Luke's variational principle

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]

  1. ^ a b J. C. Luke (1967). "A Variational Principle for a Fluid with a Free Surface". Journal of Fluid Mechanics. 27 (2): 395–397. Bibcode:1967JFM....27..395L. doi:10.1017/S0022112067000412. S2CID 123409273.
  2. ^ M. W. Dingemans (1997). Water Wave Propagation Over Uneven Bottoms. Advanced Series on Ocean Engineering. Vol. 13. Singapore: World Scientific. p. 271. ISBN 981-02-0427-2.
  3. ^ G. B. Whitham (1974). Linear and Nonlinear Waves. Wiley-Interscience. p. 555. ISBN 0-471-94090-9.
  4. ^ V. E. Zakharov (1968). "Stability of Periodic Waves of Finite Amplitude on the Surface of a Deep Fluid". Journal of Applied Mechanics and Technical Physics. 9 (2): 190–194. Bibcode:1968JAMTP...9..190Z. doi:10.1007/BF00913182. S2CID 55755251. Originally appeared in Zhurnal Prildadnoi Mekhaniki i Tekhnicheskoi Fiziki 9(2): 86–94, 1968.
  5. ^ L. J. F. Broer (1974). "On the Hamiltonian Theory of Surface Waves". Applied Scientific Research. 29: 430–446. doi:10.1007/BF00384164.
  6. ^ J. W. Miles (1977). "On Hamilton's Principle for Surface Waves". Journal of Fluid Mechanics. 83 (1): 153–158. Bibcode:1977JFM....83..153M. doi:10.1017/S0022112077001104. S2CID 121777750.