The body that displaces two things

The block the water does not lift ends by naming what is left: a body that is not in one fluid. Every earlier case has had a single density to multiply a volume by, and the whole of Archimedes’ principle is that multiplication.

Two cases break it, and they break it in different directions.

A body at an interface, and the difference that holds it. The net upward force on a ten-centimetre cube straddling the boundary between two fluids, against how far its underside sits below that boundary — each curve scaled by its own largest value so that three very different cases fit on one axis. The displaced weight has to be counted twice, once with each density, so the force is linear in the displacement with a stiffness set by gravity, the cube's cross-section and the difference of the two densities — the difference, and neither of them alone. air over water holds the cube with its underside 60 per cent of the way through, at a stiffness of 97.8 newtons per metre; oil over water holds the cube with its underside 47 per cent of the way through, at a stiffness of 14.5 newtons per metre; water over mercury holds the cube with its underside 24 per cent of the way through, at a stiffness of 1229.4 newtons per metre. Both numbers are read off the drawn curve rather than substituted. The consequence is the one worth the figure: a body at an oil–water interface is held six times less stiffly than the same body at an air–water one, because the density difference is six times smaller, and the ordinary intuition that a denser fluid holds a body more firmly is exactly wrong — what matters is the contrast across the surface the body is sitting in. — Fig. 1 The net upward force on a ten-centimetre cube straddling the boundary between two fluids, against how far its underside sits below that boundary, each curve scaled by its own largest value. The displaced weight is counted twice, once with each density, so the force is linear in the displacement with a stiffness of $gL^2(\rho_2 - \rho_1)$ . Air over water holds the cube at 98 newtons per metre; oil over water at 14.5.

Counting the volume twice

A cube floating at a boundary displaces some of the upper fluid and some of the lower. Writing $x$ for how far its underside sits below the interface, the displaced weight is

gL^2\big[\rho_2 x + \rho_1(L - x)\big]

and setting that equal to the cube’s weight gives the equilibrium at once: $x^* = L(\rho_b - \rho_1)/(\rho_2 - \rho_1)$ . The body sits at the fraction of the way through the interface that its density is of the way between the two fluids’ — which is the ordinary floating result with the upper fluid’s density subtracted from everything.

The force away from equilibrium is what matters, and it comes out as

F = gL^2(\rho_2 - \rho_1)(x - x^*)

exactly linear, with a stiffness of $gL^2(\rho_2 - \rho_1)$ .

The difference, and neither density alone. That is the substance of the argument, and it inverts the usual intuition immediately. A body floating on mercury is not held ten times more firmly than one floating on water because mercury is dense — pressure knows only the depth, and the resultant over the body’s surface knows only what is on either side of it; a body at the boundary between water and mercury is held twelve times more firmly than one at the boundary between air and water, and a body at the boundary between oil and water is held less firmly than either, because the contrast is small.

The air–water case is the one that hides this. Air’s density is a thousandth of water’s, so $\rho_2 - \rho_1$ is within a tenth of a per cent of $\rho_2$ , and every ordinary floating calculation can drop the upper fluid entirely. That habit is correct for ships and wrong for everything else.

What it means to bob

The stiffness has a consequence in a unit anybody can measure.

How long it takes to stop bobbing. The period of the vertical oscillation a floating body performs when pushed and released, against the density difference across the interface it is floating in, for three sizes of cube — both axes logarithmic. The restoring stiffness is the density difference times the waterplane area, so the period rises as the inverse square root of that difference, an exponent measured off the drawn curve as -0.5000. air over water gives 0.61 s for a ten-centimetre cube; oil over water gives 1.58 s for a ten-centimetre cube; fresh over salt gives 3.70 s for a ten-centimetre cube. So the same body bobs 2.6 times more slowly at an oil–water interface than at an air–water one, and 6.1 times more slowly at the boundary between fresh and salt water. That last case is why a layer of fresh water over salt is such a difficult place to keep anything at a depth: the restoring force exists, it is real, and it is so weak that a disturbance takes many seconds to settle out and any current at all overwhelms it. — Fig. 2 The period of the vertical oscillation a floating body performs when pushed and released, against the density difference across the interface, for three sizes of cube. The period rises as the inverse square root of that difference — an exponent measured off the curve. A ten-centimetre cube bobs in 0.61 seconds at an air–water boundary, 1.58 at an oil–water one, and 3.70 at the boundary between fresh and salt water.

