Luke's variational principle

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: • is the velocity potential, • is the fluid density, • is the acceleration by the Earth's gravity, • is the horizontal coordinate vector with components and , • and are the horizontal coordinates, • is the vertical coordinate, • is time, and • is the horizontal gradient operator, so is the horizontal flow velocity consisting of and , • is the time-dependent fluid domain with free surface. The Lagrangian \mathcal{L}, 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]\, \mathrm{d}x\; \mathrm{d}y\; \mathrm{d}z \right\} \mathrm{d}t. From Bernoulli's principle, this Lagrangian can be seen to be the integral of the fluid pressure over the whole time-dependent fluid domain . This is in agreement with the variational principles for inviscid flow without a free surface, found by Harry Bateman. This may also include moving wavemaker walls and ship motion. For the case of a horizontally unbounded domain with the free fluid surface at and a fixed bed at , 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]\; \mathrm{d}z\; +\, \frac{1}{2}\, \rho\, g\, \eta^2 \right\}\; \mathrm{d}\boldsymbol{x}\; \mathrm{d}t. The bed-level term proportional to 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 \delta\mathcal{L} = 0 in the Lagrangian with respect to variations in the velocity potential Φ(x,z,t), as well as with respect to the surface elevation , have to be zero. We consider both variations subsequently. Variation with respect to the velocity potential Consider a small variation in the velocity potential . Then the resulting variation in the Lagrangian is: \begin{align} \delta_\Phi\mathcal{L}\, &=\, \mathcal{L}(\Phi+\delta\Phi,\eta)\, -\, \mathcal{L}(\Phi,\eta) \\ &=\, -\, \int_{t_0}^{t_1} \iint \left\{ \int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},t)} \rho\, \left( \frac{\partial(\delta\Phi)}{\partial t} +\, \boldsymbol{\nabla}\Phi \cdot \boldsymbol{\nabla} (\delta\Phi) +\, \frac{\partial\Phi}{\partial z}\, \frac{\partial(\delta \Phi)}{\partial z}\, \right)\; \mathrm{d}z\, \right\}\; \mathrm{d}\boldsymbol{x}\; \mathrm{d}t. \end{align} Using Leibniz integral rule, this becomes, in case of constant density : \begin{align} \delta_\Phi\mathcal{L}\, =\, &-\, \rho\, \int_{t_0}^{t_1} \iint \left\{ \frac{\partial}{\partial t} \int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},t)} \delta\Phi\; \mathrm{d}z\; +\, \boldsymbol{\nabla} \cdot \int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},t)} \delta\Phi\, \boldsymbol{\nabla}\Phi\; \mathrm{d}z\, \right\}\; \mathrm{d}\boldsymbol{x}\; \mathrm{d}t \\ &+\, \rho\, \int_{t_0}^{t_1} \iint \left\{ \int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},t)} \delta\Phi\; \left( \boldsymbol{\nabla} \cdot \boldsymbol{\nabla}\Phi\, +\, \frac{\partial^2\Phi}{\partial z^2} \right)\; \mathrm{d}z\, \right\}\; \mathrm{d}\boldsymbol{x}\; \mathrm{d}t \\ &+\, \rho\, \int_{t_0}^{t_1} \iint \left[ \left( \frac{\partial\eta}{\partial t}\, +\, \boldsymbol{\nabla}\Phi \cdot \boldsymbol{\nabla} \eta\, -\, \frac{\partial\Phi}{\partial z} \right)\, \delta\Phi \right]_{z=\eta(\boldsymbol{x},t)}\; \mathrm{d}\boldsymbol{x}\; \mathrm{d}t \\ &+\, \rho\, \int_{t_0}^{t_1} \iint \left[ \left( \boldsymbol{\nabla}\Phi \cdot \boldsymbol{\nabla} h\, +\, \frac{\partial\Phi}{\partial z} \right)\, \delta\Phi \right]_{z=-h(\boldsymbol{x})}\; \mathrm{d}\boldsymbol{x}\; \mathrm{d}t \\ =\, &0. \end{align} The first integral on the right-hand side integrates out to the boundaries, in and , of the integration domain and is zero since the variations are taken to be zero at these boundaries. For variations which are zero at the free surface and the bed, the