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Buoyancy and Archimedes

Steel is about eight times denser than water, yet a ship made of steel floats. Understanding why is not a physics curiosity — it is the prerequisite for every stability number CargoForge-C computes. perform_analysis in src/analysis.c derives draft, KB, BM, and ultimately GM from a single foundational calculation: how much water does the ship displace, and what does that tell us?

The mental model 🧠

A floating ship is locked in a quiet tug-of-war that always ends in a draw. Gravity pulls the whole ship down with a force equal to its weight; the water pushes up with a force equal to the weight of the water the hull shoves out of the way. The ship settles to exactly the depth where those two forces match — no deeper, no shallower — and there it floats.

That is why a steel ship floats even though steel sinks: it is not the material that matters but the shape. A solid steel cube displaces only its own small volume of water and loses the tug-of-war; spread that same steel into a hull and it shoves aside many tonnes of water before it is even half submerged. Every number perform_analysis computes — draft, KB, BM, GM — grows from this one balance: weight down equals displaced-water-weight up. Load more cargo and the ship simply settles deeper until the books balance again.

Why a steel ship floats: buoyancy equals the weight of the displaced water The ship's submerged hull pushes aside a volume of water V. The upward buoyant force equals the weight of that displaced water. The ship floats at the draft where buoyancy up balances the ship's weight down. waterline displaced water · volume V weight buoyancy Buoyancy = weight of the water pushed aside (ρ·g·V) — it floats when buoyancy balances weight.

What this actually means (plain English)

No jargon — here's what the ideas in this lesson actually mean, and why they matter.

  • Buoyancy = "the upward push a fluid gives anything that shoves it aside" — a steel ship floats not because steel is light but because the hollow hull pushes aside enough water that the water's weight equals the ship's; perform_analysis opens with exactly this balance before computing anything else.
  • Displaced volume = "the underwater space the hull occupies, measured in cubic metres" — once perform_analysis divides total displacement (tonnes) by SEAWATER_DENSITY = 1.025, every stability number that follows is derived from this single volume.
  • Draft = "how deep the hull sits in the water" — the box-hull branch finds draft by dividing displaced volume by the footprint of the hull shrunk by BLOCK_COEFF = 0.75; the table branch does it by searching the hydrostatic CSV with hydro_draft_from_displacement and linear interpolation.
  • KB = "height of the buoyancy force's sweet spot above the keel" — for the box-hull approximation it is 0.53 × draft; from a real hydrostatic table it is read directly from the interpolated HydroEntry, because the centroid of the displaced water depends on hull shape.
  • BM (metacentric radius) = "how wide the waterplane is, converted into a stabilising height" — computed as waterplane inertia (\(L B^3 / 12\), adjusted by WATERPLANE_COEFF) divided by displaced volume; a wider ship cubes its beam in this formula, which is why beam is the dominant lever for initial stability.
  • GM = "the safety margin that keeps the ship from rolling over" — perform_analysis assembles it as r.kb + r.bm - r.kg; a positive value means the virtual pivot point (metacentre M) sits above the centre of gravity, so any tilt generates a righting force, and the IMO minimum is 0.15 m (IMO_GM_MIN).

Why it matters: if you get the displaced-volume calculation wrong — wrong density constant, wrong block coefficient, corrupted hydrostatic table — every downstream number (KB, BM, GM, free-surface correction) is wrong too, and the ship's certified stability margin is fiction.

The Principle

Around 250 BCE, Archimedes observed that an object submerged in a fluid experiences an upward force equal to the weight of the fluid it displaces. Written as a relationship rather than a formula:

\[\text{Buoyant force} = \text{weight of displaced fluid}\]

For a floating body in equilibrium the buoyant force exactly equals the body's own weight. The body sinks just deep enough that the water it pushes aside weighs as much as it does. If you push it deeper, the displaced water weighs more than the object and it rises back. If you push it shallower, the object outweighs the displaced water and it sinks again. Equilibrium is self-correcting.

A steel ship floats because its hull is hollow. The total weight of the steel shell, machinery, cargo, and fuel is spread over a large displaced volume — a volume whose equivalent water-weight equals that total. The density of the combined ship-plus-enclosed-air system is less than water, so it floats.

