In fluid dynamics, Luke's variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J.C. Luke, who published it in 1967.[1] This variational principle is for incompressible and inviscid potential flows, and is used to derive approximate wave models like the mild-slope equation,[2] or using the averaged Lagrangian approach for wave propagation in inhomogeneous media.[3]
Luke's Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface.[4][5][6] This is often used when modelling the spectral density evolution of the free-surface in a sea state, sometimes called wave turbulence.
Both the Lagrangian and Hamiltonian formulations can be extended to include surface tension effects, and by using Clebsch potentials to include vorticity.[1]