# Hudson's equation

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)
• KD 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) (-):
• KD = around 3 for natural quarry rock
• KD = around 10 for artificial interlocking concrete blocks
• Δ is the dimensionless relative buoyant density of rock, i.e. r / ρw - 1) = around 1.58 for granite in sea water
• ρr and ρw 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:

• Hs is the design significant wave height at the toe of the structure (m)
• Δ is the dimensionless relative buoyant density of rock, i.e. r / ρw - 1) = around 1.58 for granite in sea water
• ρr and ρw are the densities of rock and (sea)water (-)
• Dn50 is the nominal median diameter of armour blocks = (W50r)1/3 (m)
• KD 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) (-):
• KD = around 3 for natural quarry rock
• KD = 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 Ns = Hs / Δ Dn50 < 1.5 to 2, with damage rapidly increasing for Ns > 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.