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Wave set-up and set-down When waves break on a beach, they produce a set-up, a rise in the mean water level above the still-water elevation of the sea.Komarestimate both the set-down and the set-up, we need to look at an energy balance.Longuet-Higgins and Stewart did this in their original derivation of radiation-stress theory.In the cross-shore direction, the conservation of momentum becomes – ∂S ρg η h ∂x where S S dh ∂ ∂ ∂η xx g h gh − ρ η − ρ x x dx ∂x ∂ ∂ Substituting into ∂S ∂η xx − gh ρ ∂x ∂x Integrating both sides yields kH η − sinh kh Longuet-Higgins and Stewart assumed that inside the surf-zone, we have bores corresponding to the relation – H γ η h But we know that for shallow water waves, the radiation stress
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