The Velocity Triangle: Apparent Wind Is the Only Wind That Matters

Our reference ship is a Panamax bulk carrier of about 75,000 tonnes deadweight, holding her service speed of 14.0 knots. The forecast gives 18 knots of true wind on the beam. Two rigid wing sails stand on her deck, and neither sees 18 knots on the beam. They see 22.8 knots at 52 degrees off the bow. Every lift coefficient, every thrust curve and every saving in this course is indexed on that 52, not on the 90 the weather gave, and an error here is a fixed one that nothing downstream finds.

The triangle, and the two numbers it hands you

The true wind is the air velocity relative to the earth: what a moored buoy measures and what a forecast model outputs. The apparent wind is the air velocity relative to the ship, the flow the rig is immersed in and the flow the anemometer reads. They differ by the ship's own velocity, and the difference is a vector subtraction, not a correction factor.

Write V_TW for the true wind speed, V_S for the ship's speed over the ground (in still water, her speed through it), and beta_TW for the true wind angle, measured from the bow round to the direction the wind comes from, so 0 degrees is dead ahead, 90 abeam and 180 dead astern. Then

V_AW = sqrt( V_TW^2 + V_S^2 + 2 · V_TW · V_S · cos(beta_TW) ) beta_AW = arctan[ V_TW · sin(beta_TW) / ( V_TW · cos(beta_TW) + V_S ) ]

where V_AW is the apparent wind speed and beta_AW the apparent wind angle, on the same convention and units as their true counterparts. A plain arctangent returns angles only between minus 90 and plus 90 degrees, so when the denominator, V_TW · cos(beta_TW) + V_S, comes out negative (a true wind from abaft the beam carrying more speed along the ship's axis than she makes), add 180 degrees; the two-argument arctangent, atan2, does it by itself.

For our ship the cosine of 90 degrees is zero, so the cross term disappears and V_AW is the square root of 18 squared plus 14 squared: the square root of 520, or 22.80 knots, 11.73 m/s. The angle is arctan(18/14) = arctan(1.286) = 52.1 degrees.

That second line inverts easily. arctan(14/18) gives 37.9 degrees, and 52.1 plus 37.9 is 90: the wrong answer is the complement of the right one, measured from the beam instead of the bow. Nothing about it looks wrong in a spreadsheet, and it moves the rig 14 degrees along its own polar.

The limits carry the physics. Dead ahead, beta_TW is 0 and V_AW is V_TW + V_S: 32 knots across the ship's deck, all of it resistance. Dead astern, beta_TW is 180 and V_AW is V_TW − V_S: 4 knots, because the ship is running away from her own wind.

Why a fast ship is a poor wind ship

Hold the true wind at 18 knots on the beam and vary the ship. At 8 knots the apparent wind is 19.7 knots at 66.0 degrees; at 12 knots, 21.6 knots at 56.3 degrees; at 14 knots, 22.8 knots at 52.1 degrees; at 18 knots, 25.5 knots at 45.0 degrees. The speed rises, which sounds like good news; the angle draws forward, which is the bad news that outweighs it. On a beam true wind beta_AW is simply arctan(V_TW/V_S), so it depends on nothing but the speed ratio: at equal speeds it is exactly 45 degrees, on any ship, in any wind.

The general statement is sharper. The apparent wind lies abaft the beam only when

cos(beta_TW) ≤ − V_S / V_TW

For our Panamax at 14 knots in 18 knots of wind that requires beta_TW of at least 141 degrees. Slow her to 8 knots and the threshold falls to 116 degrees. Push her to 18 knots, equal to the true wind, and it reaches 180 degrees: a ship steaming at the true wind speed never sees an apparent wind abaft the beam, from any direction at all.

That matters because driving force is a product of two quantities pulling against each other: dynamic pressure is largest dead ahead and collapses when running, while the useful component of the aerodynamic force is zero dead ahead and largest near the beam. The product peaks between, and practice puts the useful window at roughly 60 to 110 degrees apparent. Treat those bounds as the rule of thumb they are; the mechanism is exact, and lesson 2 derives it. So is the consequence: raising ship speed slides the whole envelope towards the nose, and a 12-knot bulker and a 20-knot boxship in identical weather are not in the same wind.

From, or to, and which side

Wind is named for the direction it comes from; current for the direction it sets towards. That asymmetry is a real trap, because weather model output normally arrives as u and v velocity components, which describe where the air is going, and somebody has to reverse them.