A floating body pushed down and released oscillates, with a period $2\pi\sqrt{\rho_b L/(\Delta\rho\,g)}$ — the ordinary result for a spring of that stiffness and that mass. The inverse square root makes the dependence weak, and the range of density differences available makes it matter anyway: across the interfaces that occur in nature the answer moves by a factor of thirty.

The last of the three marks is the practical one. A layer of fresh water over salt differs by twenty-seven kilograms per cubic metre — under three per cent — and a body floating at that boundary is held with a stiffness under three per cent of what water alone would give. It is a real equilibrium and it is so weak that any current at all overwhelms it, which is why anything meant to sit at such a boundary has to be tethered rather than trusted to float there. It is also why a compressible body’s instability is so much easier to demonstrate than its stratified cure: the destabilising term is large and the restoring one is small.

That is also why a fresh-water lens over salt in an estuary is such a difficult place to work, why sediment collects at a density interface rather than settling through it, and why an oil slick and the water beneath it exchange almost nothing mechanically — a drop of one in the other is held so gently that surface tension, which is usually negligible at these scales, is comparable with buoyancy.

The interface as a place things collect

The weakness of the restoring force at a small density contrast has a consequence that is easier to see in what accumulates at an interface than in anything that floats there.

A particle denser than the upper fluid and lighter than the lower one has nowhere else to be: it cannot rise out of the boundary and it cannot sink through it, and the equilibrium is stable. So an interface between two fluids of similar density is a collecting surface, and it collects everything whose density lies in the narrow band between them.

In the ocean that band is where marine snow accumulates, and a pycnocline is visible on a sonar for that reason — the scattering comes from the particles held at it rather than from the density change itself. In a settling tank it is where a sludge blanket forms. In a glass of layered drinks it is where the bubbles stop.

And the narrowness of the band is the point. At an air–water boundary the band spans all densities from 1.2 to 998 kilograms per cubic metre, which is nearly everything, so nothing is distinguished — every solid sinks and every gas rises. At an oil–water boundary the band is 850 to 998, which is a seventeen per cent window, and what sits in it is a particular and rather specific class of material. Two fluids differing by three per cent collect a three per cent slice of everything there is, and hold it with a stiffness in proportion. Which is the shape deciding the answer in an unfamiliar sense: the shape in question is the density profile rather than any object’s.

The mechanism is used deliberately. Density-gradient centrifugation separates cells and organelles by spinning them in a column of graded density until each comes to rest where the density matches its own, and the resolution of the separation is set by how gently the gradient rises — the same trade a parcel displaced in a stratified column makes between how strongly it is held and how far it will travel — a steep gradient holds each band tightly and separates poorly, a gentle one separates finely and takes longer to settle. That trade is the third figure’s stiffness read as an instrument.

The continuous case, and the reversal

The other way the single-density picture fails is a column whose density changes gradually, and there the stability question changes character rather than the force.

Fig. 3 The net upward force per unit volume on a neutrally trimmed body against depth, for four arrangements, all trimmed to be exactly neutral at three hundred metres. A rigid body in a column whose density rises with depth is pushed back whichever way it is displaced. The compressible body in a uniform column is not, which is the case found for a uniform column. A real body is both, and which wins is a comparison of two gradients.

The depth past which it must sink establishes that a compressible body’s neutral depth is always unstable: push it down, it compresses, its density rises, and it goes on sinking. That result is complete for a uniform fluid and it is silent about a stratified one.

A rigid body in a stratified column does the opposite. Push it down and the surrounding fluid is denser there, so the buoyancy rises and pushes it back; lift it and the fluid is thinner and it sinks back. Its neutral depth is stable, and the restoring force is proportional to how fast the column stratifies.

So the two effects have opposite signs, and a real body — compressible, in a stratified column — is a competition between them. The criterion is a comparison of gradients: the fluid’s density must rise with depth faster than the body’s own does. Writing that out for a body of compressibility $\kappa$ gives a critical gradient of $\rho\kappa g$ , and for a body as compressible as water itself that is 4.62 parts per million per metre.

Why that number is uncomfortable

Real ocean stratification is of that order, which is the reason the criterion is worth computing rather than guessing at.

A strong thermocline — the layer where temperature falls sharply, at a few tens to a few hundred metres — has a density gradient of tens of parts per million per metre, comfortably above the critical value. A body left at neutral trim inside a thermocline is genuinely stable, and that is why submarines can and do rest on one.

