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. This variational principle is for incompressible and inviscid potential flows, and is used to derive approximate wave models like the mild-slope equation, or using the averaged Lagrangian approach for wave propagation in inhomogeneous media. 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. 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.
Luke's Lagrangian Luke's Lagrangian formulation is for non-linear surface gravity waves on an—incompressible, irrotational and inviscid—potential flow. The relevant ingredients, needed in order to describe this flow, are:
Φ(x,z,t) is the velocity potential, ρ is the fluid density, g is the acceleration by the Earth's gravity, x is the horizontal coordinate vector with components x and y, x and y are the horizontal coordinates, z is the vertical coordinate, t is time, and ∇ is the horizontal gradient operator, so ∇Φ is the horizontal flow velocity consisting of ∂Φ/∂x and ∂Φ/∂y, V(t) is the time-dependent fluid domain with free surface. The Lagrangian L {\displaystyle {\mathcal {L}}} , as given by Luke, is:
L = − ∫ t 0 t 1 { ∭ V ( t ) ρ [ ∂ Φ ∂ t + 1 2 | ∇ Φ | 2 + 1 2 ( ∂ Φ ∂ z ) 2 + g z ] d x d y d z } d t . {\displaystyle {\mathcal {L}}=-\int _{t_{0}}^{t_{1}}\left\{\iiint _{V(t)}\rho \left[{\frac {\partial \Phi }{\partial t}}+{\frac {1}{2}}\left|{\boldsymbol {\nabla }}\Phi \right|^{2}+{\frac {1}{2}}\left({\frac {\partial \Phi }{\partial z}}\right)^{2}+g\,z\right]\,\mathrm {d} x\;\mathrm {d} y\;\mathrm {d} z\right\}\mathrm {d} t.}
From Bernoulli's principle, this Lagrangian can be seen to be the integral of the fluid pressure over the whole time-dependent fluid domain V(t). This is in agreement with the variational principles for inviscid flow without a free surface, found by Harry Bateman. Variation with respect to the velocity potential Φ(x,z,t) and free-moving surfaces like z = η(x,t) results in the Laplace equation for the potential in the fluid interior and all required boundary conditions: kinematic boundary conditions on all fluid boundaries and dynamic boundary conditions on free surfaces. This may also include moving wavemaker walls and ship motion. For the case of a horizontally unbounded domain with the free fluid surface at z = η(x,t) and a fixed bed at z = −h(x), Luke's variational principle results in the Lagrangian:
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