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

ISBN: 978-1-118-84869-2 April 2014 Wiley-IEEE Press 496 Pages

## 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

1.4 Electromagnetic Waves / 23

1.4.1 Basic Field Equations and PO Backscattering / 23

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 AcousticWaves 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 − ≤ ≤ / 200

6.5.3 Bessel Interpolations for the PTD Field in the Region − ≤ ≤ / 200

6.5.4 Asymptotics for the PTD Field in the Region 2 < ≤ − Away from the GO Boundary = 2 / 201

6.5.5 Uniform Approximations for the PO Field in the Ray Region 2 ≤ ≤ − , Including the GO Boundary = 2 / 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