An Introduction to the Linear Theories and Methods of Electrostatic Waves in Plasmas

1. The Cookbook: Fourier, Laplace, and Hilbert Transforms.- 1.1. Introduction.- 1.2. A Basic Example: Electromagnetic-Wave Propagation in Vacuum.- 1.3. The Fourier-Laplace Transforms.- 1.3.1. Fourier Transform in Space.- 1.3.2. Laplace Transform in Time.- 1.3.3. Inversion of Fourier-Laplace Transforms.- 1.4. Laplace Transforms and Causality.- 1.4.1. Comparison of Fourier and Laplace Transforms.- 1.4.2. Causality and Transient Motion.- 1.4.3. Kramers-Kronig Relations.- 1.4.4. The Red-Filter Paradox.- 1.5. Hilbert Transforms.- 1.5.1. Averages.- 1.5.2 Basic Properties of Hilbert Transforms.- 1.5.2.1. The Functions, Q+, Q- and Q0.- 1.5.2.2. Analytic Continuation of Hilbert Transforms.- 1.5.2.3. Hilbert Transform as a Fourier-Laplace Transform.- 1.5.2.4. Asymptotic Expansion of Hilbert Transforms.- 1.5.3. Other Properties of Hilbert Transforms.- 1.5.3.1. Properties for Complex z.- 1.5.3.2. Properties for Real z.- 1.6. Appendix A: Functions of Complex Variables.- A.1. Definitions.- A.2. Cauchy-Riemann Conditions.- A.3. Cauchy’s Theorem, Morera’s Theorem, and Cauchy’s Formula.- A.4. Taylor Expansion of an Analytic Function.- A.5. Laurent Expansion.- A.6. Classification of Isolated Singularities.- A.7. Meromorphic Functions.- A.8. Residues.- A.9. Transformation of the Path of Integration.- A.10. Principal Part (in the Cauchy Sense).- A.11. Analytic Continuation.- 2. Waves in a Conductivity-Tensor-Defined Medium: A Cold-Plasma Example.- 2.1. Introduction.- 2.2. Waves in Idealized Media.- 2.3. Waves in Plasmas.- 2.3.1. Some General Remarks on Solving Maxwell’s Equations.- 2.3.2. Dispersion Relations.- 2.3.3. Purely Electrostatic and Purely Electromagnetic Waves.- 2.3.4. General Dispersion Relation with ?ex = 0 = jex.- 2.4. Waves in a Cold Plasma.- 2.4.1. Cold-Plasma Dielectric Tensor.- 2.4.2. Cold-Plasma Dispersion Relation.- 2.4.3. Cutoffs and Resonances.- 2.4.4. Propagation Parallel to the Applied Magnetic Field.- 2.4.5. Propagation Perpendicular to the Applied Magnetic Field.- 2.5. Applications of the Cold-Plasma-Theory Results.- 2.5.1. Using the Ordinary Wave to Measure Plasma Density.- 2.5.2. Using the Extraordinary Wave to Measure Plasma Density.- 2.5.3. Some General Comments on the Use of Microwaves to Measure Plasma Parameters.- 2.6. Selected Experiment: A Simple Transmission Experiment Using the Extraordinary Wave to Measure Plasma Density.- 3. Electrostatic Waves in a Warm Plasma: A Fluid-Theory Example.- 3.1. Introduction.- 3.2. Dispersion Relation for Purely Electrostatic Waves in a Warm Plasma.- 3.3. Electrostatic Modes in a Warm Plasma.- 3.3.1. Electron Plasma Waves.- 3.3.2. Ion-Acoustic Waves in a Two-Component Electron-Ion Plasma with Te ? Ti.- 3.3.3. Ion-Acoustic Waves in a Three-Component Electron-Positive-Ion-Negative-Ion Plasma with Te ? Ti.- 3.4. Selected Experiments.- 3.4.1. Ion-Acoustic Waves.- 3.4.2. Electron Plasma Waves.- 4. Ion-Acoustic Waves with Ion-Neutral and Electron-Neutral Collisions.- 4.1. Introduction.- 4.2. Dispersion Relation with Collisions.- 4.3. Initial-Value Problem.- 4.4. Boundary-Value Problem.- 4.5. Selected Experiment: Boundary-Value Problem for Ion-Acoustic Waves in a Collision-Dominated Discharge Plasma.- 5. Finite-Size-Geometry Effects.- 5.1. Introduction.- 5.2. Electron Plasma Waves in a Cold Plasma Supported by a Strong Magnetic Field.- 5.2.1. General Wave Equation.- 5.2.2. Propagation in a Plasma-Filled Waveguide.- 5.2.2.1. Dispersion Relation and Possible Modes.- 5.2.2.2. Electrostatic and Electromagnetic Properties.- 5.3. Ion-Acoustic Waves in a Warm Plasma Supported by a Strong Magnetic Field.- 5.3.1. General Wave Equation.- 5.3.2. Propagation in a Plasma-Filled Waveguide.- 5.3.2.1. Finite-Size Model with Constant Density.- 5.3.2.2. Infinite-Plasma Model with Realistic Density Profiles.- 5.3.2.3. Applicability of the Finite-Size Model To Electron Plasma Waves.- 5.4. Selected Experiments.