Naval Architecture Coefficients Explained

Naval architecture coefficients are the compact, dimensionless descriptors that reduce a ship’s full three-dimensional hull geometry to a handful of ratios. The lines plan specifies every curve of the hull surface, but it’s the coefficients that let designers, classification societies, model basins, and shipyards communicate hull character in a common vocabulary. Two hulls with similar coefficient sets will have broadly similar resistance, stability, and seakeeping behaviour even if their detailed geometries differ.

This article covers the complete standard set: block coefficient $C_B$ (summarised here, with full treatment at the companion block coefficient article), midship section coefficient $C_M$, prismatic coefficient $C_P$ and its Froude-number optimum, waterplane area coefficient $C_{WP}$, vertical prismatic coefficient $C_{VP}$, the longitudinal positions of buoyancy and flotation ($LCB$ and $LCF$), the displacement-length and slenderness ratios, the Admiralty coefficient for power estimation, and the coefficient-driven logic of hull form design and resistance . The companion hydrostatics and Bonjean curves article explains how the hydrostatic table from which the coefficients are read is constructed and used.

Calculators for the principal coefficients are available in the ShipCalculators.com catalogue: the block coefficient calculator, the prismatic coefficient calculator, the waterplane area coefficient calculator, the Mumford wetted-surface estimator, the Froude number calculator, and the Admiralty coefficient power estimator.

Reference dimensions and notation

Every coefficient is defined at a specific draught. The standard reference draught is the design draught (also called the scantling draught in some conventions), the draught corresponding to the full-load displacement at which the vessel is designed to operate. Coefficients at a lighter draught are numerically different and must be recalculated from the hydrostatic table.

The reference dimensions follow IMO and ITTC convention (ITTC Dictionary 75-04-01-01):

• $L$: length between perpendiculars ($L_{BP}$), measured from the forward perpendicular (fore side of the stem at the design waterline) to the aft perpendicular (centre of the rudder stock). Some formulas specify the waterline length $L_{WL}$; for most merchant vessels $L_{BP}$ is 97 to 99% of $L_{WL}$, so the numerical difference in coefficient values is small but not negligible when comparing across sources.
• $B$: moulded breadth, the maximum horizontal width of the hull excluding shell plating thickness, at the design waterline.
• $T$: moulded draught, from the design waterline to the keel reference line.
• $\nabla$: moulded underwater volume in cubic metres at the design draught.
• $\Delta$: displacement in metric tonnes, equal to $\nabla \times \rho$, where $\rho$ is seawater density, taken as 1,025 kg/m³ for standard ocean salt water unless otherwise specified.
• $A_M$: area of the midship transverse section below the design waterline.
• $A_{WP}$: waterplane area at the design draught.

The four principal coefficients are linked by:

So any three of $\{C_B, C_M, C_P\}$ determine the fourth. This identity holds exactly by definition and is a strict algebraic consequence of the definitions below; it is not an approximation.

Block coefficient $C_B$

The block coefficient is the ratio of the underwater volume to the enclosing rectangular block:

A value of 1.0 would describe a perfect rectangular barge. Real displacement ships range from about 0.45 for fast naval combatants to 0.88 for laden VLCCs. The block coefficient is the single most concise descriptor of hull fullness and propagates through resistance prediction, freeboard assignment under ICLL 1966, wake fraction estimation, structural weight scaling, and the attained EEDI and EEXI compliance calculations.

The dedicated block coefficient article covers the ICLL freeboard correction, EEDI/EEXI usage, the Telfer-Alexander CB-Froude correlation, and the precise length and draught conventions in detail. The block coefficient calculator and the struct block coefficient calculator handle the two standard contexts (stability/design and structural rules).

Midship section coefficient $C_M$

The midship section coefficient is the ratio of the midship transverse section area to the bounding rectangle of breadth and draught:

$C_M$ characterises how rectangular the midship cross-section is. A value of 1.0 means the midship section is a perfect rectangle, filling beam times draught completely. Real hulls have some rounding at the bilge, some deadrise in the bottom, or some tumblehome in the sides, so $C_M$ is always strictly less than 1.0.

