Non-linear transverse waves in a Vlasov plasma

1969 ◽  
Vol 3 (4) ◽  
pp. 577-592 ◽  
Author(s):  
S. Peter Gary
Keyword(s):  
Perturbation Expansion ◽  
Second Order ◽  
Electron Cyclotron ◽  
Wave Number ◽  
Linear Dispersion ◽  
Transverse Waves ◽  
Non Linear ◽  
Long Time ◽  
Linear Damping ◽  
Electron Cyclotron Waves

Non-linear transverse waves in a classical non-relativistic collisionless, Maxwellian electron gas with external magnetic field B0 are considered. There is assumed a small, sinusoidal variation in the initial electric and magnetic fields, corresponding to excitation of a discrete wave-number mode. The non-linear Vlasov equation is solved to second order in the long time limit via the Montgomery—Gorman perturbation expansion, and the time-independent, spatially homogeneous part of the second-order distribution function is used to modify the linear dispersion relation. For frequencies near the electron cyclotron frequency a non-linear damping decrement results such that, for many values of the parameters, the damping is less than the linear rate. Thus at sufficiently long times, the rate of damping of transverse electron cyclotron waves should decrease, a result similar to that for non-linear damping of longitudinal electron plasma waves.

Non-linear dispersion of water waves

1967 ◽  
Vol 27 (2) ◽  
pp. 399-412 ◽  
Author(s):  
G. B. Whitham
Keyword(s):  
Water Waves ◽  
Wave Train ◽  
Periodic Wave ◽  
Wave Number ◽  
Amplitude Wave ◽  
Linear Dispersion ◽  
Elliptic Case ◽  
Non Linear ◽  
Wave Trains ◽  
Linear Water Waves

The slow dispersion of non-linear water waves is studied by the general theory developed in an earlier paper (Whitham 1965b). The average Lagrangian is calculated from the Stokes expansion for periodic wave trains in water of arbitrary depth. This Lagrangian can be used for the various applications described in the above reference. In this paper, the crucial question of the ‘type’ of the differential equations for the wave-train parameters (local amplitude, wave-number, etc.) is established. The equations are hyperbolic or elliptic according to whetherkh0is less than or greater than 1.36, wherekis the wave-number per 2π andh0is the undisturbed depth. In the hyperbolic case, changes in the wave train propagate and the characteristic velocities give generalizations of the linear group velocity. In the elliptic case, modulations in the wave train grow exponentially and a periodic wave train will be unstable in this sense; thus, periodic wave trains on water will be unstable ifkh0> 1·36, The instability of deep-water waves,kh0> 1·36, was discovered in a different way by Benjamin (1966). The relation between the two approaches is explained.

Non-linear dispersion of cold plasma waves

1970 ◽  
Vol 4 (1) ◽  
pp. 109-125 ◽  
Author(s):  
Christopher K. W. Tam
Keyword(s):  
Cold Plasma ◽  
Wave Train ◽  
Plasma Waves ◽  
Periodic Wave ◽  
Wave Number ◽  
Linear Dispersion ◽  
Dispersive Waves ◽  
Elliptic Case ◽  
Non Linear ◽  
Fast Waves

The propagation of a packet of weakly non-linear dispersive cold plasma waves is studied by means of a two-time scale expansion. The effects of amplitude dispersion and the coupling to the mean plasma motion are taken into account. The governing equations are put into a conservation form. It is found that this system of equations is elliptic or hyperbolic depending on the wave-number of the dispersive waves. In the elliptic case modulations in the wave train grow exponentially in time and a periodic wave train will be unstable in this sense. In the hyperbolic case, slow variations in the wave train propagate and the characteristic velocities give a non-linear generalization of the linear group velocity. It is shown that except for waves which have their wave vectors nearly at right angle to the unperturbed magnetic field only fast waves with wave-number less than 0.585 (Ωi Ωe)½/Vα are stable; where Va is the Alfvén velocity and Ωι Ωε, are the ion and electron cyclotron frequencies respectively. A process of steepening of these waves into shocks with dissipation due to wave turbulence at the head of the wave packet is suggested.

First and Second Order Elastoplastic Analyses of Thin-Walled Metal Members using Generalized Beam Theory

2018 ◽  
Author(s):  
Miguel Abambres
Keyword(s):  
Finite Element ◽  
Stainless Steel ◽  
Cross Section ◽  
Beam Theory ◽  
Second Order ◽  
Flow Rule ◽  
Non Linear ◽  
Thin Walled ◽  
Shell Finite Element ◽  
Generalized Beam Theory

Original Generalized Beam Theory (GBT) formulations for elastoplastic first and second order (postbuckling) analyses of thin-walled members are proposed, based on the J2 theory with associated flow rule, and valid for (i) arbitrary residual stress and geometric imperfection distributions, (ii) non-linear isotropic materials (e.g., carbon/stainless steel), and (iii) arbitrary deformation patterns (e.g., global, local, distortional, shear). The cross-section analysis is based on the formulation by Silva (2013), but adopts five types of nodal degrees of freedom (d.o.f.) – one of them (warping rotation) is an innovation of present work and allows the use of cubic polynomials (instead of linear functions) to approximate the warping profiles in each sub-plate. The formulations are validated by presenting various illustrative examples involving beams and columns characterized by several cross-section types (open, closed, (un) branched), materials (bi-linear or non-linear – e.g., stainless steel) and boundary conditions. The GBT results (equilibrium paths, stress/displacement distributions and collapse mechanisms) are validated by comparison with those obtained from shell finite element analyses. It is observed that the results are globally very similar with only 9% and 21% (1st and 2nd order) of the d.o.f. numbers required by the shell finite element models. Moreover, the GBT unique modal nature is highlighted by means of modal participation diagrams and amplitude functions, as well as analyses based on different deformation mode sets, providing an in-depth insight on the member behavioural mechanics in both elastic and inelastic regimes.

Numerical analysis of trajectories in a Cassinian ion trap of second order with trap door ion inlet

2021 ◽  
Vol 27 (1) ◽  
pp. 3-12
Author(s):  
Bjoern Raupers ◽  
Hana Medhat ◽  
Juergen Grotemeyer ◽  
Frank Gunzer
Keyword(s):  
Ion Trap ◽  
Ion Traps ◽  
Second Order ◽  
Resolving Power ◽  
Periodic Pattern ◽  
Ion Motion ◽  
Trap Door ◽  
Mass Resolving Power ◽  
Long Time ◽  
The Fourier Transform

Ion traps like the Orbitrap are well known mass analyzers with very high resolving power. This resolving power is achieved with help of ions orbiting around an inner electrode for long time, in general up to a few seconds, since the mass signal is obtained by calculating the Fourier Transform of the induced signal caused by the ion motion. A similar principle is applied in the Cassinian Ion Trap of second order, where the ions move in a periodic pattern in-between two inner electrodes. The Cassinian ion trap has the potential to offer mass resolving power comparable to the Orbitrap with advantages regarding the experimental implementation. In this paper we have investigated the details of the ion motion analyzing experimental data and the results of different numerical methods, with focus on increasing the resolving power by increasing the oscillation frequency for ions in a high field ion trap. In this context the influence of the trap door, a tunnel through which the ions are injected into the trap, on the ion velocity becomes especially important.

scholarly journals Non-linear damping for scattering-passive systems in the Maxwell class

2020 ◽  
Vol 53 (2) ◽  
pp. 7458-7465
Author(s):  
Shantanu Singh ◽  
George Weiss ◽  
Marius Tucsnak
Keyword(s):  
Passive Systems ◽  
Non Linear ◽  
Linear Damping
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