- 5.4.1. Ion-Acoustic Wave Propagation in a Plasma-Filled Glass Tube.- 5.4.2. Ion-Acoustic Wave Propagation in a Cylindrical Cesium Plasma.- 6. Ion-Acoustic Waves in a Small Density Gradient.- 6.1. Introduction.- 6.2. Wave Equation.- 6.3. Wave Propagation in a Nonuniform Plasma Having a Gaussian Density Profile.- 6.4. Wave Propagation in a Nonuniform Plasma Having an Arbitrary Density Profile.- 6.5. Wave Propagation in a Uniform Plasma Having a Subsonic Density Gradient at its Edge.- 6.6. Selected Experiments.- 6.6.1. Short-Wavelength Ion-Acoustic Waves in Small Density Gradients.- 6.6.2. Reflection of Long-Wavelength Ion-Acoustic Waves in Small Density Gradients.- 7. Landau Damping: An Initial-Value Problem.- 7.1. Introduction.- 7.2. Collisionless Damping Due to Free Streaming.- 7.3. Longitudinal Oscillations in an Infinite, Homogeneous Plasma with No Applied Fields—The Electron Plasma Wave.- 7.3.1. Using Fourier-Laplace Transforms to Solve the Coupled Poisson and Vlasov Equations.- 7.3.2. Free-Streaming and Collective Contributions to the Electric Field.- 7.3.3. Time Evolution of the Electric Field.- 7.3.3.1. Inversion Procedure.- 7.3.3.2. A Cold Plasma.- 7.3.3.3. A Lorentzian Plasma.- 7.3.4. Long-Wavelength Oscillations.- 7.3.5. Maxwellian Plasma.- 7.4. Ion-Acoustic Waves.- 7.4.1. Two-Component Maxwellian Plasma.- 7.4.2. Isothermal Plasma.- 7.4.3. Te ? Ti.- 8. Kinetic Theory of Forced Oscillations in a One-Dimensional Warm Plasma.- 8.1. Introduction.- 8.2. Microscopic Theory of Forced Oscillations.- 8.2.1. The Trajectory Method.- 8.2.2. The Fourier-Laplace Method.- 8.3. Difficulties Encountered in the Forced-Oscillations Problem.- 8.4. Free-Streaming and Collective Effects.- 8.4.1 Damping Associated with Free-Streaming—Pseudowaves.- 8.4.2 Damping Associated with Collective Effects—Asymptotic Perturbations.- 8.5. Physical Meaning of Landau Damping.- 8.5.1. Kinetic Energy in a Homogeneous One-Dimensional Plasma.- 8.5.2. Energy Density Deposited in a Plasma Excited by a Dipole.- 9. Computing Techniques for Electrostatic Perturbations.- 9.1. Introduction.- 9.2. Dielectric Constant of a Maxwellian Electron Cloud.- 9.2.1. Dielectric Constant.- 9.2.2. Roots of ?+ (f, ?).- 9.2.3. Expansion of 1/?+ in Partial Fractions.- 9.2.4. Additional Properties of the Dielectric Constant.- 9.3. The Gould Technique.- 9.3.1. Principle.- 9.3.2 Some Properties of I± (z).- 9.3.3 Some Comments on the Calculation of the Laplace Transforms, I± (z).- 9.4. The Derfler-Simonen Technique.- 9.4.1. Principle.- 9.4.2. Discussion of the Method.- 9.5. The Hybrid Technique.- 9.5.1. Principle.- 9.5.2. Discussion of the Method.- 9.6. Conclusions.- 9.7. Appendix: Plasma Wave Functions.- 10. Ion-Acoustic Waves in Maxwellian Plasmas: A Boundary-Value Problem.- 10.1. Introduction.- 10.2. Dispersion Relation for Ion-Acoustic Waves.- 10.2.1. The Exact Dielectric Constant and the Boltzmann Approximation.- 10.2.2. The Roots of the Dispersion Relation.- 10.2.3. Expansion of ?± (f, n) in Partial Fractions.- 10.3. Ion-Acoustic Waves in an Isothermal Plasma.- 10.3.1. Interpretation of Gould’s Numerical Results.- 10.3.1.1. Behavior of I+ (z) Close to the Antenna.- 10.3.1.2. Contribution of the Ion-Acoustic Wave.- 10.3.1.3. Asymptotic Ion Contribution.- 10.3.1.4. Asymptotic Electron Contribution.- 10.3.2. Density Perturbations.- 10.3.3. Remarks about Wong et al.’s Experiment.- 10.3.4. Collective Effects in an Isothermal Plasma.- 10.4. Selected Experiment: Landau Damping of Ion-Acoustic Waves in a Nonisothermal Plasma.- 11. Numerical Methods.- 11.1. Introduction.- 11.2. Numerical Evaluation of Hilbert Transforms.- 11.2.1. Calculation of Q+ (z) by Integration.- 11.2.2. Series Expansion.- 11.2.3. Asymptotic Expansion.- 11.2.4. Other Methods.- 11.2.5. The Subroutine ZNDEZ.- 11.2.6. Checking ZNDEZ.- 11.3. Hunting the Roots of a Dispersion Relation.- 11.3.1. Newton’s Method.- 11.3.2. Initialization of the Roots.- 11.3.3. Numerical Solution.- 11.4. Appendix.- 11.4.1. Program ZNDEZ.- 11.4.2. Listing of the Program HUNT in BASIC.- References.