Mild-slope Equation - Derivation of The Mild-slope Equation - Luke's Variational Principle | Luke Variational Principle

Luke's Variational Principle

Luke's Lagrangian formulation gives a variational formulation for non-linear surface gravity waves. For the case of a horizontally unbounded domain with a constant density, a free fluid surface at and a fixed sea bed at Luke's variational principle uses the Lagrangian

$\mathcal{L} = \int_{t_0}^{t_1} \iint L\; \text{d}x\; \text{d}y\; \text{d}t,$

where is the horizontal Lagrangian density, given by:

$L = -\rho\, \left\{ \int_{-h(x,y)}^{\zeta(x,y,t)} \left[ \frac{\partial\Phi}{\partial t} +\, \frac{1}{2} \left( \left( \frac{\partial\Phi}{\partial x} \right)^2 + \left( \frac{\partial\Phi}{\partial y} \right)^2 + \left( \frac{\partial\Phi}{\partial z} \right)^2 \right) \right]\; \text{d}z\; +\, \frac{1}{2}\, g\, (\zeta^2\, -\, h^2) \right\},$

where is the velocity potential, with the flow velocity components being and in the, and directions, respectively. 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. Taking the variations of with respect to the potential and surface elevation leads to the Laplace equation for in the fluid interior, as well as all the boundary conditions both on the free surface as at the bed at

Read more about this topic: Mild-slope Equation, Derivation of The Mild-slope Equation