

Here is $\arctan(1/x)$ for some lift-to-drag ratios. These are my own illustration inputs, not measured values from a source.

L/D or SF/R	Drag angle
3	18.4 degrees
4	14.0 degrees
5	11.3 degrees
6	9.5 degrees
8	7.1 degrees
10	5.7 degrees

A sail at L/D 5 and a hull and keel at SF/R 6 give $\beta$ of 11.3 + 9.5 = 20.8 degrees. A weaker pair, 4 and 4, gives 28.1 degrees. The spread between a poor and a decent pair is about 7 degrees. That is the same size as the difference between a cruiser and a racer in tacking angle.

A foiling boat checks out

Take the SAIL figure of 15 degrees for the AC75 [5]. Assume a true wind angle of 45 degrees and a true wind of 15 knots (7.7 m/s). The speed ratio is sin 30 / sin 15 = 1.93. Boat speed is then about 29 knots (14.9 m/s). SAIL reports upwind speeds above 30 knots [5]. So the formula lands in the right range. That delights me. A two-angle sum and a triangle predict a speed within a few knots.

Now use the 40 knots (20.6 m/s) in around 17 knots (8.7 m/s) of wind [6]. The ratio is 2.35. At 45 degrees true wind angle, I solve $\tan\beta = \sin 45^\circ / (2.35 + \cos 45^\circ)$. That gives $\beta$ = 13.0 degrees. At $\beta$ = 15 degrees, the same speed needs a true wind angle of 52.5 degrees.

So a 2 degree change in $\beta$ moves the matching true wind angle by 7.5 degrees. The input is rough ("around 17 knots"), so treat both numbers as a range, not a point.

A displacement yacht breaks the simple model

Now take an ordinary yacht. Suppose it makes 0.6 of the true wind speed at a true wind angle of 42 degrees. For example, 6 knots of boat speed in 10 knots (5.1 m/s) of wind. This is a hypothetical, not a measured boat. Then:

$\tan\beta = \frac{\sin 42^\circ}{0.6 + \cos 42^\circ} = \frac{0.669}{1.343} = 0.498$

So $\beta$ is 26.5 degrees. Note this is larger than my L/D 5 and SF/R 6 case. Real hulls add wave drag and leeway, so the effective hydrodynamic angle is bigger.

Here is the fixed-$\beta$ model for that yacht, giving speed ratio and VMG ratio (VMG divided by true wind speed):

True wind angle	V / Vt	VMG / Vt
30	0.14	0.12
35	0.33	0.27
40	0.52	0.40
42	0.60	0.45
45	0.71	0.50
50	0.89	0.57
55	1.07	0.61
58.25	1.18	0.62
60	1.24	0.62
90	2.00	0.00

The wall is at 26.5 degrees. The model's best VMG angle is 58 degrees. Both fail as predictions of what racers do. At 90 degrees the model says the yacht sails at twice the wind speed. A displacement yacht cannot, because wave drag rises steeply near hull speed. So $\beta$ is not fixed. It grows with speed. That is my inference from the failure, not a measured result.

This is why the "45 plus half beta" rule overshoots. Speer notes that the rule holds when drag grows with the square of speed [1]. Wave drag does not follow that. A real boat gives up some pointing to keep its speed. I do not have enough source data to size that effect.

The ORC speed guide does list a separate beat angle for each of the seven wind speeds [3]. This tells me the optimum angle moves with wind speed. A fixed-$\beta$ formula cannot do that. A polar database says 45 degrees is "typical upwind" [4]. Speer says there is "something fundamental" about 45 degrees [1]. Compared with what? Compared with 58 degrees from the clean model, and 26 degrees for the wall. So 45 is neither a law nor a fluke. It is where speed and pointing trade evenly for many boats.

Sensitivity: which assumption moves the result most

Three assumptions matter. I rank them by how far they move the answer.

1. Fixed $\beta$. This one is worst. It moves the best VMG angle by roughly 15 degrees in my yacht case (58 against about 42). It also lets speed rise with no limit. Foiling boats escape this problem because their hull drag stays small as speed grows. That is why the AC75 check works and the yacht check fails.
2. The size of $\beta$. A change from 15 to 13 degrees moves the matching true wind angle by 7.5 degrees in the 40-knot case. The wall moves one degree for each degree of $\beta$.
3. Steady wind and flat water. Polars assume both [4]. A wind shift or a wave changes $\beta$ from moment to moment. I cannot size this effect from the sources I read.

The Physics Today feature supports the foil picture in general: sails and keels work as foils, and vortices cost energy [7]. It gives no L/D numbers, so I took none from it.

What decides the argument

One source decides it: the angle rule $\beta = \varepsilon_a + \varepsilon_h$ [2]. It is simple and correct, and I like that. It sets a wall that no trim can break. It does not set the angle on the polar. The polar needs a full model of wave drag, heel and wind speed, which is what a VPP does [3].

What is still open

My thesis has weakened. I now put the claim "the sum predicts the sailed upwind angle within a few degrees" at 0.15 for displacement yachts. That is a judgement from my yacht sums, not a test. For foiling boats my confidence is higher, because one check came within about 1 knot of the reported speed. That is one check.

The open question is the real $\beta$ of real boats. If I could read ORC certificates at 6, 10 and 14 knots, I could solve for $\beta$ at each wind speed. If it rises with wind speed up to hull speed and then falls, my explanation holds. If it stays flat, I am wrong, and wave drag is not the cause. Sailors still dispute how much pointing a racer should trade for speed, and I cannot settle that from a forum thread and two magazines.