For large full-form merchant ships, the midship section is nearly rectangular and $C_M$ sits between 0.98 and 0.995. The section shape is driven by cargo stowage (flat bottom for maximising hold volume), structural considerations (double-bottom tank arrangement), and the need for high deadweight. Naval architects designing tankers and bulk carriers rarely choose $C_M$ freely; the structural configuration of the cross-section largely dictates it.

Fine-form hulls carry more deadrise and bilge curvature. Fast naval frigates designed for sea-keeping in high sea states have $C_M$ in the range 0.75 to 0.88. The rounded section reduces slamming acceleration and allows the hull to move cleanly through waves. The tradeoff is reduced cargo volume for a given beam and draught.

Typical $C_M$ values by ship type:

The Holtrop-Mennen resistance regression (1982, International Shipbuilding Progress) includes $\sqrt{C_M}$ in the wetted-surface formula and $C_M$ in the bow pressure resistance coefficient, so even this apparently simple coefficient has a measurable effect on the resistance prediction output. The resistance-components-deep-dive article explains the Holtrop-Mennen method in full.

Prismatic coefficient $C_P$

The prismatic coefficient is the ratio of the underwater volume to the volume of a prism whose cross-section equals the midship area and whose length equals the ship length:

$C_P$ measures how the underwater volume is distributed along the ship’s length, independently of how full the midship section is. A high $C_P$ means the bow and stern sections are nearly as full as the midship section; the volume is spread evenly. A low $C_P$ means the volume is concentrated amidships; the ends are fine and the hull appears to taper sharply toward bow and stern.

The distinction from $C_B$ is worth being precise about. Two hulls can have the same $C_B$ but different $C_M$ and therefore different $C_P$. A hull with $C_B = 0.70$, $C_M = 0.98$, and $C_P = 0.714$ has more volume concentrated amidships than a hull with $C_B = 0.70$, $C_M = 0.92$, and $C_P = 0.761$. The second hull’s bow and stern carry more volume. The resistance behaviour of these two hulls differs substantially even though their block coefficients are identical.

$C_P$ and the Froude number

The optimum $C_P$ for minimum wave-making resistance depends on the Froude number. This relationship was established experimentally through the Series 60 systematic model tests (Todd, SNAME Transactions, 1963) and the Lap-Keller series, and it remains foundational to early-stage design. The physics is straightforward: at low Froude numbers, the dominant bow and stern wave systems have long wavelengths relative to the ship length, so a fuller end form (high $C_P$) generates less interference between bow and stern wave systems. At high Froude numbers the wave systems interact more strongly, and finer ends (lower $C_P$) reduce the wave-making penalty.

The empirical optimum (from Series 60 and confirmed by later model series):

A mismatch of 0.05 in $C_P$ from the optimum at $F_n = 0.25$ increases residuary resistance by roughly 10 to 15% for the wave-making component. Because frictional resistance typically accounts for 70 to 80% of total resistance at those Froude numbers for a full-form merchant hull, the overall resistance penalty from a $C_P$ mismatch is smaller, perhaps 3 to 8%, but it is measurable and significant over a vessel’s 25-year service life.

Use the prismatic coefficient calculator to compute $C_P$ from known $\nabla$, $A_M$, and $L$ values.

Typical $C_P$ values:

Longitudinal position of LCB and the $C_P$ optimum

The prismatic coefficient and the longitudinal centre of buoyancy (LCB) are inseparable in design. Shifting volume toward the bow raises LCB forward; shifting it aft lowers LCB aft. The Series 60 data show that the optimum LCB position for minimum resistance moves forward as $F_n$ increases. For VLCCs operating at $F_n \approx 0.16$, the optimum LCB is approximately 2 to 3% of $L_{BP}$ aft of midships. For container ships at $F_n \approx 0.25$, the optimum LCB is closer to 1 to 2% aft of midships, and for high-speed ferries at $F_n \approx 0.32$ it may be close to midships or slightly forward.

