Averaged Lagrangian Method Research Articles

Multiwave non-linear couplings in elastic structures. Part II: two-dimensional example

The non-linear resonance coupling in a thin plate in the Kirchhoff-Love approximation is selected as a two-dimensional example of mechanical systems exhibiting a rich range of resonant wave-like phenomena. This is originally examined by use of Whitham's average-Lagrangian method. In particular, the existence of three basic resonant triads between longitudinal, shear and bending modes is shown. Some of these necessarily enter cascade wave processes related to the instability of some of the mode components of the triad under small perturbations. A short comparison with Kolmogorov's cascades of turbulence is given.

Existence and modulation of traveling waves in particles chains

We consider an infinite particle chain whose dynamics are governed by the following system of differential equations: where qn(t) is the displacement of the nth particle at time t along the chain axis and denotes differentiation with respect to time. We assume that all particles have unit mass and that the interaction potential V between adjacent particles is a convex C∞ function. For this system, we prove the existence of C∞, time-periodic, traveling-wave solutions of the form qn(t) = q(wt kn + where q is a periodic function q(z) = q(z+1) (the period is normalized to equal 1), ω and k are, respectively, the frequency and the wave number, is the mean particle spacing, and can be chosen to be an arbitrary parameter. We present two proofs, one based on a variational principle and the other on topological methods, in particular degree theory. For small-amplitude waves, based on perturbation techniques, we describe the form of the traveling waves, and we derive the weakly nonlinear dispersion relation. For the fully nonlinear case, when the amplitude of the waves is high, we use numerical methods to compute the traveling-wave solution and the non-linear dispersion relation. We finally apply Whitham's method of averaged Lagrangian to derive the modulation equations for the wave parameters α, β, k, and ω. © 1999 John Wiley & Sons, Inc.

Pattern Selection in Faraday Surface Waves

Pattern selection phenomena in parametrically excited surface waves are studied by the weakly-nonlinear system of three modes derived by the average Lagrangian method. The third-order coefficients of nonlinear interaction between two line patterns intersecting at an arbitrary angle are obtained in the gravity-capillary waves of arbitrary depth. Classification, stability analysis and bifurcation study of the fixed points of the dynamical equations with linear damping are performed in the cases of three symmetric line modes and general modes. Investigating the stability of various patterns to the disturbance of line mode with an arbitrary intersecting angle and the internal stability, it is shown that squares are the most preferred pattern in capillary waves, while lines are selected in gravity waves.

Variational approach to Langmuir waves described by the Zakharov equations

Nonlinear modulational instability and the evolution of a pulse that is initially non-solitonic for Langmuir waves described by the Zakharov equations are considered. The average Lagrangian method has been used to derive the nonlinear dispersion relation for Stokes waves. It is found that the region of instability increases with the amplitude of the perturbed Langmuir field. The propagation of the pulse is studied with the help of the Rayleigh-Ritz optimization method. It is noted that the width of the pulse oscillates rapidly for high initial pulse velocity.

