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# Fundamentals of the Physical Theory of Diffraction, 2nd Edition

ISBN: 978-1-118-75366-8 May 2014 Wiley-IEEE Press 496 Pages

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## Description

The book is a complete, comprehensive description of the modern Physical Theory of Diffraction (PTD) based upon the concept of elementary edge waves. The theory is demonstrated with examples of the diffraction of acoustic and electromagnetic waves at perfectly reflecting objects.

Readers develop the skills to apply PTD to solve various scattering problems. The derived analytic expressions clearly illustrate the physical structure of the scattered field. They additionally describe all of the reflected and diffracted rays and beams, as well as the fields in the vicinity of caustics and foci. Shadow radiation, a fundamental component of PTD, is introduced and proven to contain half the total scattered power. The equivalence relationships between acoustic and electromagnetic diffracted waves are established and emphasized. Throughout the book, the author enables readers to master both the theory and its practical applications.

• Plotted numeric results supplement the theory and facilitate the visualization of individual contributions of distinct parts of the scattering objects to the total diffracted field
• Detailed comments help readers understand and implement all the critical steps of the analytic and numeric calculations
• Problem sets in each chapter give readers an opportunity to analyse and investigate the diffraction phenomena

Preface xiii

Foreword to the First Edition xv

Preface to the First Edition xix

Acknowledgments xxi

Introduction xxiii

1 Basic Notions in Acoustic and Electromagnetic Diffraction Problems 1

1.1 Formulation of the Diffraction Problem 1

1.2 Scattered Field in the Far Zone 3

1.3 Physical Optics 7

1.3.1 Definition of Physical Optics 7

1.3.2 Total Scattering Cross-Section 10

1.3.3 Optical Theorem 11

1.3.5 Shadow Contour Theorem and the Total Scattering Cross-Section 17

1.3.6 Shadow Radiation and Reflected Field in the Far Zone 20

1.3.7 Shadow Radiation and Reflection from Opaque Objects 22

1.4 Electromagnetic Waves 23

1.4.1 Basic Field Equations and PO Backscattering 23

1.4.2 PO Field Components: Reflected Field and Shadow Radiation 26

1.4.3 Electromagnetic Reflection and Shadow Radiation from Opaque Objects 28

1.5 Physical Interpretations of Shadow Radiation 31

1.5.1 Shadow Field and Transverse Diffusion 31

1.5.2 Fresnel Diffraction and Forward Scattering 32

1.6 Summary of Properties of Physical Optics Approximation 32

1.7 Nonuniform Component of an Induced Surface Field 33

Problems 36

2 Wedge Diffraction: Exact Solution and Asymptotics 49

2.1 Classical Solutions 49

2.2 Transition to Plane Wave Excitation 55

2.3 Conversion of the Series Solution to the Sommerfeld Integrals 57

2.4 The Sommerfeld Ray Asymptotics 61

2.5 The Pauli Asymptotics 63

2.6 Uniform Asymptotics: Extension of the Pauli Technique 68

2.7 Fast Convergent Integrals and Uniform Asymptotics: The “Magic Zero” Procedure 72

Problems 76

3 Wedge Diffraction: The Physical Optics Field 87

3.1 Original PO Integrals 87

3.2 Conversion of PO Integrals to the Canonical Form 90

3.3 Fast Convergent Integrals and Asymptotics for the PO Diffracted Field 94

