About Breakwater Armor Unit Calculator (Hudson Formula)
The breakwater armor unit calculator sizes the primary armor layer of a rubble-mound breakwater or revetment using the classic Hudson formula. Given a design wave height, the specific weight of the armor unit, the stability coefficient Kd for the chosen armor type and the structure slope, it returns the minimum stable armor unit weight together with an equivalent cube size and placement guidance.
Enter the design wave height H, the armor unit specific weight gamma_r, the water specific weight gamma_w, the stability coefficient Kd, and the slope cot(theta) (the horizontal run per unit rise). The tool computes the relative density Sr = gamma_r / gamma_w, evaluates W = gamma_r H^3 / (Kd (Sr - 1)^3 cot(theta)), and reports the nominal diameter, layer thickness and the number of units per 100 m^2 of slope.
How It Works
- Choose the armor type to pick a representative stability coefficient Kd, or enter Kd directly.
- Enter the design wave height H (m), the armor unit specific weight gamma_r (kN/m^3) and the slope cot(theta).
- The relative density is Sr = gamma_r / gamma_w, using a seawater specific weight of 10.05 kN/m^3 by default.
- The Hudson formula gives the required unit weight W = gamma_r H^3 / (Kd (Sr - 1)^3 cot(theta)).
- The equivalent cube edge / nominal diameter is Dn = (W / gamma_r)^(1/3); the layer thickness is r = n k_delta Dn and the placement density gives the number of units per area.
Worked Example
A rubble-mound breakwater is armored with rough angular quarrystone (gamma_r = 25.5 kN/m^3) on a 1:2 slope, so cot(theta) = 2. The design (breaking) wave height is H = 3 m and the recommended stability coefficient is Kd = 4. The seawater specific weight is gamma_w = 10.05 kN/m^3, so the relative density is Sr = 25.5 / 10.05 = 2.537 and (Sr - 1)^3 = 1.537^3 = 3.633. The Hudson formula gives W = 25.5 x 3^3 / (4 x 3.633 x 2) = 688.5 / 29.07 = 23.7 kN, i.e. about 2.41 tonnes per stone. The equivalent cube edge is Dn = (23.7 / 25.5)^(1/3) = 0.976 m, the two-layer thickness is r = 2 x 1.0 x 0.976 = 1.95 m, and roughly 132 stones are needed per 100 m^2 of slope.
Formulas
- Relative density
Sr = gamma_r / gamma_w- Hudson formula - armor unit weight
W = gamma_r * H^3 / (Kd * (Sr - 1)^3 * cot(theta))- Equivalent cube / nominal diameter
Dn = (W / gamma_r)^(1/3)- Armor layer thickness
r = n * k_delta * Dn- Number of units per area
N/A = n * k_delta * (1 - P/100) * (gamma_r / W)^(2/3)
Standards & References
- Hudson formula (Hudson, 1959)
- Shore Protection Manual (SPM, USACE 1984)
- Coastal Engineering Manual (CEM, EM 1110-2-1100)
- CIRIA/CUR Rock Manual
Frequently Asked Questions
What is the stability coefficient Kd in the Hudson formula?
Kd is an empirical coefficient that captures how well an armor unit interlocks and resists wave attack. It depends on the unit shape, placement, number of layers and whether the design wave breaks on the structure. Rough angular quarrystone has Kd around 2 to 4, while interlocking concrete units such as tetrapods, tribars and dolosse have much higher Kd and therefore need far lighter units.
Why does armor weight grow with the cube of wave height?
The Hudson formula has W proportional to H^3, so doubling the design wave height increases the required armor unit weight eightfold. This strong sensitivity is why selecting a realistic design wave height, and a denser or more interlocking armor unit, has such a large effect on the size of the armor stones or concrete units.
Should I use the Hudson formula or the Van der Meer formulae?
Hudson is a simple, widely taught method that is well suited to preliminary sizing and checks. Modern coastal-engineering practice (SPM/CEM and the Rock Manual) recommends the Van der Meer formulae for detailed design because they include wave period, storm duration, permeability and a defined damage level. Use this calculator for first-pass sizing and verify with Van der Meer for final design.
What relative density and water unit weight should I use?
Use the actual specific weight of the armor material (about 25.5 kN/m^3 for typical rock and 23 to 24 kN/m^3 for concrete) and the specific weight of the water the structure sits in. Seawater is about 10.05 kN/m^3 (1025 kg/m^3) and fresh water about 9.81 kN/m^3, giving a relative density Sr that must exceed 1 for the formula to be valid.