Filled with new approaches and basic results connected with the discontinuities of the electromagnetic field, this new book offers an important resource for graduate and undergraduate students. The discontinuities in question may be one of three things: The bounded jump discontinuities on the interfaces between two media or on the material sheets which model very thin layers or; Unbounded values at the edge of wedge type structures; Unbounded values at the tips of conical structures. The book contains many examples as well as problems and exercises that challenge the readers.
Filled with new approaches and basic results connected with the discontinuities of the electromagnetic field, this new book offers an important resource for graduate and undergraduate students. The discontinuities in question may be one of three things: The bounded jump discontinuities on the interfaces between two media or on the material sheets which model very thin layers or; Unbounded values at the edge of wedge type structures; Unbounded values at the tips of conical structures. The book contains many examples as well as problems and exercises that challenge the readers.
M. Mithat Idemen, PhD, is Professor in the Mathematics Department of Yeditepe University (Istanbul, Turkey) and honorary member of the Turkish Academy of Sciences (TüBA). He has served as editor and associate editor for various scientific journals, including the International Series of Monographs on Advanced Electromagnetics.
Inhaltsangabe
Preface ix 1. Introduction 1 2. Distributions and Derivatives in the Sense of Distribution 7 2.1 Functions and Distributions 7 2.2 Test Functions. The Space C infinity 0 9 2.3 Convergence in D 14 2.4 Distribution 16 2.5 Some Simple Operations in D 21 2.5.1 Multiplication by a Real Number or a Function 21 2.5.2 Translation and Rescaling 21 2.5.3 Derivation of a Distribution 22 2.6 Order of a Distribution 26 2.7 The Support of a Distribution 31 2.8 Some Generalizations 33 2.8.1 Distributions on Multidimensional Spaces 33 2.8.2 Vector-Valued Distributions 38 3. Maxwell Equations in the Sense of Distribution 49 3.1 Maxwell Equations Reduced into the Vacuum 49 3.1.1 Some Simple Examples 53 3.2 Universal Boundary Conditions and Compatibility Relations 54 3.2.1 An Example. Discontinuities on a Combined Sheet 57 3.3 The Concept of Material Sheet 59 3.4 The Case of Monochromatic Fields 62 3.4.1 Discontinuities on the Interface Between Two Simple Media that Are at Rest 64 4. Boundary Conditions on Material Sheets at Rest 67 4.1 Universal Boundary Conditions and Compatibility Relations for a Fixed Material Sheet 67 4.2 Some General Results 69 4.3 Some Particular Cases 70 4.3.1 Planar Material Sheet Between Two Simple Media 70 4.3.2 Cylindrically or Spherically Curved Material Sheet Located Between Two Simple Media 91 4.3.3 Conical Material Sheet Located Between Two Simple Media 93 5. Discontinuities on a Moving Sheet 109 5.1 Special Theory of Relativity 110 5.1.1 The Field Created by a Uniformly Moving Point Charge 112 5.1.2 The Expressions of the Field in a Reference System Attached to the Charged Particle 114 5.1.3 Lorentz Transformation Formulas 115 5.1.4 Transformation of the Electromagnetic Field 118 5.2 Discontinuities on a Uniformly Moving Surface 120 5.2.1 Transformation of the Universal Boundary Conditions 123 5.2.2 Transformation of the Compatibility Relations 126 5.2.3 Some Simple Examples 126 5.3 Discontinuities on a Nonuniformly Moving Sheet 138 5.3.1 Boundary Conditions on a Plane that Moves in a Direction Normal to Itself 139 5.3.2 Boundary Conditions on the Interface of Two Simple Media 143 6. Edge Singularities on Material Wedges Bounded by Plane Boundaries 149 6.1 Introduction 149 6.2 Singularities at the Edges of Material Wedges 153 6.3 The Wedge with Penetrable Boundaries 154 6.3.1 The H Case 156 6.3.2 The E Case 171 6.4 The Wedge with Impenetrable Boundaries 174 6.5 Examples. Application to Half-Planes 175 6.6 Edge Conditions for the Induced Surface Currents 176 7. Tip Singularities at the Apex of a Material Cone 179 7.1 Introduction 179 7.2 Algebraic Singularities of an H-Type Field 185 7.2.1 Contribution of the Energy Restriction 185 