Problems 100

4 Wedge Diffraction: Radiation by Fringe Components of Surface Sources 103

4.1 Integrals and Asymptotics 104

4.2 Integral Forms of Functions f (1) and g(1) 112

4.3 Oblique Incidence of a Plane Wave at a Wedge 114

4.3.1 Acoustic Waves 114

4.3.2 Electromagnetic Waves 118

Problems 120

5 First-Order Diffraction at Strips and Polygonal Cylinders 123

5.1 Diffraction at a Strip 124

5.1.1 Physical Optics Part of the Scattered Field 124

5.1.2 Total Scattered Field 128

5.1.3 Numerical Analysis of the Scattered Field 132

5.1.4 First-Order PTD with Truncated Scattering Sources j(1)h 135

5.2 Diffraction at a Triangular Cylinder 140

5.2.1 Symmetric Scattering: PO Approximation 141

5.2.2 Backscattering: PO Approximation 143

5.2.3 Symmetric Scattering: First-Order PTD Approximation 145

5.2.4 Backscattering: First-Order PTD Approximation 148

5.2.5 Numerical Analysis of the Scattered Field 150

Problems 152

6 Axially Symmetric Scattering of Acoustic Waves at Bodies of Revolution 157

6.1 Diffraction at a Canonical Conic Surface 158

6.1.1 Integrals for the Scattered Field 159

6.1.2 Ray Asymptotics 160

6.1.3 Focal Fields 166

6.1.4 Bessel Interpolations for the Field u(1)s,h 167

6.2 Scattering at a Disk 169

6.2.1 Physical Optics Approximation 169

6.2.2 Relationships Between Acoustic and Electromagnetic PO Fields 171

6.2.3 Field Generated by Fringe Scattering Sources 172

6.2.4 Total Scattered Field 173

6.3 Scattering at Cones: Focal Field 176

6.3.1 Asymptotic Approximations for the Field 176

6.3.2 Numerical Analysis of Backscattering 179

6.4 Bodies of Revolution with Nonzero Gaussian Curvature: Backscattered Focal Fields 183

6.4.1 PO Approximation 184

6.4.2 Total Backscattered Focal Field: First-Order PTD Asymptotics 186

6.4.3 Backscattering from Paraboloids 186

6.4.4 Backscattering from Spherical Segments 192

6.5 Bodies of Revolution with Nonzero Gaussian Curvature: Axially Symmetric Bistatic Scattering 196

6.5.1 Ray Asymptotics for the PO Field 196

6.5.2 Bessel Interpolations for the PO Field in the Region 𝜋 𝜔 𝜗 𝜋 200

6.5.3 Bessel Interpolations for the PTD Field in the Region 𝜋 𝜔 𝜗 𝜋 200

6.5.4 Asymptotics for the PTD Field in the Region 2𝜔 < 𝜗 𝜋 𝜔 Away from the GO Boundary 𝜗 = 2𝜔 201

6.5.5 Uniform Approximations for the PO Field in the Ray Region 2𝜔 𝜗 𝜋 𝜔, Including the GO Boundary 𝜗 = 2𝜔 202

6.5.6 Approximation of the PO Field in the Shadow Region for Reflected Rays 205

Problems 207

7 Elementary Acoustic and Electromagnetic Edge Waves 211

7.1 Elementary Strips on a Canonical Wedge 212

7.2 Integrals for j(1)s,h on Elementary Strips 213

7.3 Triple Integrals for Elementary Edge Waves 217

7.4 Transformation of Triple Integrals into One-Dimensional Integrals 220

7.5 General Asymptotics for Elementary Edge Waves 225

7.6 Analytic Properties of Elementary Edge Waves 230

7.7 Numerical Calculations of Acoustic Elementary Fringe Waves 234

7.8 Electromagnetic Elementary Edge Waves 237

7.8.1 Electromagnetic EEWs on the Diffraction Cone Outside the Wedge 241

7.8.2 Electromagnetic EEWs on the Diffraction Cone Inside the Wedge 243

7.8.3 Numerical Calculations of Electromagnetic Elementary Fringe Waves 245

7.9 Improved Theory of Elementary Edge Waves: Removal of the Grazing Singularity 245

7.9.1 Acoustic EEWs 248

7.9.2 Electromagnetic EEWs Generated by the Modified Nonuniform Current 253