Standing radial cross-waves

Standing radial cross-waves in an annular wave tank are investigated using Whitham's average-Lagrangian method. For the simplest case, in which a single radial cross-wave is excited, energy is transferred from the wavemaker to the cross-wave through the spatial mean motion of the free surface, as described by Garrett (1970) for a purely transverse cross-wave in a rectangular tank. In addition, energy is transferred through spatial coupling since, in contrast to the purely transverse cross-wave in the rectangular tank, the (non-axisymmetric) radial cross-wave is three-dimensional. It is shown in an Appendix that this spatial coupling does occur for a three-dimensional cross-wave in a rectangular tank. The equations that govern this single-mode resonance are isomorphic to those that govern the Faraday resonance of surface waves in a basin of fluid subjected to vertical excitation (Miles 1984a).It is found that the second-order Stokes-wave expansion for deep-water, standing gravity waves, which is regular for rectangular containers, may become singular for circular containers (Mack (1962) noted these resonances for finite-depth, standing gravity waves in circular containers). The evolution equations that govern two distinct types of resonant behaviour are derived: (i) 2:1 resonance between a radial cross-wave and a resonantly forces axisymmetric wave, corresponding to approximate equality among the driving frequency, a natural frequency of the directly forced wave, and twice the natural frequency of a cross-wave; (ii) 2:1 internal resonance between a radial cross-wave and a non-axisymmetric second harmonic, corresponding to approximate equality among the driving frequency, the natural frequency of a non-axisymmetric wave of even azimuthal wavenumber, and twice the natural frc.quency of the cross-wave. The axisymmetric, directly forced wave in (i) is resonantly excited and exchanges energy with the subharmonic cross-wave through spatial coupling, whereas the cross-wave in (ii) is parametrically excited and exchanges energy with the non-axisymmetric second harmonic through spatial coupling. The equations governing case (i) are shown to exhibit chaotic motions; those governing (ii) are shown to be isomorphic to the equations governing 2:1 internal resonance in the Faraday problem (Miles 1984a, §6), which have been shown to exhibit chaotic motions (Gu & Sethna 1987).Preliminary experiments on standing radial cross-waves are reported in an Appendix, and theoretical predictions of mode stability are in qualitative agreement with these experiments. For the single-mode theory, the interaction coefficient that is a measure of the energy exchange between the wavemaker and the cross-wave is evaluated numerically for a particular wavemaker. The maximum interaction coefficient for a fixed azimuthal wavenumber of the cross-wave typically occurs for that radial mode number for which the turning point of the cross-wave radial profile is nearest the wavemaker. The present experiments for standing radial cross-waves are compared with those of Tatsuno, Inoue & Okabe (1969) for progressive radial cross-waves.

Nonlinear dynamics of two-mode interactions in parametric excitation of surface waves

Parametric excitation of surface waves in a container under vertical forcing is investigated in detail, by an averaged Lagrangian method due to John Miles, and a system of evolution equations of third-order nonlinearity is presented for the case that the forcing frequency is chosen to be near twice the frequencies of two nearly degenerate free modes. The system of first-order differential equations in four variables which are derived from an averaged Hamiltonian is considered in a unified fashion, and the analytical results are compared with three experimental observations. It is found with the help of numerical integration that this dynamical system yields not only excitation of a single-mode state, but also interaction between two modes in which each mode oscillates either periodically or chaotically. These results are in good agreement with the observations, except for one case in which nonlinearity is considered to be too strong. As a fourth case, homoclinic chaos in the Hamiltonian system of two-degrees of freedom without damping is studied numerically. It is suggested that the chaotic mode competition observed in the experiments is different from the homoclinic chaos.

Pseudo-variational principles and non-linear surface waves in a dissipative fluid

This paper is concerned with the evolution of non-linear surface waves in a dissipative fluid. A pseudo-variational principle and the ideas regarding the evolution of a slowly varying wavetrain in a dispersive and dissipative medium are used to develop an averaged Lagrangian method and then to treat the evolution of a modulated finite-amplitude gravity wave on the surface of a dissipative fluid. A modified non-linear Schrödinger equation is derived to describe the modulation of the wave-amplitude in a dissipative medium. It is shown that the effect of dissipation is to stabilize the modulation by reducing the growth rate.

Nonlinear Dynamics and Chaos in Parametrically Excited Surface Waves

Surface waves in a closed container subject to vertical oscillation are studied. Nonlinear dynamical equations of two nearly degenerate subharmonic modes responding to the external forcing are derived, using the averaged Lagrangian method for slowly varying amplitudes. Stability and bifurcation diagrams are shown for the system with linear damping. Period-doubling bifurcation and chaotic solutions with one positive Lyapunov characteristic exponent are obtained numerically. It is shown that some of the period-doubling bifurcations are related to the symmetry of the dynamical system.

Magnetized Langmuir wave packets excited by a strong beam-plasma interaction.

