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| '''Hudson's Equation''', also known as '''Hudson's Formula''', is an [[equation]] used by [[coastal management|coastal engineers]] to calculate the minimum size of [[riprap]] (rock armour blocks) required to provide ''satisfactory'' stability characteristics for [[rubble]] structures such as [[Breakwater_(structure)|breakwaters]] under attack from storm [[wave]] conditions.
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| 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).
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| ==Initial Equation==
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| The equation itself is:
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| :<math>W =\frac{\gamma_r H^3}{K_D \Delta^3\cot\theta}</math>
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| where:
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| *''W'' is the design weight of the riprap armour (Newton)
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| *''<math>\gamma_r</math>'' is the [[specific weight]] of the armour blocks (N/m<sup>3</sup>)
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| *''H'' is the design wave height at the toe of the structure (m)
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| *''K''<sub>''D''</sub> 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) (-):
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| :* ''K''<sub>''D''</sub> = around 3 for natural quarry rock
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| :* ''K''<sub>''D''</sub> = around 10 for artificial interlocking concrete blocks
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| *''Δ'' is the dimensionless relative buoyant density of rock, i.e. ''(ρ<sub>r</sub> / ρ<sub>w</sub> - 1)'' = around 1.58 for granite in sea water
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| *''ρ''<sub>''r''</sub> and ''ρ''<sub>''w''</sub> are the [[density|densities]] of rock and (sea)water (-)
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| *''θ'' is the angle of revetment with the horizontal
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| ==Updated Equation==
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| This equation was rewritten as follows in the nineties:
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| :<math>\frac{H_s}{\Delta D_{n50}}= \frac{(K_D cot\theta)^{1/3}}{1.27}</math>
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| where: | |
| *''H''<sub>''s''</sub> is the design significant wave height at the toe of the structure (m)
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| *''Δ'' is the dimensionless relative buoyant density of rock, i.e. ''(ρ<sub>r</sub> / ρ<sub>w</sub> - 1)'' = around 1.58 for granite in sea water
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| *''ρ''<sub>''r''</sub> and ''ρ''<sub>''w''</sub> are the [[density|densities]] of rock and (sea)water (-)
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| *''D''<sub>''n50''</sub> is the nominal median diameter of armour blocks = ''(W<sub>50</sub>/ρ<sub>r</sub>)<sup>1/3</sup>'' (m)
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| *''K''<sub>''D''</sub> 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) (-):
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| :* ''K''<sub>''D''</sub> = around 3 for natural quarry rock | |
| :* ''K''<sub>''D''</sub> = around 10 for artificial interlocking concrete blocks
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| *''θ'' is the angle of revetment with the horizontal
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| The armour blocks may be considered stable if the ''stability number'' ''N<sub>s</sub> = H<sub>s</sub> / Δ D<sub>n50</sub>'' < 1.5 to 2, with damage rapidly increasing for N<sub>s</sub> > 3.
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| 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.
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| ==See also==
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| * [[Breakwater (structure)]]
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| * [[Coastal erosion]]
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| * [[Coastal management]]
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| * [[Riprap]]
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| ==References==
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| * US Army Corps of Engineers (1984). [http://books.google.com/books?id=Nf5RAAAAMAAJ&q=shore+protection+manual&dq=shore+protection+manual&lr=&as_brr=0&pgis=1 "Shore Protection Manual."] Vol. II.
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| * Ciria-CUR (2007) - [http://www.ciria.org/service/Web_Site/AM/ContentManagerNet/ContentDisplay.aspx?Section=Web_Site&ContentID=9003 Rock Manual - The use of rock in hydraulic engineering].
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| {{coastal management}}
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| [[Category:Equations]]
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| [[Category:Coastal engineering]]
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| [[Category:Coastal erosion]]
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