Averaged Lagrangian
In continuum mechanics, Whitham's averaged Lagrangian method – or in short Whitham's method – is used to study the Lagrangian dynamics of slowly-varying wave trains in an inhomogeneous (moving) medium.
It is for instance used in the modelling of surface gravity waves on fluid interfaces,[1][2] and in plasma physics.
[3][4] In case a Lagrangian formulation of a continuum mechanics system is available, the averaged Lagrangian methodology can be used to find approximations for the average dynamics of wave motion – and (eventually) for the interaction between the wave motion and the mean motion – assuming the envelope dynamics of the carrier waves is slowly varying.
(1) This equation states the conservation of wave action – a generalization of the concept of an adiabatic invariant to continuum mechanics – with[6]
(3) The first consistency equation is known as the conservation of wave crests, and the second states that the wavenumber field
Using an ansatz on the form of the wave part of the motion, the Lagrangian is phase averaged.
Originally the averaged Lagrangian method was developed by Whitham for slowly-varying dispersive wave trains.
which is the second-order partial differential equation describing the dynamics of
Higher-order partial differential equations require the inclusion of higher than first-order derivatives in the Lagrangian.
the system is linear and the classical one-dimensional Klein–Gordon equation is obtained.
Whitham developed several approaches to obtain an averaged Lagrangian method.
[14][17] The simplest one is for slowly-varying linear wavetrains, which method will be applied here.
[14] The slowly-varying wavetrain –without mean motion– in a linear dispersive system is described as:[18]
are expressed as the time derivative and gradient of the wave phase
These two consistency relations denote the "conservation of wave crests", and the irrotationality of the wavenumber field.
Because of the assumption of slow variations in the wave train – as well as in a possible inhomogeneous medium and mean motion – the quantities
[14][17] In the averaged Lagrangian method, the above-given assumptions on the field
is now proposed by Whitham to follow the average variational principle:[14]
follow the dynamical equations for the slowly-varying wave properties.
Applying the averaged Lagrangian principle, variation with respect to the wave phase
and the nonlinear dispersion relation follows from variation with respect to the amplitude
(these derivatives being, by definition, the angular frequency and wavenumber) and does not depend on the wave phase itself.
So the solutions will be independent of the choice of the zero level for the wave phase.
Consequently – by Noether's theorem – variation of the averaged Lagrangian
with respect to the wave phase results in a conservation law:
Conservation of wave action is also found by applying the generalized Lagrangian mean (GLM) method to the equations of the combined flow of waves and mean motion, using Newtonian mechanics instead of a variational approach.
[21] Pure wave motion by linear models always leads to an averaged Lagrangian density of the form:[14]
More generally, for weakly nonlinear and slowly modulated waves propagating in one space dimension and including higher-order dispersion effects – not neglecting the time and space derivatives
is a small modulation parameter – the averaged Lagrangian density is of the form:[22]
An overview can be found in the book: Some publications by Whitham on the method are:
