# 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 so-called 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]}

## Contents

## 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,is the horizontal coordinate vector with components**x***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.^{[7]}

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.

^{[8]}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*(

*), Luke's variational principle results in the Lagrangian:*

**x**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.

### Derivation of the flow equations resulting from Luke's variational principle

The variation in the Lagrangian with respect to variations in the velocity potential *Φ*(* x*,

*z*,

*t*), as well as with respect to the surface elevation

*η*(

*,*

**x***t*), have to be zero. We consider both variations subsequently.

#### Variation with respect to the velocity potential

Consider a small variation *δΦ* in the velocity potential *Φ*.^{[8]} Then the resulting variation in the Lagrangian is:

Using Leibniz integral rule, this becomes, in case of constant density *ρ*:^{[8]}

The first integral on the right-hand side integrates out to the boundaries, in * x* and

*t*, of the integration domain and is zero since the variations

*δΦ*are taken to be zero at these boundaries. For variations

*δΦ*which are zero at the free surface and the bed, the second integral remains, which is only zero for arbitrary

*δΦ*in the fluid interior if there the Laplace equation holds:

with Δ=∇·∇ + ∂^{2}/∂*z*^{2} the Laplace operator.

If variations *δΦ* are considered which are only non-zero at the free surface, only the third integral remains, giving rise to the kinematic free-surface boundary condition:

Similarly, variations *δΦ* only non-zero at the bottom *z* = -*h* result in the kinematic bed condition:

#### Variation with respect to the surface elevation

Considering the variation of the Lagrangian with respect to small changes *δη* gives:

This has to be zero for arbitrary *δη*, giving rise to the dynamic boundary condition at the free surface:

This is the Bernoulli equation for unsteady potential flow, applied at the free surface, and with the pressure above the free surface being a constant — which constant pressure is taken equal to zero for simplicity.

## Hamiltonian formulation

The Hamiltonian structure of surface gravity waves on a potential flow was discovered by Vladimir E. Zakharov in 1968, and rediscovered independently by Bert Broer and John Miles:^{[4]}^{[5]}^{[6]}

where the surface elevation *η* and surface potential *φ* — which is the potential *Φ* at the free surface *z*=*η*(* x*,

*t*) — are the canonical variables. The Hamiltonian is the sum of the kinetic and potential energy of the fluid:

The additional constraint is that the flow in the fluid domain has to satisfy Laplace's equation with appropriate boundary condition at the bottom *z*=-*h*(* x*) and that the potential at the free surface

*z*=

*η*is equal to

*φ*:

### Relation with Lagrangian formulation

The Hamiltonian formulation can be derived from Luke's Lagrangian description by using Leibniz integral rule on the integral of ∂*Φ*/∂*t*:^{[6]}

with the value of the velocity potential at the free surface, and the Hamiltonian density — sum of the kinetic and potential energy density — and related to the Hamiltonian as:

The Hamiltonian density is written in terms of the surface potential using Green's third identity on the kinetic energy:^{[9]}

where *D*(*η*) *φ* is equal to the normal derivative of ∂*Φ*/∂*n* at the free surface. Because of the linearity of the Laplace equation — valid in the fluid interior and depending on the boundary condition at the bed *z*=-*h* and free surface *z*=*η* — the normal derivative ∂*Φ*/∂*n* is a *linear* function of the surface potential *φ*, but depends non-linear on the surface elevation *η*. This is expressed by the Dirichlet-to-Neumann operator *D*(*η*), acting linearly on *φ*.

The Hamiltonian density can also be written as:^{[6]}

with *w*(* x*,

*t*) = ∂

*Φ*/∂

*z*the vertical velocity at the free surface

*z*=

*η*. Also

*w*is a

*linear*function of the surface potential

*φ*through the Laplace equation, but

*w*depends non-linear on the surface elevation

*η*:

^{[9]}

with *W* operating linear on *φ*, but being non-linear in *η*. As a result, the Hamiltonian is a quadratic functional of the surface potential *φ*. Also the potential energy part of the Hamiltonian is quadratic. The source of non-linearity in surface gravity waves is through the kinetic energy depending non-linear on the free surface shape *η*.^{[9]}

Further ∇*φ* is not to be mistaken for the horizontal velocity ∇*Φ* at the free surface:

Taking the variations of the Lagrangian with respect to the canonical variables and gives:

provided in the fluid interior *Φ* satisfies the Laplace equation, Δ*Φ*=0, as well as the bottom boundary condition at *z*=-*h* and *Φ*=*φ* at the free surface.

## References and notes

- ^
^{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. **^**M. W. Dingemans (1997).*Water Wave Propagation Over Uneven Bottoms*. Advanced Series on Ocean Engineering.**13**. Singapore: World Scientific. p. 271. ISBN 981-02-0427-2.**^**G. B. Whitham (1974).*Linear and Nonlinear Waves*. Wiley-Interscience. p. 555. ISBN 0-471-94090-9.- ^
^{a}^{b}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. Originally appeared in*Zhurnal Prildadnoi Mekhaniki i Tekhnicheskoi Fiziki***9**(2): 86–94, 1968. - ^
^{a}^{b}L. J. F. Broer (1974). "On the Hamiltonian Theory of Surface Waves".*Applied Scientific Research*.**29**: 430–446. doi:10.1007/BF00384164. - ^
^{a}^{b}^{c}^{d}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. **^**H. Bateman (1929). "Notes on a Differential Equation Which Occurs in the Two-Dimensional Motion of a Compressible Fluid and the Associated Variational Problems".*Proceedings of the Royal Society of London A*.**125**(799): 598–618. Bibcode:1929RSPSA.125..598B. doi:10.1098/rspa.1929.0189.- ^
^{a}^{b}^{c}G. W. Whitham (1974).*Linear and Nonlinear Waves*. New York: Wiley. pp. 434–436. ISBN 0-471-94090-9. - ^
^{a}^{b}^{c}D. M. Milder (1977). "A note on: 'On Hamilton's principle for surface waves'".*Journal of Fluid Mechanics*.**83**(1): 159–161. Bibcode:1977JFM....83..159M. doi:10.1017/S0022112077001116.