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

Modern plasma physics, encompassing wave-particle interactions and collec- tive phenomena characteristic of the collision-free nature of hot plasmas, was founded in 1946 when 1. D. Landau published his analysis of linear (small- amplitude) waves in such plasmas. It was not until some ten to twenty years later, however, with impetus from the then rapidly developing controlled- fusion field, that sufficient attention was devoted, in both theoretical and experimental research, to elucidate the importance and ramifications of Landau's original work. Since then, with advances in laboratory, fusion, space, and astrophysical plasma research, we have witnessed important devel- opments toward the understanding of a variety of linear as well as nonlinear plasma phenomena, including plasma turbulence. Today, plasma physics stands as a well-developed discipline containing a unified body of powerful theoretical and experimental techniques and including a wide range of appli- cations. As such, it is now frequently introduced in university physics and engineering curricula at the senior and first-year-graduate levels. A necessary prerequisite for all of modern plasma studies is the under- standing oflinear waves in a temporally and spatially dispersive medium such as a plasma, including the kinetic (Landau) theory description of such waves. Teaching experience has usually shown that students (seniors and first-year graduates), when first exposed to the kinetic theory of plasma waves, have difficulties in dealing with the required sophistication in multidimensional complex variable (singular) integrals and transforms.
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Table of Contents

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.- The Functions, Q+, Q- and Q0.- Analytic Continuation of Hilbert Transforms.- Hilbert Transform as a Fourier-Laplace Transform.- Asymptotic Expansion of Hilbert Transforms.- 1.5.3. Other Properties of Hilbert Transforms.- Properties for Complex z.- 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.- Dispersion Relation and Possible Modes.- 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.- Finite-Size Model with Constant Density.- Infinite-Plasma Model with Realistic Density Profiles.- 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.- Inversion Procedure.- A Cold Plasma.- 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.- Behavior of I+ (z) Close to the Antenna.- Contribution of the Ion-Acoustic Wave.- Asymptotic Ion Contribution.- 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.

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