The nonlinear dispersion relation of geodesic acoustic modes

On the example of the Poynting–Thomson–Zener rheological model for solids, which exhibits both dissipation and wave propagation, with nonlinear dispersion relation, we introduce and investigate a finite difference numerical scheme. Our goal is to demonstrate its properties and to ease the computations in later applications for continuum thermodynamical problems. The key element is the positioning of the discretized quantities with shifts by half space and time steps with respect to each other. The arrangement is chosen according to the spacetime properties of the quantities and of the equations governing them. Numerical stability, dissipative error, and dispersive error are analyzed in detail. With the best settings found, the scheme is capable of making precise and fast predictions. Finally, the proposed scheme is compared to a commercial finite element software, COMSOL, which demonstrates essential differences even on the simplest—elastic—level of modeling.

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Nonlinear Bloch waves and balance between hardening and softening dispersion

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2018.0173 ◽

2018 ◽

Vol 474 (2217) ◽

pp. 20180173 ◽

Cited By ~ 10

Author(s):

M. I. Hussein ◽

R. Khajehtourian

Keyword(s):

Dispersion Relation ◽

Cell Size ◽

Unit Cell ◽

Homogeneous Layer ◽

Nonlinear Dispersion ◽

Elastic Band ◽

Bloch Waves ◽

Unit Cell Size ◽

Nonlinear Dispersion Relation ◽

Hardening And Softening

The introduction of nonlinearity alters the dispersion of elastic waves in solid media. In this paper, we present an analytical formulation for the treatment of finite-strain Bloch waves in one-dimensional phononic crystals consisting of layers with alternating material properties. Considering longitudinal waves and ignoring lateral effects, the exact nonlinear dispersion relation in each homogeneous layer is first obtained and subsequently used within the transfer matrix method to derive an approximate nonlinear dispersion relation for the overall periodic medium. The result is an amplitude-dependent elastic band structure that upon verification by numerical simulations is accurate for up to an amplitude-to-unit-cell length ratio of one-eighth. The derived dispersion relation allows us to interpret the formation of spatial invariance in the wave profile as a balance between hardening and softening effects in the dispersion that emerge due to the nonlinearity and the periodicity, respectively. For example, for a wave amplitude of the order of one-eighth of the unit-cell size in a demonstrative structure, the two effects are practically in balance for wavelengths as small as roughly three times the unit-cell size.

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Nonlinear dispersion relation in anharmonic periodic mass-spring and mass-in-mass systems

Journal of Sound and Vibration ◽

10.1016/j.jsv.2019.114929 ◽

2019 ◽

Vol 462 ◽

pp. 114929 ◽

Cited By ~ 1

Author(s):

R. Zivieri ◽

F. Garescì ◽

B. Azzerboni ◽

M. Chiappini ◽

G. Finocchio

Keyword(s):

Dispersion Relation ◽

Nonlinear Dispersion ◽

Nonlinear Dispersion Relation ◽

Mass Spring

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Envelope solitons of surface waves in a plasma column

Journal of Plasma Physics ◽

10.1017/s0022377800012691 ◽

1987 ◽

Vol 38 (3) ◽

pp. 427-437 ◽

Cited By ~ 21

Author(s):

D. Grozev ◽

A. Shivarova ◽

A. D. Boardman

Keyword(s):

Dispersion Relation ◽

Surface Waves ◽

Plasma Column ◽

Soliton Solutions ◽

Elliptic Functions ◽

Nonlinear Dispersion ◽

Dark Solitons ◽

Envelope Solitons ◽

Nonlinear Dispersion Relation ◽

Universal Procedure

The problem of envelope solitons of surface waves is considered on the basis of results for the nonlinear dispersion relation of the waves in a plasma column. The soliton solutions are derived as particular cases of the general solutions obtained by a universal procedure and expressed in terms of Jacobi elliptic functions. Since the two types of interactions, namely the (ω + ω) – ω and the (ω – ω) + ω interactions (where ω is the frequency of the carrier wave) included in the nonlinear dispersion relation act in opposite ways, existence both of bright and dark solitons is shown to be possible. The effect of the ponderomotive force that in our case is expressed through the contribution of the (ω – ω) + ω interaction leads to the formation of dark solitons. The effect of the losses is also considered.

