Rowing crew scale

Why does an eight travel faster than a single sculler when every rower in it produces no more power than the sculler does?

Put eight rowers in a boat and it goes faster than one rower alone. Nobody finds that surprising. But look at what has actually been added: eight bodies, each producing about the same power as the single sculler, and eight bodies' worth of weight for the water to carry. The crew is not stronger per person. It is not more efficient per stroke. Every extra rower brings their own drag with them.

So why does the boat go faster at all — and, more revealingly, why does it go faster by so little? An eight is not eight times quicker than a single. It is not twice as quick. What sets that number?

Start with what the water actually charges for. A racing shell is astonishingly slender — a long narrow needle, not a boat shape — and that slenderness is deliberate: it makes very little wave. So the dominant cost is not pushing water aside but dragging it along the hull's skin. Skin friction goes roughly as the wetted surface area times the square of the speed.

Now, how does wetted area grow when you add rowers? Not directly. It grows through displacement. The hull floats by displacing its own weight in water, so the volume submerged is set by the mass aboard, which is essentially the crew: V is proportional to n, the number of rowers.

And area? If the shells are geometric copies of one another — longer, wider and deeper in proportion, which is close to true across racing classes — then area scales as volume to the two-thirds. So wetted area goes as n to the two-thirds. That is the whole trick in one line: crew grows, drag surface grows more slowly.

Now put power on both sides. Power delivered scales with the crew: proportional to n. Power demanded is drag times speed, and drag is area times speed squared, so demand is proportional to n^(2/3) times *v*³.

Set them equal, because in a steady race they are equal:

n ∝ n^(2/3) · v*³, so *v*³ ∝ *n^(1/3), and therefore v ∝ n^(1/9).

A ninth root. That is a spectacularly weak dependence, and it answers both halves of the question at once. Adding rowers does help, because power rises faster than drag surface. But it helps almost imperceptibly, because the surplus is being spent on speed, which drag punishes as a cube.

Put numbers to it. Eight to the power of one ninth is about 1.26 — an eight should be around a quarter faster than a single. Two to the power of one ninth is about 1.08, so every doubling of the crew, whether one to two or four to eight, should buy about the same eight per cent. I recall the actual spread of boat-class times sitting close to that prediction, within a couple of percent; I will not quote race times to support it, because those vary by course, conditions and era in ways that would make the agreement look more precise than it is.

The ninth-root result is a scaling argument, not a prediction about any actual race, and it leans on four assumptions worth stating plainly.

It assumes geometric similarity between boat classes — close, but not exact. It assumes skin friction dominates; racing shells are slender enough that this is a fair approximation, but wave-making is not zero, and since longer boats run at lower Froude number, the bigger boat is quietly favoured beyond what the model grants. It assumes every rower delivers the same power, which is where the comparison is at its weakest: a single sculler holds two oars while an eight sweeps one each, a different movement with different mechanics, and crews are not selected the way singles are. And it ignores the energy lost to the boat surging and slowing within each stroke, which differs by class.

So the honest claim is narrower than it first appears: the exponent is a physical result; any particular race margin is not. And notice what the argument does not say. It does not say rowers cooperate to become individually stronger, or that technique multiplies. Nothing here is synergy at all. It is bookkeeping about surfaces.

What would falsify it? The sharpest test is the constancy of the exponent. The model insists that doubling crew size buys the same fractional speed increase wherever you double — one to two, two to four, four to eight, always about eight per cent. If the four-to-eight step delivered substantially more than the one-to-two step, a fixed exponent would be dead and something unmodelled would be doing the work.

There is a second, cleaner test. A coxswain adds mass without power. Take a four with a cox and a four without: the extra mass raises displacement, hence wetted area, while crew power stays fixed, so the model predicts the coxed boat is slower by a few per cent — a figure that follows from the two-thirds and one-third exponents alone. Coxed and coxless fours race the same distance, so the comparison is directly available. Were the two classes to record indistinguishable times, the claim that displacement is what sets drag would be in serious trouble.

• Why racing shells are built so extreme in slenderness, and what the hull would look like if wave drag mattered more.
• How the within-stroke surge of the boat wastes power, and why it differs between a single and an eight.

Every term the collection defines is gathered in the glossary.