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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 so-called 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.〔 Originally appeared in ''Zhurnal Prildadnoi Mekhaniki i Tekhnicheskoi Fiziki'' 9(2): 86–94, 1968.〕〔 〕〔 〕 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. ==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 , as given by Luke, is: : 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: : The bed-level term proportional to ''h''2 in the potential energy has been neglected, since it is a constant and does not contribute in the variations. Below, Luke's variational principle is used to arrive at the flow equations for non-linear surface gravity waves on a potential flow. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Luke's variational principle」の詳細全文を読む スポンサード リンク
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