Seawater Density — the One Constant that Connects Weight to Volume

Salt water at 15 °C has a density of approximately 1.025 tonnes per cubic metre. That number appears at the top of src/analysis.c as a named constant:

/* from src/analysis.c */
#define SEAWATER_DENSITY    1.025f  /* t/m3 at 15C */

It is the conversion factor between mass (how heavy the ship is) and volume (how much water must be moved aside to keep it afloat). Given a displacement in tonnes:

\[V_{displaced} = \frac{\Delta}{\rho_{SW}} = \frac{\Delta}{1.025}\]

where \(\Delta\) is displacement in tonnes and \(V\) is in cubic metres. This is the only physical constant hard-coded into the analysis; every other number is either measured (from the hydrostatic table) or derived from it.

Fresh water vs. salt water

Fresh water is 1.000 t/m³. A ship in a fresh-water river sits slightly deeper than in open sea because fresh water is less dense — the same displaced volume weighs less, so the hull must go deeper to displace more of it. CargoForge-C uses the salt-water constant throughout.

From Weight to Displacement to Draft

perform_analysis (in src/analysis.c) builds displacement by summing all mass contributions:

/* from src/analysis.c, perform_analysis() */
float displacement_kg = ship->lightship_weight + r.total_cargo_weight_kg;

/* Include tank weight in displacement if tanks are loaded */
float tank_weight_t = 0.0f;
if (ship->tanks && ship->tanks->count > 0) {
    tank_weight_t = calculate_tank_weight(ship->tanks);
    displacement_kg += tank_weight_t * 1000.0f;
}

Then it converts to tonnes and finds the displaced volume:

float displacement_t  = displacement_kg / 1000.0f;
float displaced_vol   = displacement_t / SEAWATER_DENSITY;

With the displaced volume known, the question becomes: at what draft does the hull enclose that volume? The answer depends on the hull's shape.

Box-Hull Fallback

When no hydrostatic table is loaded (ship->hydro is NULL), CargoForge-C approximates the hull as a rectangular box scaled by a block coefficient:

/* from src/analysis.c, box-hull branch */
r.draft = displaced_vol / (ship->length * ship->width * BLOCK_COEFF);

The block coefficient BLOCK_COEFF = 0.75 accounts for the fact that real ship hulls are not perfect boxes — they taper at bow and stern. A value of 0.75 means the actual underwater volume is 75% of the bounding rectangular box of the same length, width, and draft. So the hull must go a little deeper than a pure box would to displace the required volume.

Table-Based Hydrostatics

Real ships are measured in dry-dock at many drafts and the resulting data is published in a hydrostatic table. src/hydrostatics.c reads that table from a CSV file and provides two look-up functions:

  • hydro_draft_from_displacement — given a displacement in tonnes, finds the matching draft by searching the table and interpolating between rows.
  • hydro_interpolate — given a draft, returns all hydrostatic properties (KB, BM, TPC, MTC, …) at that draft.

Both use a simple linear interpolation helper:

/* from src/hydrostatics.c */
static float lerp(float a, float b, float t) {
    return a + t * (b - a);
}

When a displacement falls between two table rows, t is the fractional position within that interval, and every field of the output HydroEntry is blended proportionally. This is exactly the same mathematics as reading between ruled lines on a graph.

KB — Where Is the Centre of Buoyancy?

The buoyant force does not act at the waterline; it acts at the centroid of the displaced volume, called the centre of buoyancy (B). Its height above the keel is written \(KB\).

For the box-hull approximation, \(KB\) is estimated as a fixed fraction of the draft:

r.kb = KB_FACTOR * r.draft;   /* KB_FACTOR = 0.53 */

For a uniform rectangular cross-section, the centroid of the submerged rectangle would be exactly at half the draft (\(KB = 0.5 \times T\)). Real hull forms are fuller near the keel and taper upward, so their centroid is slightly higher — the constant 0.53 is a well-established approximation for typical cargo ships.

When a hydrostatic table is available, KB is read directly from the interpolated row:

r.kb = he.kb;   /* from HydroEntry at interpolated draft */

BM — Metacentric Radius and the Source of Initial Stability

Archimedes' principle explains that a ship floats; it does not immediately explain why it rights itself when tilted. That second question is answered by the metacentre.

When a ship heels by a small angle, the waterplane (the cross-section at the waterline) rotates. One side dips deeper, the other rises. The displaced volume shifts to the lower side, moving the centre of buoyancy B sideways. This sideward shift generates a righting couple — as long as the new position of B is above G (the centre of gravity). The height of this "virtual pivot point" above B is called \(BM\), the metacentric radius:

\[BM = \frac{I_T}{V}\]

where \(I_T\) is the second moment of area of the waterplane about its centreline (a measure of how wide the waterplane is) and \(V\) is the displaced volume.

