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. The method is applicable to both linear and non-linear systems. As a direct consequence of the averaging used in the method, wave action is a conserved property of the wave motion. In contrast, the wave energy is not necessarily conserved, due to the exchange of energy with the mean motion. However the total energy, the sum of the energies in the wave motion and the mean motion, will be conserved for a time-invariant Lagrangian. Further, the averaged Lagrangian has a strong relation to the dispersion relation of the system.
- 1 Resulting equations for pure wave motion
- 2 Method
- 3 Conservation of wave action
- 4 Conservation of energy and momentum
- 5 Connection to the dispersion relation
- 6 References
Resulting equations for pure wave motion
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. Phase averaging of the Lagrangian results in an averaged Lagrangian, which is always independent of the wave phase itself (but depends on slowly varying wave quantities like wave amplitude, frequency and wavenumber). By Noether's theorem, variation of the averaged Lagrangian with respect to the invariant wave phase then gives rise to a conservation law:
( 1 )
( 2 )
and both are assumed to be slowly varying. Due to this definition, and have to satisfy the consistency relations:
( 3 )
The averaged Lagrangian approach applies to wave motion – possibly superposed on a mean motion – that can be described in a Lagrangian formulation. Using an ansatz on the form of the wave part of the motion, the Lagrangian is phase averaged. Since the Lagrangian is associated with the kinetic energy and potential energy of the motion, the oscillations contribute to the Lagrangian, although the mean value of the wave's oscillatory excursion is zero (or very small).
The resulting averaged Lagrangian contains wave characteristics like the wavenumber, angular frequency and amplitude (or equivalently the wave's energy density or wave action). But the wave phase itself is absent due to the phase averaging. Consequently, through Noether's theorem, there is a conservation law called the conservation of wave action.
Originally the averaged Lagrangian method was developed by Whitham for slowly-varying dispersive wave trains. Several extensions have been made, e.g. to interacting wave components, Hamiltonian mechanics, higher-order modulational effects, dissipation effects.
The averaged Lagrangian method requires the existence of a Lagrangian describing the wave motion. For instance for a field , described by a Lagrangian density the principle of stationary action is:
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.
( 4 )
This Euler–Lagrange equation emerges from the Lagrangian density:
( 5 )
Slowly-varying linear waves
The slowly-varying wavetrain – without mean motion – in a linear dispersive system is described as:
- with and
where is the real-valued wave phase, denotes the absolute value of the complex-valued amplitude while is its argument and denotes its real part. The real-valued amplitude and phase shift are denoted by and respectively.
As a consequence, and have to satisfy the consistency relations:
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 and all vary slowly in space and time – but the wave phase itself does not vary slowly. Consequently, derivatives of and are neglected in the determination of the derivatives of for use in the averaged Lagrangian:
Next these assumptions on and its derivatives are applied to the Lagrangian density
Slowly-varying non-linear waves
Several approaches to slowly-varying non-linear wavetrains are possible. One is by the use of Stokes expansions, used by Whitham to analyse slowly-varying Stokes waves. A Stokes expansion of the field can be written as:
where the amplitudes etc. are slowly varying, as are the phases etc. As for the linear wave case, in lowest order (as far as modulational effects are concerned) derivatives of amplitudes and phases are neglected, except for derivatives and of the fast phase
These approximations are to be applied in the Lagrangian density , and its phase average
Averaged Lagrangian for slowly-varying waves
For pure wave motion the Lagrangian is expressed in terms of the field and its derivatives. In the averaged Lagrangian method, the above-given assumptions on the field – and its derivatives – are applied to calculate the Lagrangian. The Lagrangian is thereafter averaged over the wave phase 
As a last step, this averaging result can be expressed as the averaged Lagrangian density – which is a function of the slowly varying parameters and and independent of the wave phase itself.
From the variations of follow the dynamical equations for the slowly-varying wave properties.