The deep ocean below about a kilometre is very nearly uniform in potential density, with gradients of a part per million per metre or less — below the critical value. A body left at neutral trim there is unstable, exactly as the uniform-column result says, and has to be held by active control.

The two regimes are separated by a number that is a property of the hull rather than of the ocean, and a hull made stiffer than water — a thick pressure hull with little compressible volume — lowers its own gradient and moves the boundary. That is a design consideration and it is not usually described in these terms; what it is, is the choice to make the second gradient in the comparison smaller.

There is a corresponding statement about the water itself. Seawater’s own compressibility means a parcel moved down is compressed and warmed, so comparing densities at different depths requires a correction — which is why oceanography works in potential density, the density a parcel would have if brought adiabatically to a reference pressure, rather than in the density it actually has. The stability criterion in that variable is simply that potential density rise with depth, and the correction that makes it simple is exactly the gradient this figure is comparing against.

The same fact drawn as a landscape. The potential energy of the compressible body against depth, obtained by integrating the net force of the previous figure. A body with 4.0% of its volume as gas has a turning point at 150.0 metres, and the turning point is a maximum. That is the whole story in one shape. An equilibrium at a maximum of the potential is unstable in both directions: a body nudged up floats away to the surface with increasing force, and a body nudged down sinks with increasing force, and neither returns. Nothing about the fluid is unusual and nothing about the body is badly made. The instability is a consequence of the gas being more compressible than the water — if the two compressed equally the curve would be flat, and if the body were the less compressible of the two the turning point would be a minimum and the depth would hold itself. The deep ocean has bodies of the third kind in it, which is why a swim bladder is an organ that needs continuous control and an oil-filled float does not. — Fig. 4 The uniform-fluid case for comparison, from the uniform-fluid case: net force against depth for a body whose displaced volume falls with pressure. It crosses zero once and the crossing slopes the wrong way, so a small displacement grows. Everything here is about what a density gradient in the surrounding fluid does to that slope.

What a shape has to do with it

The stability discussed so far is vertical. A body at an interface has a second stability question — whether it stays upright — and the interface changes that one too.

The righting arm against heel. The righting arm of a rectangular hull 3 m in the beam with its centre of gravity 1.05 m above the keel, measured off the solved waterline at each angle. The straight line is the small-angle rule GM·sin φ with GM = 0.278 m. The measured arm is the larger of the two from the first degree onward, because a wall-sided hull gains a second term growing as the square of the tangent of the heel — so the small-angle rule is conservative here rather than merely approximate. — Fig. 5 The righting moment of a floating body against how far it has heeled, for a body in one fluid. The restoring moment comes from the waterplane’s second moment of area multiplied by the density of what is being displaced — so at an interface it is multiplied by the *difference* of two densities instead, and a hull that is comfortably stable in water is marginal at a boundary between two similar fluids.

A ship comes back upright because heeling moves the centre of buoyancy sideways, and how far it moves depends on the waterplane’s second moment of area. That derivation assumes one density below and nothing above.

At an interface, the same geometry produces a restoring moment proportional to $(\rho_2-\rho_1)$ rather than to $\rho_2$ . So a body’s metacentric height at an oil–water boundary is about a seventh of what the same body has in water, and a hull designed with a comfortable margin in water has almost none there — the metacentre having moved for a reason that is nothing to do with the geometry it is computed from.

That is not a hypothetical. A submerged body at a density interface, a subsea structure being lowered through a halocline, and a barge in water with a thick layer of froth or slurry on it are all in that situation, and the last is a known hazard — the apparent waterline is in a low-density layer and the stability is much worse than the draught suggests.

How deep a floating body sits. A block of relative density 0.92 floating in water. The fraction submerged is 0.92 — the density ratio and nothing else — so the waterline cuts the block at 92 per cent of its height, whatever the block is made of and whatever its size. — Fig. 6 And the ordinary case the others are departures from: a body floating in one fluid at the draught where the displaced weight equals its own. Everything here is that picture with the upper fluid given a density, or with the lower fluid given a gradient — and in each case the answer is not a correction to this one but a different statement about stability.

What the two failures have in common

The two cases here look unrelated — one is a discontinuity in the density and the other a gradient — and they are the same correction.