7.2.2 Contribution of the Boundary Conditions 186 7.3 Algebraic Singularities of an E-Type Field 191 7.4 The Case of Impenetrable Cones 193 7.5 Confluence and Logarithmic Singularities 195 7.6 Application to some Widely used Actual Boundary Conditions 197 7.7 Numerical Solutions of the Transcendental Equations Satisfied by the Minimal Index 200 7.7.1 The Case of Very Sharp Tip 200 7.7.2 The Case of Real-Valued Minimal v 201 7.7.3 A Function-Theoretic Method to Determine Numerically the Minimal v 203 8. Temporal Discontinuities 209 8.1 Universal Initial Conditions 209 8.2 Linear Mediums in the Generalized Sense 211 8.3 An Illustrative Example 212 References 215 Index 219 IEEE Press Series on Electromagnetic Wave Theory
Preface ix 1. Introduction 1 2. Distributions and Derivatives in the Sense of Distribution 7 2.1 Functions and Distributions 7 2.2 Test Functions. The Space C infinity 0 9 2.3 Convergence in D 14 2.4 Distribution 16 2.5 Some Simple Operations in D 21 2.5.1 Multiplication by a Real Number or a Function 21 2.5.2 Translation and Rescaling 21 2.5.3 Derivation of a Distribution 22 2.6 Order of a Distribution 26 2.7 The Support of a Distribution 31 2.8 Some Generalizations 33 2.8.1 Distributions on Multidimensional Spaces 33 2.8.2 Vector-Valued Distributions 38 3. Maxwell Equations in the Sense of Distribution 49 3.1 Maxwell Equations Reduced into the Vacuum 49 3.1.1 Some Simple Examples 53 3.2 Universal Boundary Conditions and Compatibility Relations 54 3.2.1 An Example. Discontinuities on a Combined Sheet 57 3.3 The Concept of Material Sheet 59 3.4 The Case of Monochromatic Fields 62 3.4.1 Discontinuities on the Interface Between Two Simple Media that Are at Rest 64 4. Boundary Conditions on Material Sheets at Rest 67 4.1 Universal Boundary Conditions and Compatibility Relations for a Fixed Material Sheet 67 4.2 Some General Results 69 4.3 Some Particular Cases 70 4.3.1 Planar Material Sheet Between Two Simple Media 70 4.3.2 Cylindrically or Spherically Curved Material Sheet Located Between Two Simple Media 91 4.3.3 Conical Material Sheet Located Between Two Simple Media 93 5. Discontinuities on a Moving Sheet 109 5.1 Special Theory of Relativity 110 5.1.1 The Field Created by a Uniformly Moving Point Charge 112 5.1.2 The Expressions of the Field in a Reference System Attached to the Charged Particle 114 5.1.3 Lorentz Transformation Formulas 115 5.1.4 Transformation of the Electromagnetic Field 118 5.2 Discontinuities on a Uniformly Moving Surface 120 5.2.1 Transformation of the Universal Boundary Conditions 123 5.2.2 Transformation of the Compatibility Relations 126 5.2.3 Some Simple Examples 126 5.3 Discontinuities on a Nonuniformly Moving Sheet 138 5.3.1 Boundary Conditions on a Plane that Moves in a Direction Normal to Itself 139 5.3.2 Boundary Conditions on the Interface of Two Simple Media 143 6. Edge Singularities on Material Wedges Bounded by Plane Boundaries 149 6.1 Introduction 149 6.2 Singularities at the Edges of Material Wedges 153 6.3 The Wedge with Penetrable Boundaries 154 6.3.1 The H Case 156 6.3.2 The E Case 171 6.4 The Wedge with Impenetrable Boundaries 174 6.5 Examples. Application to Half-Planes 175 6.6 Edge Conditions for the Induced Surface Currents 176 7. Tip Singularities at the Apex of a Material Cone 179 7.1 Introduction 179 7.2 Algebraic Singularities of an H-Type Field 185 7.2.1 Contribution of the Energy Restriction 185 7.2.2 Contribution of the Boundary Conditions 186 7.3 Algebraic Singularities of an E-Type Field 191 7.4 The Case of Impenetrable Cones 193 7.5 Confluence and Logarithmic Singularities 195 7.6 Application to some Widely used Actual Boundary Conditions 197 7.7 Numerical Solutions of the Transcendental Equations Satisfied by the Minimal Index 200 7.7.1 The Case of Very Sharp Tip 200 7.7.2 The Case of Real-Valued Minimal v 201 7.7.3 A Function-Theoretic Method to Determine Numerically the Minimal v 203 8. Temporal Discontinuities 209 8.1 Universal Initial Conditions 209 8.2 Linear Mediums in the Generalized Sense 211 8.3 An Illustrative Example 212 References 215 Index 219 IEEE Press Series on Electromagnetic Wave Theory
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