second integral remains, which is only zero for arbitrary in the fluid interior if there the Laplace equation holds: \Delta \Phi\, =\, 0 \qquad \text{ for } -h(\boldsymbol{x})\, with the Laplace operator. If variations 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: \frac{\partial\eta}{\partial t}\, +\, \boldsymbol{\nabla}\Phi \cdot \boldsymbol{\nabla} \eta\, -\, \frac{\partial\Phi}{\partial z}\, =\, 0. \qquad \text{ at } z\, =\, \eta(\boldsymbol{x},t). Similarly, variations only non-zero at the bottom result in the kinematic bed condition: \boldsymbol{\nabla}\Phi \cdot \boldsymbol{\nabla} h\, +\, \frac{\partial\Phi}{\partial z}\, =\, 0 \qquad \text{ at } z\, =\, -h(\boldsymbol{x}). Variation with respect to the surface elevation Considering the variation of the Lagrangian with respect to small changes gives: \delta_\eta\mathcal{L}\, =\, \mathcal{L}(\Phi,\eta+\delta\eta)\, -\, \mathcal{L}(\Phi,\eta) =\, -\, \int_{t_0}^{t_1} \iint \left[ \rho\, \delta\eta\, \left( \frac{\partial\Phi}{\partial t} +\, \frac12\, \left| \boldsymbol{\nabla}\Phi \right|^2\, +\, \frac12\, \left( \frac{\partial\Phi}{\partial z} \right)^2 +\, g\, \eta \right)\, \right]_{z=\eta(\boldsymbol{x},t)}\; \mathrm{d}\boldsymbol{x}\; \mathrm{d}t\, =\, 0. This has to be zero for arbitrary , giving rise to the dynamic boundary condition at the free surface: \frac{\partial\Phi}{\partial t} +\, \frac12\, \left| \boldsymbol{\nabla}\Phi \right|^2\, +\, \frac12\, \left( \frac{\partial\Phi}{\partial z} \right)^2 +\, g\, \eta\, =\, 0 \qquad \text{ at } z\, =\, \eta(\boldsymbol{x},t). 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: H\, =\, \frac 1 2 \, \rho\, \sqrt{ 1\, +\, \left| \boldsymbol{\nabla} \eta \right|^2}\;\; \varphi\, \bigl( D(\eta)\; \varphi \bigr)\, +\, \frac12\, \rho\, g\, \eta^2, where is equal to the normal derivative of 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 and free surface — the normal derivative is a linear function of the surface potential , but depends non-linear on the surface elevation . This is expressed by the Dirichlet-to-Neumann operator , acting linearly on . The Hamiltonian density can also be written as: H\, =\, \frac 1 2 \, \rho\, \varphi\, \Bigl[ w\, \left( 1\, +\, \left| \boldsymbol{\nabla} \eta \right|^2 \right) -\, \boldsymbol{\nabla}\eta \cdot \boldsymbol{\nabla}\, \varphi \Bigr]\, +\, \frac 1 2 \, \rho\, g\, \eta^2, with the vertical velocity at the free surface . Also is a linear function of the surface potential through the Laplace equation, but depends non-linear on the surface elevation : w\, =\, W(\eta)\, \varphi, with operating linear on , but being non-linear in . As a result, the Hamiltonian is a quadratic functional of the surface potential . 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 . Further is not to be mistaken for the horizontal velocity at the free surface: \boldsymbol{\nabla}\varphi\, =\, \boldsymbol{\nabla} \Phi\bigl(\boldsymbol{x},\eta(\boldsymbol{x},t),t\bigr)\, =\, \left[ \boldsymbol{\nabla}\Phi\, +\, \frac{\partial\Phi}{\partial z}\, \boldsymbol{\nabla}\eta \right]_{z=\eta(\boldsymbol{x},t)}\, =\, \Bigl[ \boldsymbol{\nabla}\Phi \Bigr]_{z=\eta(\boldsymbol{x},t)}\, +\, w\, \boldsymbol{\nabla}\eta. Taking the variations of the Lagrangian \mathcal{L}_H with respect to the canonical variables \varphi(\boldsymbol{x},t) and \eta(\boldsymbol{x},t) gives: \begin{align} \rho\, \frac{\partial\eta}{\partial t}\, &=\, +\, \frac{\delta\mathcal{H}}{\delta\varphi},\\ \rho\, \frac{\partial\varphi}{\partial t}\, &=\, -\, \frac{\delta\mathcal{H}}{\delta\eta}, \end{align} provided in the fluid interior satisfies the Laplace equation, , as well as the bottom boundary condition at and at the free surface. ==References and notes==