The physics of beam-plasma interaction, which has been investigated for a long time mostly in relation with solar bursts, is now more widely invoked in various astrophysical contexts such as pulsars, active galactic nuclei, close binaries, cataclysmic variables, \ensuremath{\gamma} bursters, and so on. In these situations the interaction is more likely in the spirit of strong Langmuir turbulence rather than in the spirit of quasilinear theory. Many investigations have been done for two opposite extremes, namely, in very weak and in very strong magnetic fields. Very few properties of the strong Langmuir turbulence are known in the most usual astrophysical situation where the magnetic field plays a significant role but is not strong enough to force the electrons into one-dimensional motion. For this case, we analyze the dynamics of Langmuir wave packets and provide new results about the stability of the solitons against transverse perturbations. It turns out that both the averaged Lagrangian method and the adiabatic perturbation method derived from the inverse scattering transform give exactly the same results (which is not obvious in soliton perturbation theory). In particular, they predict the stability of the solitons as long as the electron gyrofrequency is greater than the plasma frequency (strong magnetic field) and their instability against transverse self-modulation in the opposite case (weak magnetic field); moreover, they allow one to deduce the self-similar collapsing oblate cavitons in the latter case. The laws governing the collapse of the wave packets determine the relaxation of the beam in the surrounding medium and we derive a useful formula giving the power loss of the beam. We outline the astrophysical consequences of this investigation.

Three-wave resonance for free-surface flows over flexible boundaries

Inviscid flow with a free surface over a flexible boundary is considered. The dispersion relation is obtained, and the conditions for linear instability are investigated. The linear dispersion relation is then used to show that the conditions for nonlinear three-wave resonance are often met. In some circumstances, the resonance may be of the “explosive” sort, involving waves of opposite energy sign; but the most common resonant configurations are non-explosive ones. Next, the wave-amplitude evolution equations for three-wave resonance are derived, firstly by a “direct” approach, and then via a variational (averaged Lagrangian) method. Results are in agreement with those of Case and Chin for capillary-gravity waves, and Craik and Adam, for three-layer fluid flow, on taking the appropriate limits. We also consider a nonlinear model for the flexible boundary.

Nonlinear dispersion relation for nonlinear Schrödinger equation

By using the average-Lagrangian method (average variational principle), a nonlinear dispersion relation has been derived for the cubic nonlinear Schrödinger equation. It is found that the size of the instability region in wavenumber space decreases with increasing field amplitude in comparison with the linear theory.

Parametrically excited, standing cross-waves

Luke's (1967) variational formulation for surface waves is extended to incorporate the motion of a wavemaker and applied to the cross-wave problem. Whitham's average-Lagrangian method then is invoked to obtain the evolution equations for the slowly varying complex amplitude of the parametrically excited cross-wave that is associated with symmetric excitation of standing waves in a rectangular tank of width π/k, lengthland depthdfor whichkl=O(1) andkd[Gt ] 1. These evolution equations are Hamiltonian and isomorphic to those for parametric excitation of surface waves in a cylinder that is subjected to a vertical oscillation, for which phase-plane trajectories, stability criteria and the effects of damping are known (Miles 1984a). The formulation and results differ from those of Garrett (1970) in consequence of his linearization of the boundary condition at the wavemaker and his neglect of self-interaction of the cross-waves in the free-surface conditions (although Garrett does incorporate self-interaction in his calculation of the equilibrium amplitude of the cross-waves). These differences have only a small effect on the criterion for the stability of plane waves, but the self-interaction is crucial for the determination of the stability of the cross-waves.

Three-dimensional nonlinear Schrödinger equation for finite-amplitude gravity waves in a fluid (erratum and addendum)

A few more remarks about the origin of the discrepancies between the results obtained by the method of multiple scales and the averaged Lagrangian method are in order. I t must be mentioned that the latter method does not apparently compare well on matters of rigor because the notion of slow variation is never quantified there precisely. On this account, it is quite probable that the averaged Lagrangian method has underestimated the amplitude dispersion in the transverse direction, and may be this is the cause of the forementioned discrepancies. Nonetheless, the evolution equation produced by the averaged Lagrangian method does have some interesting properties, as demonstrated in the above paper, and as such it could be accepted as another model equation in its own right.