Selecting $C_P$ sets the envelope of achievable LCB positions: a hull with high $C_P$ can distribute volume almost anywhere along the length, while a hull with low $C_P$ has its volume so concentrated amidships that the naval architect has less freedom to shift LCB longitudinally.

Waterplane area coefficient $C_{WP}$

The waterplane area coefficient is the ratio of the actual waterplane area to the bounding rectangle of length and breadth:

$C_{WP}$ measures how much of the waterline rectangle is occupied by the actual waterplane. A value of 1.0 would mean the waterplane is a perfect rectangle with no tapering at bow or stern. Real hulls taper to a point at the bow (in ship-shaped forms) or to a relatively blunt transom stern (in modern merchant ships), so $C_{WP}$ is always below 1.0.

The waterplane coefficient is critical for two reasons.

First, it drives the transverse metacentric radius $BM$:

where $I_T$ is the second moment of area of the waterplane about its centreline. $I_T$ is proportional to $C_{WP}$ and to $L \cdot B^3$, so a higher $C_{WP}$ gives greater $BM$ and thus higher metacentric height $GM$, improving initial transverse stability. This is the direct channel from waterplane shape to intact stability, covered in the metacentric height and intact stability articles.

Second, the waterplane coefficient governs the rate at which displacement changes with draught (the tonnes-per-centimetre immersion, TPC):

This is directly relevant to draught survey calculations and trim computations described in the trim and list article.

Use the waterplane area coefficient calculator to compute $C_{WP}$ from Simpson’s rule integration of the waterplane half-breadths.

There is a rough empirical correlation between $C_{WP}$ and $C_B$, reflecting that fuller hulls tend to have fuller waterplanes:

This is Munro-Smith’s approximation, widely quoted in early-stage design texts (Schneekluth and Bertram, Ship Design for Efficiency and Economy, 2nd ed., 1998). It is valid to within about 0.03 for typical merchant hull forms. Fine-form vessels with high flare deviate more substantially.

Typical $C_{WP}$ values:

Vertical prismatic coefficient $C_{VP}$

The vertical prismatic coefficient is the ratio of the underwater volume to the product of waterplane area and draught:

It can also be written as:

$C_{VP}$ measures how the underwater volume is distributed vertically. A value of 1.0 would mean the hull is a perfect prism with the waterplane cross-section constant from keel to waterline. Real hulls taper vertically toward the keel, so $C_{VP}$ is less than 1.0. A high $C_{VP}$ means the hull retains much of its waterplane area down to the keel; a low $C_{VP}$ means the underwater body tapers significantly as you move deeper.

The practical importance of $C_{VP}$ lies in stability. It determines how the centre of buoyancy $B$ moves vertically as draught changes. A hull with high $C_{VP}$ has a centre of buoyancy close to $T/2$; a hull with low $C_{VP}$ has its centre of buoyancy at a relatively higher fraction of draught because the volume is concentrated in the upper part of the underwater body. The vertical rise of $B$ per unit draught increase directly affects the metacentric radius $BM$ and therefore $GM$.

In the ITTC terminology (Dictionary 75-04-01-01), $C_{VP}$ is defined exactly as above. Some older texts use the symbol $C_V$ for the volumetric coefficient $\nabla / L^3$ instead, so the context matters when reading literature. This article follows the ITTC convention throughout.

Typical $C_{VP}$ values for merchant ships fall between 0.72 and 0.82. Fine-form naval and fast passenger vessels may reach 0.65 to 0.72. VLCCs and bulkers typically sit near 0.78 to 0.82.

Longitudinal centre of buoyancy ($LCB$) and longitudinal centre of flotation ($LCF$)

$LCB$ and $LCF$ are not dimensionless ratios but positional parameters, typically expressed as a percentage of $L_{BP}$ from midships (positive forward, negative aft) or from the aft perpendicular.

