The wave that holds a ship back

In August 1893 Fridtjof Nansen’s ship Fram, making for the ice north of Siberia, ran into water in which it could barely move. The engine was running and the sea was calm, and the ship, which made four or five knots under steam, was held to about one. Nansen described the vessel as if gripped, and noticed that the effect disappeared when the ship broke free into open sea. Sailors in Norwegian fjords had a name for it, dødvand — dead water — and an older tradition, repeated by Pliny, put the blame on a small fish that clamped itself to ships’ hulls.

The explanation, worked out by Vagn Walfrid Ekman in laboratory tanks in 1904, is a wave, and it is one of the cleanest demonstrations that a wave the eye cannot see can carry away most of a ship’s power. The Fram was sailing in a layer of fresh meltwater lying on top of salt water, floating on it exactly as a body floats on the fluid it displaces, because it is lighter. The boundary between the two layers carries waves of its own, very slow ones, and a ship moving at about their speed spends its effort making them.

A wave on a hidden boundary #

The layer a parcel cannot leave found that a fluid layered by density resists vertical displacement, and the wave that is required to stand still found the waves that a steady flow over an obstacle excites in such a fluid. The simplest layered fluid is not a continuous gradient but two layers — light water of depth $h_1$ on top of denser water of depth $h_2$ — and the boundary between them carries waves exactly as the surface of a pond does, with one difference that changes everything.

At the surface of a pond the restoring force on a raised hump of water is its full weight. At an interface between two waters of slightly different densities, a raised hump of the lower water is surrounded by upper water almost as heavy, and the restoring force is only the difference in weight. Gravity is replaced by reduced gravity, $g' = g\,\Delta\rho/\rho$, and for a density difference of two per cent that is a fiftieth of ordinary gravity.

The fastest waves on the interface are the longest, and their speed is $c_0 = \sqrt{g' h_1 h_2 / (h_1 + h_2)}$, which for a thin upper layer is close to $\sqrt{g' h_1}$ — the shallow-water speed of waves whose speed depends on their length, with reduced gravity in place of gravity. For a metre and a half of water two per cent lighter than the water beneath, it is 0.52 metres per second: 1.02 knots, slower than a person walks. For a density difference of 0.2 per cent it is a third of a knot.

The waves are enormous and invisible. The restoring force is so weak that the interface moves easily and far, and the surface above hardly moves at all: an interfacial wave that lifts the boundary by a metre raises the surface by roughly the density difference times that metre, two centimetres, spread over tens of metres. A ship in dead water can be dragging a wave as tall as its own draft along the boundary beneath it while the sea around it looks flat.

What a hull does to the interface #

A hull that reaches down towards the interface pushes it down, and as the hull moves the depression moves with it. What happens next depends on whether the interface’s waves can keep up.

The displacements come from transforming the interface’s response to the moving hull back into space. Moving at 0.95 of the fastest wave speed, the hull leaves a train of waves 53 metres long trailing behind it — the length at which an interfacial wave travels exactly as fast as the hull, so that the pattern stands still relative to the ship, which is the steady-wave condition a mountain imposes on the wind. The wavelength was measured from the drawn profile and matches the one the waves’ dispersion relation gives. Every crest in that train holds energy, and the train lengthens as the ship advances, so the ship is paying continuously for waves it leaves behind.

At 1.2 times the wave speed there is no such train. No interfacial wave travels that fast, so no pattern can keep pace with the hull, and the interface only bulges under the hull and settles behind it. The bulge is upward rather than downward: a flow faster than its own waves rises over an obstacle, the way a fast stream humps over a stone, where a slow one dips.

Why the train grows behind the ship #

The standing wave behind the hull has a phase speed equal to the ship’s by construction; that is what makes it stand still in the ship’s frame. Its energy does not travel at that speed. Interfacial waves are dispersive — shorter ones are slower — and in any dispersive wave the energy moves at the group velocity rather than the speed of the crests. For these waves the group velocity is less than the phase velocity, so the energy the hull puts into the interface falls behind the hull, and the train of waves lengthens astern at the difference between the two speeds.