For the box-hull fallback, CargoForge-C computes this directly:

/* from src/analysis.c, box-hull branch */
float inertia_t = (ship->length * powf(ship->width, 3) / 12.0f) * WATERPLANE_COEFF;
r.bm = inertia_t / displaced_vol;

The \(L \times B^3 / 12\) factor is the second moment of area of a rectangle about its long centreline. WATERPLANE_COEFF = 0.85 reduces it to account for the real waterplane being slightly smaller than the bounding rectangle. When a hydrostatic table is used, BM comes directly from the measured he.bm.

Why width cubed?

Beam (width) appears cubed in \(I_T = L B^3 / 12\). This means doubling the beam multiplies BM by eight. Wider ships are dramatically more stable in initial heel — which is why loaded barges can look absurdly stable and narrow racing yachts are tender.

GM — Putting It All Together

With KB and BM known, the transverse metacentric height GM is:

\[GM = KB + BM - KG\]

In perform_analysis:

/* from src/analysis.c */
r.gm = r.kb + r.bm - r.kg;

A positive GM means the metacentre M is above the centre of gravity G — the ship is stable and will right itself. A negative GM means G is above M — the ship will capsize. The IMO sets a minimum of 0.15 m (the constant IMO_GM_MIN in src/analysis.c). CargoForge-C uses gm_corrected — GM after subtracting any free-surface correction — for all subsequent stability criteria.

Symbol Meaning Source (box-hull) Source (table)
\(KB\) Height of centre of buoyancy \(0.53 \times T\) Interpolated from CSV
\(BM\) Metacentric radius \(I_T / V\) Interpolated from CSV
\(KG\) Height of centre of gravity Moment sum / displacement Same
\(GM\) Metacentric height \(KB + BM - KG\) Same

The Hydrostatic Table in Practice

The CSV format expected by parse_hydro_table in src/hydrostatics.c requires at least seven comma-separated columns per row in ascending draft order:

# draft_m, displacement_t, km_m, kb_m, bm_m, tpc_t_per_cm, mtc_tm_per_cm
4.0,  8500,  6.80, 2.12, 4.68, 18.5, 142.3
4.5,  9800,  6.75, 2.38, 4.37, 19.1, 148.7
5.0, 11200,  6.71, 2.65, 4.06, 19.8, 155.2

parse_hydro_table enforces strictly ascending drafts — if a row's draft is not greater than the previous row's, parsing aborts with an error. This matters because hydro_draft_from_displacement relies on monotonically increasing displacement as well; non-monotonic tables would silently produce wrong interpolations.

Table boundaries

Both interpolation functions clamp to the nearest boundary row when the requested draft or displacement falls outside the table range. They do not extrapolate. If a fully-loaded ship's displacement exceeds the table's heaviest entry, CargoForge-C returns the hydrostatic values at maximum tabulated draft — a conservative approximation, but not a crash.

Recap

  • Archimedes' principle: a floating body displaces a volume of water whose weight exactly equals the body's own weight. A hollow steel ship floats because its average density (hull plus air) is less than seawater.
  • CargoForge-C uses SEAWATER_DENSITY = 1.025 t/m³ to convert displacement (tonnes) to displaced volume (m³): \(V = \Delta / 1.025\).
  • Draft is found from displaced volume either via the box-hull formula (using BLOCK_COEFF = 0.75) or by inverse interpolation in a hydrostatic CSV table loaded through src/hydrostatics.c.
  • KB (height of centre of buoyancy) and BM (metacentric radius, \(I_T / V\)) come from the same table row or from box-hull constants.
  • All three feed into the fundamental stability equation \(GM = KB + BM - KG\), computed in perform_analysis in src/analysis.c.
  • IMO requires \(GM \geq 0.15\ \text{m}\); CargoForge-C enforces this as part of its six-criterion intact stability check.

Check yourself

A solid steel cube sinks, but a steel ship floats. What's actually different?

Not the material — the shape. A solid cube displaces only its own small volume of water and loses the buoyancy tug-of-war. Spread that same steel into a hollow hull and it shoves aside far more water before it's even half submerged, generating enough upward force to balance its weight.

In one sentence, what determines the draft a ship actually settles at?

The depth at which the weight of the water it has displaced exactly equals the ship's own total weight — the point where the tug-of-war between gravity and buoyancy is a draw.

Next: Displacement, density, and draft.