Longitudinal centre of buoyancy $LCB$

$LCB$ is the centroid of the underwater volume. For a vessel floating freely at rest with no external moment, $LCB$ must be vertically above $LCG$ (the longitudinal centre of gravity). Any longitudinal separation between $LCB$ and $LCG$ produces a trimming moment and a trim angle, which the hull accommodates until the new underwater volume’s centroid realigns with $LCG$. This is the physical basis for trim and list calculations.

$LCB$ expressed as a percentage of $L_{BP}$ forward of midships typically ranges from +1% to +3% aft of midships (negative values) for full-form tankers and bulk carriers. Container ships’ optimum $LCB$ sits around 1% aft of midships at their design Froude number of 0.23 to 0.28. This optimal position comes from Series 60 resistance data, which showed minimum wave-making resistance when LCB is positioned within a Froude-number-dependent band.

In practice, the naval architect adjusts the hull form iteratively in parametric CAD tools (NAPA, Maxsurf) until the designed $LCB$ matches both the required resistance optimum and the loading condition’s expected $LCG$.

Longitudinal centre of flotation $LCF$

$LCF$ is the centroid of the waterplane area. It is the pivot point about which the vessel trims when mass is added or removed without changing displacement. When a mass is placed directly above $LCF$, the vessel sinks bodily without trimming; when placed forward or aft of $LCF$, the vessel trims toward that end.

$LCF$ is always used in trim calculations and draught surveys. The moment to change trim 1 cm (MCT1cm or MCTC) is:

where $GML$ is the longitudinal metacentric height, itself derived from the longitudinal second moment of the waterplane about $LCF$.

For full-form vessels, $LCF$ is typically 1 to 3% of $L_{BP}$ aft of midships. It shifts forward at lighter draughts as the waterplane shape changes.

Displacement-length ratio and slenderness ratio

Displacement-length ratio

The displacement-length ratio ($\Delta_L$) has two common forms, depending on the unit tradition.

The US naval architecture convention (from SNAME PNA) uses long tons and feet:

where $\Delta_{LT}$ is displacement in long tons and $L_{ft}$ is $L_{BP}$ in feet. This ratio is dimensionless in the sense that it scales by $(L/100)^3$, and typical values for displacement vessels fall in the range 50 to 500.

The equivalent dimensionless volumetric coefficient $C_\nabla$ uses SI units directly:

For merchant ships, $C_\nabla$ typically ranges from about 0.002 for very fine fast craft to 0.012 for very full slow ships. The two representations carry the same information; the US displacement-length ratio is simply a scaled version of $C_\nabla$.

High displacement-length ratio means the vessel is heavy relative to its length, a characteristic of full-form, slow ships optimised for cargo capacity. Low displacement-length ratio means the vessel is slender relative to its displacement, a characteristic of fast ships where wave-making resistance rises steeply with speed.

Slenderness ratio (length-displacement ratio)

The slenderness ratio $L/\nabla^{1/3}$ inverts this logic and measures how long the vessel is relative to its linear dimension of displacement:

Typical values: 4 to 5 for full-form tankers and bulk carriers; 6 to 7 for container ships; 7 to 9 for naval frigates and fast passenger ferries. Higher slenderness ratios generally correlate with lower wave-making resistance per unit displacement at high Froude numbers, which is why destroyers and frigates are long and slender relative to their displacement.

Hull dimension ratios: $L/B$, $B/T$, $L/D$

These ratios are not form coefficients in the strict sense but are integral to the dimensional context in which the coefficients are interpreted. The ITTC Dictionary and SNAME PNA treat them as primary hull descriptors alongside the non-dimensional coefficients.

Length-to-beam ratio $L/B$

$L/B$ characterises longitudinal slenderness. A higher ratio means a narrower hull relative to its length. Canal constraints impose hard limits: the Panama Canal New Locks limit beam to 49 m, and the Suez Canal limits beam to about 77.5 m for safe transit.

The triple-E class container ships (Maersk, 2013 onwards, 399 m x 59 m) have $L/B \approx 6.8$, a deliberately broad beam to maximise TEU capacity while keeping Froude number low at slow-steaming speeds.