Continuing on the example of the nonlinear Klein–Gordon equation, see equations 4 and 5, and applying the above approximations for and (for this 1D example) in the Lagrangian density, the result after averaging over is:
where it has been assumed that, in big-O notation, and . Variation of with respect to leads to So the averaged Lagrangian is:
For linear wave motion the averaged Lagrangian is obtained by setting equal to zero.
Set of equations emerging from the averaged Lagrangian
Applying the averaged Lagrangian principle, variation with respect to the wave phase leads to the conservation of wave action:
since and while the wave phase does not appear in the averaged Lagrangian density due to the phase averaging. Defining the wave action as and the wave action flux as the result is:
The wave action equation is accompanied with the consistency equations for and which are:
Variation with respect to the amplitude leads to the dispersion relation
Continuing with the nonlinear Klein–Gordon equation, using the average variational principle on equation 6, the wave action equation becomes by variation with respect to the wave phase
and the nonlinear dispersion relation follows from variation with respect to the amplitude
So the wave action is and the wave action flux The group velocity is
Mean motion and pseudo-phase
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Conservation of wave action
The averaged Lagrangian is obtained by integration of the Lagrangian over the wave phase. As a result, the averaged Lagrangian only contains the derivatives of the wave phase (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:
Note that in general the energy of the wave motion does not need to be conserved, since there can be an energy exchange with a mean flow. The total energy – the sum of the energies of the wave motion and the mean flow – is conserved (when there is no work by external forces and no energy dissipation).
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.
Conservation of energy and momentum
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Connection to the dispersion relation
Pure wave motion by linear models always leads to an averaged Lagrangian density of the form:
Consequently, the variation with respect to amplitude: gives
So this turns out to be the dispersion relation for the linear waves, and the averaged Lagrangian for linear waves is always the dispersion function times the amplitude squared.
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 and of the amplitude when taking derivatives, where is a small modulation parameter – the averaged Lagrangian density is of the form:
with the slow variables and
- Grimshaw (1984)
- Janssen (2004, pp. 16–24)
- Dewar (1970)
- Craik (1988, p. 17)
- Whitham (1974, pp. 395–397)
- Bretherton & Garrett (1968)
- Whitham (1974, p. 382)
- Whitham (1965)
- Simmons (1969)
- Willebrand (1975)
- Hayes (1973)
- Yuen & Lake (1975)
- Jimenez & Whitham (1976)
- Whitham (1974, pp. 390–397)
- Whitham (1974, pp. 522–523)
- Whitham (1974, p. 487)
- Whitham (1974, pp. 491–510)
- Whitham (1974, p. 385)
- Whitham (1974, p. 498)
- Whitham (1974, §§16.6–16.13)
- Andrews & McIntyre (1978)
- Whitham (1974, pp. 522–526)
Publications by Whitham on the method
An overview can be found in the book:
Some publications by Whitham on the method are:
- Whitham, G.B. (1965), "A general approach to linear and non-linear dispersive waves using a Lagrangian", Journal of Fluid Mechanics, 22 (2): 273–283, Bibcode:1965JFM....22..273W, doi:10.1017/S0022112065000745
- —— (1967a). "Non-linear dispersion of water waves". Journal of Fluid Mechanics. 27 (2): 399–412. Bibcode:1967JFM....27..399W. doi:10.1017/S0022112067000424.
- —— (1967b), "Variational methods and applications to water waves", Proceedings of the Royal Society of London A: Mathematical and Physical Sciences, 299 (1456): 6–25, Bibcode:1967RSPSA.299....6W, doi:10.1098/rspa.1967.0119
- —— (1970), "Two-timing, variational principles and waves" (PDF), Journal of Fluid Mechanics, 44 (2): 373–395, Bibcode:1970JFM....44..373W, doi:10.1017/S002211207000188X
- Jimenez, J.; Whitham, G.B. (1976), "An averaged Lagrangian method for dissipative wavetrains", Proceedings of the Royal Society of London A: Mathematical and Physical Sciences, 349 (1658): 277–287, Bibcode:1976RSPSA.349..277J, doi:10.1098/rspa.1976.0073
- Andrews, D.G.; McIntyre, M.E. (1978), "On wave-action and its relatives" (PDF), Journal of Fluid Mechanics, 89 (4): 647–664, Bibcode:1978JFM....89..647A, doi:10.1017/S0022112078002785
- Badin, G.; Crisciani, F. (2018). Variational Formulation of Fluid and Geophysical Fluid Dynamics - Mechanics, Symmetries and Conservation Laws -. Springer. p. 218. doi:10.1007/978-3-319-59695-2. ISBN 978-3-319-59694-5.
