Bramble–Hilbert lemma

Hudson's Equation, also known as Hudson's Formula, is an equation used by coastal engineers to calculate the minimum size of riprap (rock armour blocks) required to provide satisfactory stability characteristics for rubble structures such as breakwaters under attack from storm wave conditions.

The equation was developed by the United States Army Corps of Engineers, Waterways Experiment Station (WES), following extensive investigations by Hudson (1953, 1959, 1961a, 1961b) (see Shore Protection Manual and Rock Manual referenced below).

Initial Equation

The equation itself is:

$W={\frac {\gamma _{r}H^{3}}{K_{D}\Delta ^{3}\cot \theta }}$

where:

• W is the design weight of the riprap armour (Newton)
• $\gamma _{r}$ is the specific weight of the armour blocks (N/m³)
• H is the design wave height at the toe of the structure (m)
• K_D is a dimensionless stability coefficient, deduced from laboratory experiments for different kinds of armour blocks and for very small damage (a few blocks removed from the armour layer) (-):

• K_D = around 3 for natural quarry rock
• K_D = around 10 for artificial interlocking concrete blocks

• Δ is the dimensionless relative buoyant density of rock, i.e. (ρ_r / ρ_w - 1) = around 1.58 for granite in sea water
• ρ_r and ρ_w are the densities of rock and (sea)water (-)
• θ is the angle of revetment with the horizontal

Updated Equation

This equation was rewritten as follows in the nineties:

${\frac {H_{s}}{\Delta D_{n50}}}={\frac {(K_{D}cot\theta )^{1/3}}{1.27}}$

where:

• H_s is the design significant wave height at the toe of the structure (m)
• Δ is the dimensionless relative buoyant density of rock, i.e. (ρ_r / ρ_w - 1) = around 1.58 for granite in sea water
• ρ_r and ρ_w are the densities of rock and (sea)water (-)
• D_n50 is the nominal median diameter of armour blocks = (W₅₀/ρ_r)^1/3 (m)
• K_D is a dimensionless stability coefficient, deduced from laboratory experiments for different kinds of armour blocks and for very small damage (a few blocks removed from the armour layer) (-):

• K_D = around 3 for natural quarry rock
• K_D = around 10 for artificial interlocking concrete blocks

• θ is the angle of revetment with the horizontal

The armour blocks may be considered stable if the stability number N_s = H_s / Δ D_n50 < 1.5 to 2, with damage rapidly increasing for N_s > 3.

Obviously, these equations may be used for preliminary design, but scale model testing (2D in wave flume, and 3D in wave basin) is absolutely needed before construction is undertaken.

See also

• Breakwater (structure)
• Coastal erosion
• Coastal management
• Riprap

References

• US Army Corps of Engineers (1984). "Shore Protection Manual." Vol. II.
• Ciria-CUR (2007) - Rock Manual - The use of rock in hydraulic engineering.