Beam-to-draught ratio $B/T$

$B/T$ drives transverse stability through its effect on the waterplane second moment of inertia. A higher $B/T$ gives greater waterplane inertia and thus higher $BM$ and $GM$. For vessels with high centres of gravity (container ships with high stacks, cruise ships with tall superstructures, LNG carriers with large tanks above the waterline), $B/T$ is set partly by the need for adequate $GM$.

LNG carriers have anomalously high $B/T$ relative to their displacement because the membrane tank arrangement dictates a wide, relatively shallow hull with a high centre of gravity. The LNG carrier article discusses the structural and stability implications of this geometry.

Length-to-depth ratio $L/D$

$L/D$ is the principal structural parameter. It drives the sagging and hogging bending moments in the hull girder: the primary bending stress at midship section is roughly proportional to $(L/D)^2$. The IACS Common Structural Rules (CSR BC&OT, 2024 edition) set minimum section modulus requirements that depend explicitly on $L/D$, among other parameters.

Typical values:

Ships with $L/D > 14$ face progressively severe bending stress at the midship section and require extensive structural reinforcement in the upper deck and bottom shell plating. The rule limits in IACS CSR cap $L/D$ implicitly through minimum section modulus requirements.

Coefficient identity and mutual constraints

The fundamental identity is:

Three corollaries follow directly:

1. For a merchant ship with $C_M = 0.99$, knowing $C_B = 0.82$ fixes $C_P = C_B / C_M = 0.828$. The prismatic coefficient can’t be independently chosen once $C_B$ and $C_M$ are set.
2. A designer who wants a specific $C_P$ for resistance optimisation, given a fixed $C_B$ (from cargo capacity requirements), must accept the $C_M$ that results: $C_M = C_B / C_P$. For a container ship with $C_B = 0.67$ and a target $C_P = 0.70$, this gives $C_M = 0.957$.
3. The waterplane coefficient $C_{WP}$ is not algebraically constrained by this identity. It depends on the longitudinal distribution of waterplane half-breadths and is only empirically correlated with $C_B$ and $C_P$.

A second useful relation connects the vertical prismatic coefficient to the other coefficients:

This is exact by definition. For a VLCC with $C_B = 0.84$ and $C_{WP} = 0.90$, $C_{VP} = 0.933$: nearly the full waterplane area is maintained from keel to waterline, consistent with the nearly rectangular box shape of a laden crude tanker.

Admiralty coefficient for power estimation

The Admiralty coefficient $C_A$ (sometimes written $AC$ in older texts) is an empirical power-estimation parameter defined as:

where $\Delta$ is displacement in tonnes, $V$ is speed in knots, and $P_D$ is delivered shaft power in kW (some older references use shaft horsepower, which produces a different numerical range for $C_A$). The formula rearranges to:

The Admiralty coefficient works because both $\Delta^{2/3}$ (a linear scale related to wetted surface area) and $V^3$ (from the Froude scaling of wave-making resistance) track the dominant resistance components across geometrically similar hulls. For a given hull form and hull condition, $C_A$ is approximately constant across a moderate speed range.

The method has two practical applications:

1. Sister-ship scaling. If a vessel’s sea-trial power is known at a given displacement and speed, $C_A$ is computed and then used to estimate power at a different displacement (different loading condition) or different speed.
2. Early-design sizing. Typical $C_A$ values by ship type allow quick power estimates before a detailed resistance analysis is available.

Typical $C_A$ values (kW basis, displacement in metric tonnes, speed in knots):

The Admiralty coefficient power estimator and the Admiralty constant delivered-power calculator implement both the direct $P_D$ calculation and the sister-ship scaling variant.

The Admiralty coefficient carries significant assumptions: it treats hull condition (fouling, surface roughness), propulsive efficiency, and loading condition as constant. Variation in any of these shifts $C_A$ by 5 to 20%. It is a screening tool, not a substitute for a full Holtrop-Mennen or CFD resistance analysis at detailed design stage.