That lag is the drag. The rate at which the ship does work against the waves equals the rate at which wave energy is left behind in the lengthening train: the energy per unit length of the train times the speed at which the train grows. Near the fastest interfacial wave speed, where the standing wave is very long, the group velocity approaches the phase velocity and the train grows slowly — but the waves are also at their largest, because the hull presses on them with its full length, and the product peaks there.

A thin upper layer makes the effect worse in a second way. The interfacial wave speed rises as the square root of the upper layer’s depth, so the thinner the layer of fresh water, the slower the trap. A metre of meltwater over the sea, not unusual near a glacier front or where a river spills into a fjord, puts the hump below a knot; half a metre, lower still.

The drag of making those waves #

The energy carried away by the lee wave is a drag on the hull, and it can be computed from the interface’s response: it is set by how strongly the hull presses on the one wavelength that can stand still behind it.

For a slow hull the standing wave is short, and a hull much longer than the wave presses on it feebly, because the push from one end of the hull cancels the push from the other. As the hull speeds up the standing wave lengthens, the hull presses on it more effectively, and the drag climbs. As the speed approaches the fastest wave speed the standing wave becomes very long, the hull presses on it with its whole length, and the drag is largest. Above that speed there is no standing wave, and the drag falls away.

The calculation includes a little damping of the interfacial waves, standing for the mixing and viscosity of a real boundary. Without any, a two-dimensional calculation gives a drag that stops at exactly the wave speed as if cut by a knife, and the damped drag was checked against that undamped result below the wave speed, where the two must agree. The damping turns the knife-edge into a hump that peaks just below the wave speed, as ships actually experience. A short hull, which presses on short waves as easily as long ones, peaks at 0.82 of the wave speed; a 15-metre hull at 0.94; a 40-metre hull at 0.99, closest and most sharply.

Trapped below the hump #

The hump is what makes dead water a trap rather than a nuisance.

A ship under a steady thrust settles where the thrust equals the resistance. Friction alone rises smoothly with speed and meets any thrust once. The hump of wave drag gives the resistance curve a maximum and a minimum, so a thrust between the two meets it three times. For the weaker thrust drawn, the crossings are at speeds below, near and above the hump; for the stronger, at 0.94, 1.00 and 1.82 knots.

Which of those speeds a ship reaches depends on how it got there. Starting from rest and accelerating, a ship climbs the resistance curve from the bottom and stops gaining speed at the first crossing, below the hump, because beyond it the resistance exceeds the thrust. The steady state at 1.82 knots exists and is stable, and the ship cannot reach it from below without either a thrust greater than the top of the hump or some other push — a gust, a current, a tow — that carries it over. A ship already moving fast that enters layered water stays on the fast branch. That dependence on history is the signature of dead water, and it explains why Nansen found the effect vanishing abruptly once the Fram escaped the layered water: the hump, not the thrust, set the speed.

The same curve says how a ship gets out. One way is over the top: a temporary extra push, from a gust in the sails or a burst of power, carries the ship past the maximum of the resistance, after which the same steady thrust holds it on the fast branch. The other is under it: slowing right down, to a speed at which the standing interfacial wave is short and the hull presses on it feebly, costs little in wave drag and wastes no effort fighting the hump. What does not work is the natural response of simply demanding more from a steady engine that cannot supply the peak — the ship then sits at the top of the hump, burning fuel into a wave nobody can see.

The same hump in other waters #

The hump is not peculiar to hidden interfaces. Any vessel making waves in a medium whose waves have a maximum speed meets a rise in wave drag as it approaches that speed, and three familiar cases are the same mechanism.

A ship in shallow water meets it at the surface. In water of depth $d$ the fastest surface wave travels at $\sqrt{gd}$, and a ship approaching that speed — a depth Froude number near one — makes a long, strong wave, sinks lower in the water, and needs sharply more power. Canal boats and ships in shallow channels have always been handled with this in mind, and at speeds above it the resistance falls again, which is why a horse-drawn barge that was forced past it in the 1830s was found to run more easily on the far side — the observation that led John Scott Russell to his solitary wave.

