Breakwater Design Calculator
Size rubble-mound breakwater armor units using the Hudson formula. Computes armor weight, layer thickness, unit count, and stability number.
About this calculator
The Hudson formula, W = (ρ_a·g·H³) / (K_D·(S_r − 1)³·cot α), is the classic (if empirical) way to size the individual rock or concrete units armoring a rubble-mound breakwater's slope, and this calculator applies it directly: given the design wave height, slope angle, armor density, and the Hudson stability coefficient K_D for the chosen unit type, it solves for the minimum weight a single armor unit needs to resist being dislodged. From that weight it derives a nominal diameter (the equivalent cube side length), then uses standard rubble-mound relations to estimate armor layer thickness (number of layers times a packing coefficient times the nominal diameter) and the number of units needed per square meter of slope, accounting for the layer's porosity. It also reports the Hudson stability number Ns = H/(Dn·(Sr−1)) as a sanity check and a rule-of-thumb crest freeboard of 1.5 times the wave height.
The biggest source of error in practice is K_D itself: it varies enormously by armor type and wave condition — roughly 4 for rough angular rock, up to 8 for tetrapods, and 16 or more for interlocking units like Dolos — and mixing up breaking versus non-breaking wave K_D values, or the wrong armor-unit category, changes the required weight by a large factor since it's a cubed relationship with wave height. This formula also doesn't capture overtopping, toe scour, or underlayer/filter design, all of which need separate checks per the Shore Protection Manual or CIRIA Rock Manual.
Inputs
Results
Armor unit weight (N)
29,369.9
Armor unit mass (tonnes)
2.99
How to Use This Calculator
- Enter design significant wave height Hs (m) at the breakwater toe.
- Enter armor slope angle (degrees), Hudson stability coefficient K_D, armor unit density (kg/m3), and water density (kg/m3); optionally adjust number of armor layers, packing porosity, and layer coefficient k_Δ.
- Read required armor unit weight (N) and equivalent mass (tonnes) from the Hudson formula.
- Use armor unit weight, nominal diameter Dn, and stability number to select appropriate rock class or concrete armor unit type.
- Review armor layer thickness, units per square meter of slope, and recommended crest freeboard for the design wave height.
How the result changes with Water density (kg/m³)
| Water density (kg/m³) | Armor unit weight (N) | Armor unit mass (tonnes) |
|---|---|---|
| 920 | 17,600.1 | 1.79 |
| 970 | 22,525.7 | 2.3 |
| 1,030 | 30,078.6 | 3.07 |
| 1,080 | 38,094.5 | 3.88 |
What each input means
- Design wave height Hs (m)
- Significant design wave height at the breakwater toe.
- Armor slope angle (°)
- Angle of the armor slope from horizontal. 33.7° ≈ 1V:1.5H.
- Stability coefficient K_D
- Hudson stability coefficient. Rough angular rock ~4, Dolos ~16, Tetrapods ~8.
- Armor unit density (kg/m³)
- Mass density of the armor material. Granite ~2650, concrete ~2400.
- Water density (kg/m³)
- Seawater density, typically 1025 kg/m³.
- Number of armor layers
- Number of armor unit layers (typically 2).
- Packing porosity (%)
- Porosity of the armor layer (rock ~37%, Dolos ~56%).
- Layer coefficient k_Δ
- Layer thickness coefficient. Smooth rock ~1.02, rough rock ~1.00.
What each result means
- Armor unit weight (N)
- Minimum required weight of each armor unit from the Hudson formula.
- Armor unit mass (tonnes)
- Equivalent mass per armor unit.
- Nominal diameter Dn (m)
- Equivalent cube side length of one armor unit.
- Armor layer thickness (m)
- Total thickness of the armor layer (n layers).
- Units per m² of slope
- Estimated number of armor units per square meter of slope surface.
- Stability number Ns
- Hs / (Dn * (Sr-1)). Lower is more stable.
- Recommended crest freeboard (m)
- Rule-of-thumb crest height above SWL (1.5 * Hs).
How this is calculated
Worked example, using the default values
- Identify Input Parameters4 parametersDesign wave height Hs (m) = 3, Armor slope angle (°) = 33.7, Stability coefficient K_D = 4, Armor unit density (kg/m³) = 2650 = 8 input(s) provided
- Calculate Armor unit weightArmor unit weight = (rhoArmor * g * Hs * Hs * Hs) / (KD * pow(Sr - 1, 3) * cotAlpha)29369.9 = 29369.9
- Calculate Armor unit massArmor unit mass = massKg / 10002.994 = 2.994
- Calculate Nominal diameter DnNominal diameter Dn = pow(massKg / rhoArmor, 1 / 3)1.042 = 1.042
- Calculate Armor layer thicknessArmor layer thickness = numLayers * kDelta * Dn2.125 = 2.125
Engine last updated . Checked against 2 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Why does the required armor weight change so drastically with wave height?
The Hudson formula puts wave height in the numerator raised to the third power, W = (ρ_a·g·H³)/(K_D·(S_r−1)³·cot α), so doubling the design wave height increases the required armor weight by a factor of eight, not two. This cubic sensitivity is why getting the design wave height right — including whether it represents a significant, maximum, or extreme-return-period event — matters far more to the final armor size than most of the formula's other inputs.
Why does choosing the wrong stability coefficient K_D matter so much?
K_D sits in the denominator alongside the cubed specific-gravity term, and it varies enormously by armor unit type and wave-breaking condition — around 4 for rough angular rock but up to 16 or more for interlocking concrete units like Dolos. Since the calculator divides by K_D directly, picking a coefficient for the wrong unit type, or using a non-breaking-wave value where the site actually sees breaking waves, changes the computed required weight by several times without any change to the wave conditions themselves.
What does the Hudson stability number Ns tell me that the armor weight doesn't?
Ns = H/(Dn·(Sr−1)) restates the same physics in dimensionless form, relating wave height to the armor unit's nominal diameter and its density relative to water. It's useful as a cross-check against published design curves and case-history data that are often expressed in terms of Ns rather than raw weight, letting you sanity-check the calculated armor size against comparable built breakwaters independent of unit conversions.
What does this calculator not check that I still need to verify separately?
It only sizes the primary armor layer's individual unit weight, layer thickness, and unit count — it doesn't check wave overtopping over the crest, scour at the toe of the structure, or the design of the underlayer and filter layers beneath the armor that prevent the core material from washing out. All of these need separate verification per the Shore Protection Manual or CIRIA Rock Manual before a breakwater cross-section is considered complete.
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