Soliton emission in the forced non-linear Schrödinger equation

The plasma waves generated by resonant absorption of light in the vicinity of the critical density of laser-produced plasmas, are modelled by a non-linear Schrödinger equation with additional terms accounting for the presence of a source and the inhomogeneity of the medium.We use an average lagrangian method to describe the behaviour of the solutions of this equation in the range of parameters where periodic soliton generation occurs. An iterating scheme describing the successive emission of solitons yields values for this range of parameters which are in reasonable agreement with those found from direct numerical simulations of the non-linear Schrödinger equation.

Three-dimensional nonlinear Schrödinger equation for finite-amplitude gravity waves in a fluid

The averaged Lagrangian method is used to derive a new nonlinear Schrodinger equation in order to describe the evolution of three-dimensional modulations superposed on finite-amplitude gravity waves in a fluid. An interesting feature of this equation is that the wave energy initially confined to a narrow band of wave numbers continues to be confined to a limited region in the wave number space. Unlike the three-dimensional nonlinear evolution equation available in the literature, the equation obtained in this paper may, therefore, be adequate for the description of the evolution of the three-dimensional modulations.

An average Lagrangian formulation of ponderomotive forces in Vlasov plasmas

Whitham's averaged Lagrangian principle, and the Low Lagrangian are used to derive the averaged Lagrangian for slowly varying electrostatic and electromagnetic wave packets in dispersive Vlasov plasmas. From the averaged Lagrangian, the stress tensor and wave-momentum densities are derived for the total wave-particle system. For a specific division of these total quantities into wave and background parts, time-dependent ponderomotive forces in Vlasov plasmas are derived from a momentum conservation theorem. We also show that the wave-induced magnetization current can be derived by the averaged Lagrangian method.

A nonlinear analogue of the WKB method and the method of the averaged Lagrangian. II

A nonlinear analogue of the WKB method and the method of the averaged Lagrangian. II

Nonlinear Resonant Interaction of Two Longitudinal Waves and a Transverse Wave in a Hot Magnetized Plasma

AbstractBy Whitham's method of averaged Lagrangian and using Low's form of Lagrangian, coupled mode equations and coupling coefficients are derived for resonant nonlinear interaction of two longitudinal and one transverse wave in a magnetized plasma, in which the later wave propagates along the external uniform magnetic field. The limiting form of these coupling coefficients are obtained when the external magnetic field vanishes.

Current-waveform dependence of punchthrough probability in a Josephson tunnel junction

The so-called punchthrough phenomenon occurring in a logic gate using a Josephson tunnel junction is studied theoretically. An analytical solution of the Josephson equation describing the dynamic behavior of the Josephson gate in its resetting process is obtained with the modified averaged Lagrangian method. The solution leads to an expression for the punchthrough probability, which is capable of quantitative discussion and is applicable to arbitrary waveforms of the gate current. From the obtained expression, a study is made of the waveform dependence of the punchthrough probability, as well as its dependence on junction parameters. It is shown that the present analytical results agree well with those of computer simulations.

On the theory of electron motion due to superposed em waves and its application to the free-electron laser with linear polarization

The relativistic motion of a charged particle in an electromagnetic field depending only on one coordinate is shown to be exactly equivalent to the unidimensional motion of a particle having a time and space-varying mass. By means of the averaged Lagrangian method an approximate solution is obtained for the relativistic electron motion in two oppositely moving waves of any polarization, one of them being weaker than the other one, thus generalizing the results previously obtained for circular polarization. A comparison with numerical calculations shows a good agreement. A pendulum-like motion in an effective periodic potential is shown to exist not only for two waves of the same frequency, but more generally when the frequency of the weaker wave is an odd harmonic of the frequency of the stronger one.