Archimedes’ principle is the statement that the pressure over a closed surface integrates to the weight of the fluid that would fill it. That derivation is a divergence theorem and it needs the fluid to have a definite density inside the body’s own volume, because that is what the volume integral is over. Where the density is not definite — because the volume contains two fluids, or because it contains a range of one — the integral has to be performed rather than summarised, and the summary’s failure is not an error but an absence of the quantity it summarises.

So both cases are one statement: the displaced weight is an integral, and reading it as a product requires the density to be constant over the volume displaced. Where it is not, the force is still correct and the derivative of the force — which is what stability is about — depends on how the density varies, which the product form has thrown away.

That is the same structure as the earlier lesson about what surface an integral was taken over, one level up: there the surface had to still be there, and here the integrand has to be constant. Each is a condition the summarised form hides.

What surface tension and a sharp interface leave out

Surface tension is ignored throughout. At a fluid–fluid interface the surface tension is much smaller than at a liquid–air one, but so is the buoyant stiffness, and the ratio of the two does not scale the obvious way. For a body of a few millimetres at an oil–water boundary the two are comparable, and the contact line’s behaviour decides the equilibrium.

The interface is treated as sharp. A real halocline or thermocline is a gradual change over metres, so a body of comparable size is not at an interface at all but in a gradient — which is the third figure’s case rather than the first’s, and the crossover between the two descriptions happens when the body’s dimension matches the interface’s thickness.

The stratified column is taken as linear in depth. Real ocean stratification varies enormously with depth and place, and the critical gradient has to be compared with the local one rather than with an average. The figure’s numbers are for an order of magnitude rather than for a location.

Every fluid here is at rest. A density interface is also a surface that supports waves of its own, and a body sitting at one is coupled to them: it is not bobbing in a still boundary but exchanging energy with an internal wave field whose restoring force is the same weak one. That coupling is what a wave on a density interface is, and it is entirely absent from these figures.

And the body’s compressibility is taken as a single number. A real submerged body is a rigid hull with compressible contents — trapped air, foam, a ballast tank — and its effective compressibility depends on what fraction of its volume each occupies. A hull that is ninety-six per cent solid and four per cent air behaves very differently from one that is half air, and the design question is which side of the critical gradient that puts it on.

An interface that is a region rather than a line

The first figure draws a force against a displacement and cannot show the interface. The boundary between two liquids is not a line: it is a region a few molecules thick, it is deformable, and a body sitting in it deforms it — pulling a meniscus up or down, which changes the displaced volumes by an amount the straight-line calculation does not contain. For a large body that correction is negligible and for a small one it is the whole answer.

The stratification figure draws four straight lines and hides that two of them describe the same body. The rigid case and the compressible case are not two different objects; they are the same object with and without a term that is always present, and which of the two descriptions applies is decided by a comparison the figure performs and does not draw.

And none of the figures can show what a body at a density interface actually looks like, which is often nothing like a floating block. A drop of oil at a water–air boundary spreads or beads according to three surface tensions rather than to any density; a solid particle at an interface is held by its contact line — a boundary whose behaviour depends on the history of the surface — far more strongly than by buoyancy and is the basis of froth flotation, with a surface that is not a skin and behaves like one doing the holding; and a body less dense than both fluids simply rises through the interface without ever straddling it.

Still open: what happens when the body is the size of the gradient

Every figure here treats the body as small compared with the scale over which the fluid’s density changes, so that the density at the body’s own depth is the only thing that matters.

That assumption fails for the cases that matter most. A submarine is a hundred metres long and a thermocline can be thirty metres thick, so the vessel spans a substantial part of the gradient: its bow and its stern are in water of different densities, the buoyancy is distributed rather than concentrated, and it acquires a trimming moment that nothing in a point-body calculation contains. Submariners know the effect and manage it by trimming; predicting it requires integrating the density over the hull rather than evaluating it at a point.

The same difficulty arrives more severely for a body crossing an interface it is comparable with. The displaced volume is then a function of the interface’s shape as well as the body’s, and the interface’s shape depends on the body — a coupled problem with no closed solution. Numerical treatments exist, and the general behaviour of a large body at a diffuse interface, including whether it can be trapped there and how strongly, is not a solved question.

The habit worth carrying away is the one the first figure is. When a result depends on a quantity, check whether it actually depends on a difference. Archimedes’ principle is usually written with one density because one of the two fluids is air, and air’s density is a thousandth of water’s — so a subtraction has been performed and forgotten. Every argument that treats the upper fluid as nothing is making that approximation, and the approximation is excellent in the one case it is usually met in and useless in all the others.

The body that displaces two things