- Bretherton, F.P.; Garrett, C.J.R. (1968), "Wavetrains in inhomogeneous moving media", Proceedings of the Royal Society of London A: Mathematical and Physical Sciences, 302 (1471): 529–554, Bibcode:1968RSPSA.302..529B, doi:10.1098/rspa.1968.0034
- Craik, A.D.D. (1988), Wave interactions and fluid flows, Cambridge University Press, ISBN 9780521368292
- Dewar, R.L. (1970), "Interaction between hydromagnetic waves and a time‐dependent, inhomogeneous medium", Physics of Fluids, 13 (11): 2710–2720, Bibcode:1970PhFl...13.2710D, doi:10.1063/1.1692854, ISSN 0031-9171
- Grimshaw, R. (1984), "Wave action and wave–mean flow interaction, with application to stratified shear flows", Annual Review of Fluid Mechanics, 16: 11–44, Bibcode:1984AnRFM..16...11G, doi:10.1146/annurev.fl.16.010184.000303
- Hayes, W.D. (1970), "Conservation of action and modal wave action", Proceedings of the Royal Society of London A: Mathematical and Physical Sciences, 320 (1541): 187–208, Bibcode:1970RSPSA.320..187H, doi:10.1098/rspa.1970.0205
- Hayes, W.D. (1973), "Group velocity and nonlinear dispersive wave propagation", Proceedings of the Royal Society of London A: Mathematical and Physical Sciences, 332 (1589): 199–221, Bibcode:1973RSPSA.332..199H, doi:10.1098/rspa.1973.0021
- Holm, D.D. (2002), "Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics", Chaos, 12 (2): 518–530, Bibcode:2002Chaos..12..518H, doi:10.1063/1.1460941, PMID 12779582
- Janssen, P.A.E.M. (2004), The Interaction of Ocean Waves and Wind, Cambridge University Press, ISBN 9780521465403
- Radder, A.C. (1999), "Hamiltonian dynamics of water waves", in Liu, P.L.-F. (ed.), Advances in Coastal and Ocean Engineering, 4, World Scientific, pp. 21–59, ISBN 9789810233105
- Sedletsky, Y.V. (2012), "Addition of dispersive terms to the method of averaged Lagrangian", Physics of Fluids, 24 (6): 062105 (15 , pp.), Bibcode:2012PhFl...24f2105S, doi:10.1063/1.4729612
- Simmons, W.F. (1969), "A variational method for weak resonant wave interactions", Proceedings of the Royal Society of London A: Mathematical and Physical Sciences, 309 (1499): 551–577, Bibcode:1969RSPSA.309..551S, doi:10.1098/rspa.1969.0056
- Willebrand, J. (1975), "Energy transport in a nonlinear and inhomogeneous random gravity wave field", Journal of Fluid Mechanics, 70 (1): 113–126, Bibcode:1975JFM....70..113W, doi:10.1017/S0022112075001929
- Yuen, H.C.; Lake, B.M. (1975), "Nonlinear deep water waves: Theory and experiment", Physics of Fluids, 18 (8): 956–960, Bibcode:1975PhFl...18..956Y, doi:10.1063/1.861268
- Yuen, H.C.; Lake, B.M. (1980), "Instabilities of waves on deep water", Annual Review of Fluid Mechanics, 12: 303–334, Bibcode:1980AnRFM..12..303Y, doi:10.1146/annurev.fl.12.010180.001511