7.10 Some References Related to Elementary Edge Waves 256

Problems 257

8 Ray and Caustic Asymptotics for Edge Diffracted Waves 261

8.1 Ray Asymptotics 261

8.1.1 Acoustic Waves 261

8.1.2 Electromagnetic Waves 266

8.1.3 Comments on Ray Asymptotics 267

8.2 Caustic Asymptotics 269

8.2.1 Acoustic waves 269

8.2.2 Electromagnetic Waves 274

8.3 Relationships between PTD and GTD 275

Problems 276

9 Multiple Diffraction of Edge Waves: Grazing Incidence and Slope Diffraction 285

9.1 Statement of the Problem and Related References 285

9.2 Grazing Diffraction 286

9.2.1 Acoustic Waves 286

9.2.2 Electromagnetic Waves 290

9.3 Slope Diffraction in Configuration of Figure 9.1 292

9.3.1 Acoustic Waves 292

9.3.2 Electromagnetic Waves 295

9.4 Slope Diffraction: General Case 296

9.4.1 Acoustic Waves 296

9.4.2 Electromagnetic Waves 299

Problems 302

10 Diffraction Interaction of Neighboring Edges on a Ruled Surface 305

10.1 Diffraction at an Acoustically Hard Surface 306

10.2 Diffraction at an Acoustically Soft Surface 309

10.3 Diffraction of Electromagnetic Waves 312

10.4 Test Problem: Secondary Diffraction at a Strip 314

10.4.1 Diffraction at a Hard Strip 314

10.4.2 Diffraction at a Soft Strip 317

Problems 318

11 Focusing of Multiple Acoustic Edge Waves Diffracted at a Convex Body of Revolution with a Flat Base 325

11.1 Statement of the Problem and its Characteristic Features 325

11.2 Multiple Hard Diffraction 327

11.3 Multiple Soft Diffraction 328

Problems 330

12 Focusing of Multiple Edge Waves Diffracted at a Disk 333

12.1 Multiple Hard Diffraction 334

12.2 Multiple Soft Diffraction 336

12.3 Multiple Diffraction of Electromagnetic Waves 340

Problems 341

13 Backscattering at a Finite-Length Cylinder 343

13.1 Acoustic Waves 343

13.1.1 PO Approximation 343

13.1.2 Backscattering Produced by the Nonuniform Component j(1) 347

13.1.3 Total Backscattered Field 352

13.2 Electromagnetic Waves 354

13.2.1 E-polarization 354

13.2.2 H-polarization 360

Problems 362

14 Bistatic Scattering at a Finite-Length Cylinder 365

14.1 Acoustic Waves 365

14.1.1 PO Approximation 366

14.1.2 Shadow Radiation as a Part of the Physical Optics Field 368

14.1.3 PTD for Bistatic Scattering at a Hard Cylinder 370

14.1.4 Beams and Rays of the Scattered Field 376

14.1.5 PO Shooting-Through Rays and Their Cancellation by Fringe Rays 381

14.1.6 Refined Asymptotics for the Specular Beam Reflected from the Lateral Surface 382

14.2 Electromagnetic Waves 386

14.2.1 E-Polarization 386

14.2.2 H-Polarization 388

14.2.3 Refined Asymptotics for the Specular Beam Reflected from the Lateral Surface 390

Problems 393

Conclusion 397

References 399

Appendix to Chapter 4: MATLAB Codes for Two-Dimensional Fringe Waves and Figures (F. Hacivelioglu and L. Sevgi) 411

Appendix to Chapter 6: MATLAB Codes for Axial Backscattering at Bodies of Revolution (F. Hacivelioglu and L. Sevgi) 431

Appendix to Section 7.7: MATLAB Codes for Diffraction Coefficients of Acoustic Elementary Fringe Waves (F. Hacivelioglu and L. Sevgi) 439

Appendix to Section 7.8.3: MATLAB Codes for Diffraction Coefficients of Electromagnetic Elementary Fringe Waves (F. Hacivelioglu and L. Sevgi) 443

Appendix to Section 7.9.2: Field d⃗E (0) mod Radiated by Modified Uniform Currents ⃗J (0) mod Induced on Elementary Strips (P. Ya. Ufimtsev) 447

Index 451