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Plasma maser theory of the extraordinary mode in the presence of Langmuir turbulence

Journal of Plasma Physics ◽

10.1017/s0022377800016159 ◽

1991 ◽

Vol 46 (2) ◽

pp. 331-346 ◽

Cited By ~ 11

Author(s):

S. N. Sarma ◽

K. K. Sarma ◽

M. Nambu

Keyword(s):

Growth Rate ◽

Electron Beam ◽

Dispersion Relation ◽

Space Plasma ◽

Conversion Process ◽

Nonlinear Dispersion ◽

Langmuir Turbulence ◽

Nonlinear Dispersion Relation ◽

Maser Effect ◽

Scope Of Application

The emission of extraordinary mode radiation in a plasma with Langmuir turbulence driven by an electron beam is considered. The process of emission considered in this paper is the plasma maser effect, which is essentially an energy up-conversion process. The energy necessary for the growth of the extraordinary mode is derived from the Langmuir turbulence. The nonlinear dispersion relation of the extraordinary mode in the presence of Langmuir turbulence is obtained and its growth rate calculated. The scope of application of the results to space-plasma observation is then stressed.

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Parametric decay in a two-electron-temperature plasma

Journal of Plasma Physics ◽

10.1017/s0022377800014896 ◽

1990 ◽

Vol 43 (3) ◽

pp. 451-456

Author(s):

S. Guha ◽

Meenu Asthana

Keyword(s):

Growth Rate ◽

Dispersion Relation ◽

Electron Temperature ◽

Pump Wave ◽

Nonlinear Dispersion ◽

Hybrid Wave ◽

Temperature Plasma ◽

Parametric Decay ◽

Electromagnetic Pump ◽

Nonlinear Dispersion Relation

Nonlinear decay of an ordinary electromagnetic pump wave into an electro-acoustic wave and an upper-hybrid wave in a two-electron-temperature plasma has been investigated analytically. In contrast with the work of Sharma, Ramamurthy & Yu (1984), it is found that the decay can take place in the absence of the restrictive condition Ti ≫ Te and the plasma be magnetized. Using a hydrodynamical model of the plasma, the nonlinear dispersion relation and growth rate are obtained. A comparison of the present investigation is made with earlier work, and its possible application to the ELMO bumpy torus is discussed.

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Parametric instabilities with well separated frequencies in magnetized plasmas

Journal of Plasma Physics ◽

10.1017/s0022377800022947 ◽

1981 ◽

Vol 25 (1) ◽

pp. 73-79 ◽

Cited By ~ 4

Author(s):

Yu-Mei Chou ◽

Shih-Tung Tsai

Keyword(s):

Dispersion Relation ◽

Ponderomotive Force ◽

Dipole Approximation ◽

Magnetized Plasma ◽

Nonlinear Dispersion ◽

Anomalous Absorption ◽

Force Method ◽

Nonlinear Dispersion Relation ◽

Parametric Instabilities ◽

The Right

A nonlinear dispersion relation is derived for a homogeneous, magnetized plasma by the conventional ‘ponderomotive force’ method. By using this dispersion relation, various parametric instabilities (stimulated anomalous absorptions and scatterings) are discussed. The results are compared in detail with those obtained by other authors. In the validity region of this model the results derived by previous authors can be recovered. In addition to obtaining a few instabilities which have not been published in the literature, we also find that in general (i) the ‘dipole approximation’ can give the right results for the stimulated anomalous absorption problems and (ii) the ambient magnetic field may enhance the plasma stability.

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