Hudson's equation

Hudson's equation, also known as Hudson formula, is an equation used by coastal engineers to calculate the minimum size of riprap 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, following extensive investigations by Hudson

Initial equation

The equation itself is:
where:

W is the design weight of the riprap armor
is the specific weight of the armor blocks
H is the design wave height at the toe of the structure
K_D is a dimensionless stability coefficient, deduced from laboratory experiments for different kinds of armour blocks and for very small damage :S_r = = around 1.58 for granite in sea water
ρ_r and ρ_w are the densities of rock and water
θ is the angle of revetment with the horizontal

Updated equation

This equation was rewritten as follows in the nineties:
where:

H_s is the design significant wave height at the toe of the structure
Δ is the dimensionless relative buoyant density of rock, i.e. = around 1.58 for granite in sea water
ρ_r and ρ_w are the densities of rock and water
D_n50 is the nominal median diameter of armor blocks = ^1/3
K_D is a dimensionless stability coefficient, deduced from laboratory experiments for different kinds of armor blocks and for very small damage :
θ is the angle of revetment with the horizontal

The armourstone 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. This formula has been for many years the US standard for the design of rock structures under influence of wave action Obviously, these equations may be used for preliminary design, but scale model testing is absolutely needed before construction is undertaken.
The drawback of the Hudson formula is that it is only valid for relatively steep waves. Also it is not valid for breakwaters and shore protections with an impermeable core. It is not possible to estimate the degree of damage on a breakwater during a storm with this formula. Therefore nowadays for armourstone the Van der Meer formula or a variant of it is used. For concrete breakwater elements often a variant of the Hudson formula is used.