Coefficient ranges by ship type: comparative table

The table below consolidates the standard coefficient ranges for the principal commercial and naval vessel types. Ranges reflect typical newbuild designs; individual vessels may fall outside these bounds for specific design trade-offs.

Sources for the ranges: SNAME PNA Vol. II (1988); Schneekluth & Bertram, Ship Design for Efficiency and Economy (1998); IACS CSR BC&OT (2024); DNV Rules for Classification of Ships (2024 edition).

How the coefficients drive hull form design

The design sequence

A typical early-stage hull form design sequence (following SNAME PNA Vol. II, Chapter 1) proceeds in roughly this order:

The design speed and payload requirement set the displacement $\Delta$ and the required service speed $V_s$. From $V_s$ and the estimated $L_{BP}$, the designer computes the design Froude number $F_n = V_s / \sqrt{g L_{BP}}$. The optimum $C_P$ and $C_B$ are then read from the Froude-number correlations. The canal and port constraints bound $B$ and $T$, which, combined with $\Delta$ and $C_B$, determine $L_{BP}$ iteratively. The stability target (minimum $GM$) constrains $C_{WP}$ and $B/T$ through the metacentric radius calculation. The structural target constrains $L/D$.

The design process is iterative because the parameters are not independent. Lengthening the vessel to achieve a lower $F_n$ and higher $C_B$ may violate port entry constraints or raise structural weight. Widening the beam to improve stability may push $B/T$ to values that create excessive slamming loads in head seas. The coefficients are the handles the designer adjusts while checking all constraints simultaneously. Modern tools such as NAPA Design or Maxsurf Modeller embed this iteration in a parametric framework.

$C_B$ vs Froude number: the speed-form tradeoff

The relationship between $C_B$ and $F_n$ is the central tradeoff in hull form design. Wave-making resistance is approximately proportional to $C_B^2$ at a given Froude number, so doubling $C_B$ from 0.40 to 0.80 increases wave-making resistance by a factor of four for the same $F_n$. At low Froude numbers, wave-making resistance is a small fraction of total resistance (perhaps 5 to 15% for a VLCC), so the penalty is manageable and the capacity advantage of high $C_B$ dominates. At high Froude numbers ($F_n > 0.30$), wave-making resistance can exceed 40% of total resistance, and the penalty from excess $C_B$ becomes the dominant cost driver over the vessel’s life.

The resistance components deep dive article quantifies the individual resistance components and the tradeoffs in detail.

During the slow-steaming era (2008 to 2019), many container lines reduced service speed from 24 to 26 knots ($F_n \approx 0.27$ to 0.28) to 18 to 21 knots ($F_n \approx 0.20$ to 0.23). At those lower Froude numbers, the optimum $C_B$ for their hulls moved from about 0.64 to 0.68 upward toward 0.72 to 0.76. Hulls designed for the higher speed operated at sub-optimal $C_P$ during slow steaming, which contributed measurable excess resistance compared to what a purpose-designed slow-speed hull would achieve. This is why newer container ships ordered from 2015 onward show somewhat fuller hull forms than the pre-2008 designs.

Wetted surface and frictional resistance

The frictional resistance component is:

where $S$ is the wetted surface area and $C_F$ is the friction coefficient from the ITTC 1957 line:

The Mumford formula gives a quick estimate of $S$ from the principal dimensions and $C_B$:

A higher $C_B$ increases $S$ for the same $L$, $B$, $T$. The Holtrop-Mennen formula provides a more precise estimate that incorporates $C_M$, $C_{WP}$, and the bulbous bow area:

where $A_{BT}$ is the transverse area of the bulbous bow, set to zero if absent. The Mumford wetted-surface estimator implements the simpler formula, while the full Holtrop-Mennen method is described in the resistance components deep dive and the ship resistance and powering articles.

The wetted surface area article covers both methods and the direct integration approach from offset tables or a 3D hull model.