Take the ship heading due north at 14 knots and a forecast reporting 340 degrees at 18 knots. Read as a meteorological direction, the wind comes from 340, 20 degrees off the port bow, so beta_TW is 20 degrees and the triangle gives 31.5 knots at 11.3 degrees off the bow: a near headwind, no thrust, both wings feathered. Read the same field as the direction the air is going and the wind comes from 160 degrees, giving 6.8 knots at 115.3 degrees, a gentle broad reach. One record, two answers, neither obviously wrong.

The sign matters as much as the number, because it decides which way a rotor spins and which way a wing is sheeted. A rotor commanded the wrong way does not merely fail to help; its driving component points aft. The ship's anemometer sidesteps the question: it reads apparent wind relative to the ship's head, which is what a device controller wants. True wind is the derived quantity, needed for reporting and routing, not for control.

The wind aloft is not the wind at the anemometer

Wind speed grows with height above the sea, and ship performance work moves a reading between heights with a power law:

V(z) / V(z_ref) = ( z / z_ref )^(1/n)

with z the height above the free surface in metres and z_ref a reference height, conventionally 10 metres. Which exponent to use matters. The familiar one-seventh law is an onshore convention, and ISO 15016's 2015 edition used it. ISO 15016:2025 replaced it with one-ninth, a profile in which the wind grows less with height, which fits offshore wind better, and the ITTC's speed and power trials procedure, 7.5-04-01-01.1 in its 2024 revision, prescribes the same. IMO's EEDI survey guidelines, as amended in 2025, name both, and accepted the 2015 edition alongside them only until 1 May 2026. This course uses one-ninth throughout. At 20 metres it gives 1.080 times the 10-metre wind and at 40 metres 1.166, and since dynamic pressure goes as the square of speed, factors of 1.17 and 1.36 on the force available. On the old one-seventh those last two would read 1.22 and 1.49, so the choice of exponent is worth about ten per cent of the force at masthead heights, which is why it is worth getting right.

Two consequences follow. First, readings and rigs must be reduced to a common height, and the law belongs to the true wind alone, because the ship's own velocity is the same at every height. So strip the ship's velocity out of the anemometer's apparent reading, scale the true wind, then add the ship back. The ship's anemometer sits at 40 metres and her wing sails carry their centre of effort at 19.5, so the factor on the true wind is (19.5/40) to the one-ninth, or 0.923: 18 knots of true wind at the masthead is 16.6 knots at the rig. Used raw, the masthead true wind overstates the true wind's dynamic pressure at the rig by (40/19.5) to the two-ninths, a factor of 1.17. Add the ship's 14 knots back on a beam wind and the rig meets 21.7 knots of apparent wind rather than 22.8, so its dynamic pressure is overstated by about 10 per cent, on every record and always in the same direction. The shortcut to avoid is scaling the apparent reading itself, which shrinks the ship's speed along with the weather. Everywhere else in this course the reference 18 knots is already the true wind at the rig's centre of effort, so the triangle can be worked without a further reduction.

Second, the apparent wind angle twists with height, because only one leg of the triangle grows: ship speed is the same at the masthead as on deck, the true wind is not. Over a 30-metre wing sail on the beam, foot at the deck 5.5 metres above the water and head at 35.5, the twist is about 6 degrees; at a true wind angle of 140 degrees it is about 15. That is why a wing sail is built twisted rather than flat.

Be honest about what the law is. Any single exponent describes a neutrally stratified marine boundary layer and nothing more; the real profile moves with stability, fetch and sea state, which is why the standard could change the number at all. It is a convention for moving a reading between heights, not a measurement. Nor does any exponent fix the second problem, that the anemometer stands in air already disturbed by mast, funnel and accommodation, which a rig forward of the bridge disturbs further.

Monday morning

Establish three facts about your ship's wind data before trusting any performance figure built on it: whether the columns hold true or apparent values, whether the direction is where the wind comes from or where it goes, and the anemometer's height above the waterline. Then re-plot a month of records against apparent wind angle instead of true, and count how much of that month the ship spent between 60 and 110 degrees. If you run a slow ship and a fast one, do both, because the gap will be wider than their speeds suggest. Lesson 2 turns the apparent wind you now have into two forces, one that drives the ship and one the ship has to pay for.