A displacement hull in deep water meets it in a different form. Deep-water waves have no maximum speed, but the wave a hull makes becomes as long as the hull itself at a speed of about $1.34\sqrt{L}$ knots, with $L$ in feet, and beyond that the hull climbs its own bow wave: the “hull speed” beyond which a displacement vessel cannot be pushed efficiently, set by how a water wave’s speed depends on its length.

And an aircraft meets it in the air. The speed of sound is the fastest disturbance the air can carry, and an aircraft approaching it compresses the air ahead into a pattern that cannot run away, with a sharp rise in drag near the speed of sound and a cone of disturbance trailing behind once it is exceeded. Dead water is the same problem for a ship whose relevant waves happen to be twenty times slower than the ones on the surface above them.

How strong the layering must be #

The hump moves with the layering. For a density difference of 0.2 per cent it peaks at 0.30 knots, too slow to trouble anything but a drifting boat; for 1.0 per cent at 0.68 knots; for 2.0 per cent at 0.96 knots. It also grows, because depressing the interface by the same amount takes more force when the density difference is larger, and more force makes more wave. Sharp fresh-over-salt layering — meltwater over sea water in a summer fjord or off a glacier front, or a river plume spreading over the sea — puts the hump at a knot or two, which is exactly where sailing vessels in light wind, rowing boats and small steamers of the nineteenth century travelled. Modern ships, making ten knots or more, sail far above any interfacial wave speed the ocean supplies and meet dead water only when manoeuvring slowly.

Beyond two layers and two dimensions #

The calculation is two-dimensional. A real hull has a beam as well as a length, and makes interfacial waves that spread sideways in a wedge, as a ship’s surface wake does. In three dimensions the drag does not stop at the wave speed even without damping, because obliquely travelling waves can keep pace with a faster hull, and the hump is broader and its fall more gradual than the figures draw.

The hull is a pressure on the interface. A real hull displaces the upper layer, and its effect on the interface depends on its draft relative to the layer’s depth; a hull whose keel reaches through the interface is a different problem. The figures stand for a hull whose effect reaches the interface without crossing it.

The interface is sharp and the layers uniform. Real stratification has a transition layer of finite thickness, and a continuous gradient beneath it, which support many internal wave modes rather than one. The first mode dominates when the transition is thin, as in meltwater over salt, and the others add their own humps at lower speeds.

The ship’s speed is steady. Laboratory experiments in dead-water tanks have found ships whose speed oscillates rather than settling: the hull outruns its wave, the wave builds and catches it, the drag rises, the hull slows, and the cycle repeats. The steady picture captures where the trap is and not how a ship moves within it.

A tall wave under a flat sea #

The interface profiles are scaled so that the wave can be seen, and the scaling hides the one fact that makes dead water mysterious. The real interface in the Fram’s situation would have moved by metres while the sea surface moved by centimetres. A figure that showed both at true scale would show a flat line above a tall wave, which is what a sailor saw and what made the effect so hard to explain.

Nor do the pictures show where the energy goes. The wave train behind the hull carries it away along the interface, where it can travel for kilometres before steepening until it breaks and mixes the two layers — so a ship caught in dead water is not only slowed but is, in a small way, stirring the ocean.

Still open: how much of the ocean’s mixing is done at interfaces like this #

Dead water is the most visible case of a general process: energy put into a layered fluid at its surface travelling down into waves on its internal boundaries, and being carried away from where it was put in before it breaks and mixes. The same physics governs the wind blowing over a stratified sea, the tides flowing over sills at the mouths of fjords, and the wakes of submarines moving through the ocean’s thermocline. How much of the mixing that maintains the ocean’s density structure is done by such waves breaking far from where they were made, rather than near their source, is measured only roughly, and the tide’s own internal waves are the best-studied piece of a budget that is still uncertain by a factor of about two.

The habit worth carrying away is to ask what a moving body can keep pace with. A body faster than every wave its medium supports cannot leave a steady wave behind it, and one slower than some of them can — and the boundary between the two is where the drag of wave-making peaks, whether the medium is the sea surface, an interface nobody can see, or the air around an aircraft approaching the speed of sound.