Coefficients in IACS structural rules

The IACS CSR BC&OT (2024 edition) use $C_B$, $C_M$, and the dimension ratios $L/B$, $L/D$, and $B/T$ explicitly in setting the design wave loads and minimum section modulus requirements. The wave bending moment $M_W$ in the CSR is proportional to $C_B$ (larger $C_B$ means more buoyancy variation between sagging and hogging positions) and to $L^2 B$. The shear force distribution along the hull similarly depends on the $C_B$ distribution. These rules are the formal channel through which the form coefficients constrain structural design rather than merely characterising it.

Coefficients in EEDI and EEXI

IMO’s Energy Efficiency Design Index (EEDI, MEPC.212(63), in force January 2013) and the Energy Efficiency Existing Ship Index (EEXI, MEPC.328(76), in force November 2022) both incorporate the Admiralty coefficient implicitly: the reference speed $V_{ref}$ at which EEDI is computed depends on the hull resistance at 75% MCR, which the Holtrop-Mennen method estimates from $C_B$, $C_P$, $C_M$, and the dimension ratios. Vessels with higher $C_B$ carry more deadweight per unit length but also generate more wave-making resistance at a given speed, so the EEDI penalises high-speed, fine-form vessels (which generate proportionally more wave resistance for their capacity) while benefiting slow, full-form bulk carriers and tankers. The IMO energy efficiency regulations page gives the current reduction factors by phase and ship type.

Limitations of coefficient-based analysis

Form coefficients are compact descriptors, but their compactness is also their limitation. Several important caveats apply.

Coefficients do not uniquely define a hull. Two hulls with identical $C_B$, $C_M$, $C_P$, and $C_{WP}$ can have significantly different resistance, seakeeping, or stability behaviour if their detailed section shapes, bow geometry, stern geometry, or waterplane distribution differ. Coefficients constrain the design space; they don’t fully specify it. CFD analysis is required to evaluate detailed form effects, especially bow wave interaction, stern flow separation, and propeller-hull interaction.

Froude-number correlations are empirical and were derived from specific model series. The optimum $C_P$ correlations from Series 60 (Todd, 1963) are valid for monohull displacement vessels in the $F_n$ range 0.14 to 0.40 and $C_B$ range 0.60 to 0.80. Extrapolation outside these ranges, or application to multihulls, SWATH vessels, or semi-planing craft, requires independent model testing or CFD.

The Admiralty coefficient is a screening method only. It assumes geometric similarity (the same form coefficients and proportional dimensions). Applied to hulls of different form, $C_A$ can vary by 20 to 40%. It does not account for propulsive efficiency differences, fouling, shaft arrangement, or auxiliary loads.

Coefficients are defined at a single draught. A vessel’s $C_B$ at its summer load line draught differs from its $C_B$ in ballast by 3 to 8% for a typical bulker or tanker. Resistance and stability calculations at a different draught require the hydrostatic table, not just the design-draught coefficients. The hydrostatics and Bonjean curves article explains how the draught-dependence of all the coefficients is tabulated and used.

Empirical correlations for $C_{WP}$ and $LCB$ have scatter. The Munro-Smith $C_{WP}$ approximation carries a standard deviation of about 0.025 around the regression mean. Using it in stability calculations without cross-checking against the actual waterplane integration from the lines plan can produce $GM$ errors of 0.2 to 0.5 m for large vessels, which is significant relative to typical intact stability requirements.

ITTC and SNAME conventions differ in length and displacement units. The displacement-length ratio from SNAME PNA uses long tons and feet; the ITTC slenderness ratio uses metric tonnes and metres. Mixing conventions without conversion produces errors of up to 3% in the computed ratio. Always state the unit convention when quoting a displacement-length ratio.

Related Articles

• Block coefficient
• Hull form design
• Hydrostatics and Bonjean curves
• Resistance components deep dive
• Ship resistance and powering
• Wetted surface area
• Metacentric height
• Intact stability
• GZ curve and righting arm
• Trim and list
• Trim optimisation
• Freeboard and reserve buoyancy
• Damage stability
• Ship motions
• Marine propeller
• Bulbous bow retrofits
• Bulk carrier
• Container ship
• LNG carrier
• Oil tanker
• Classification society