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:
![\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]\; \text{d}x\; \text{d}y\; \text{d}z\; \right\}\; \text{d}t.](http://upload.wikimedia.org/math/6/f/2/6f22f30d9905e7ba8250d5e793e8c39d.png)
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:
![\mathcal{L} = -\, \int_{t_0}^{t_1} \iint \left\{ \int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},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 \right]\; \text{d}z\; +\, \frac{1}{2}\, \rho\, g\, \eta^2 \right\}\; \text{d}\boldsymbol{x}\; \text{d}t.](http://upload.wikimedia.org/math/6/4/c/64c700096c3e600a7c52bd91046a9e89.png)
The bed-level term proportional to h2 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.
Read more about this topic: Luke's Variational Principle
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