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TEAM LinG

Fundamentals of the

Physical Theory of

Diffraction

TEAM LinG

Fundamentals of the

Physical Theory of

Diffraction

Pyotr Ya. Uﬁmtsev

TEAM LinG

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/ permission.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and speciﬁcally disclaim any implied warranties of merchantability or ﬁtness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of proﬁt or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

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Library of Congress Cataloging-in-Publication Data.

Uﬁmstev, Pyotr Yakovlevich.

Fundamentals of the physical theory of diffraction / by Pyotr Uﬁmstev. p. cm.

“A Wiley–Interscience publication.”

Includes bibliographical references and index. ISBN: 978-0-470-09771-7

1. Electromagnetic waves- -Diffraction. 2. Diffractive scattering. I. Title.

QC665.D5U35 2007
621.382 4- -dc22	2006049684

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

TEAM LinG

Contents

Foreword					xi
Preface					xv

Acknowledgments					xvii

Introduction					1

1. Basic Notions in Acoustic and Electromagnetic Diffraction Problems					5

1.1	Formulation of the Diffraction Problem			5
1.2	Scattered Field in the Far Zone		7
1.3	Physical Optics	11

		1.3.1 Deﬁnition of the Physical Optics						11
		1.3.2	Total Scattering Cross-Section					13
		1.3.3	Optical	Theorem		15
		1.3.4 Introducing the Notion of					“Shadow Radiation”			16
		1.3.5 Shadow Contour Theorem and theTotal Scattering
			Cross-Section		21
		1.3.6 Summary of Properties of Physical
			Optics Approximation				24
1.4	Nonuniform Component of Induced Surface Field								25
1.5	Electromagnetic Waves				27
Problems		31
2. Wedge Diffraction: Exact Solution and Asymptotics											33

2.1	Classical Solutions			33
2.2	Transition to the Plane Wave Excitation							38
2.3	Conversion of the Series Solution to the Sommerfeld Integrals										40
2.4	The Sommerfeld Ray Asymptotics						44
2.5	The Pauli Asymptotics				47
2.6	Uniform Asymptotics: Extension of the Pauli Technique									51
2.7	Comments on Alternative Asymptotics							55
Problems		56

TEAM LinG

vi Contents
3. Wedge Diffraction: The Physical Optics Field				59

3.1	Original PO Integrals	59
3.2	Conversion of the PO Integrals to the Canonical Form			61
3.3	Ray Asymptotics for the PO Diffracted Field		67

Problems		68
4. Wedge Diffraction: Radiation by the Nonuniform Component
of Surface Sources										71

4.1	Integrals and Asymptotics				71
4.2	Integral Form of Functions f (1) and g(1)						76
4.3	Oblique Incidence of a Plane Wave at a Wedge							78
Problems		82
5. First-Order Diffraction at Strips and Polygonal Cylinders										83

5.1	Diffraction at a Strip			83
		5.1.1	Physical Optics Part of the Scattered Field					85
		5.1.2	Total Scattered Field			87
		5.1.3	Numerical Analysis of the Scattered Field					92
		5.1.4	First-Order PTD with Truncated Scattering Sources jh(1)							95
5.2	Diffraction at a Triangular Cylinder					99
		5.2.1	Symmetric Scattering: PO Approximation					101
		5.2.2	Backscattering: PO Approximation				102
		5.2.3	Symmetric Scattering: First-Order PTD Approximation							104
		5.2.4	Backscattering: First-Order PTD Approximation						107
		5.2.5	Numerical Analysis of the Scattered Field					110
Problems		112
6. Axially Symmetric Scattering of Acoustic Waves at
Bodies of Revolution										115

6.1	Diffraction at a Canonical Conic Surface						115
		6.1.1	Integrals for the Scattered Field				117
		6.1.2	Ray Asymptotics		118
		6.1.3	Focal Fields		124
		6.1.4	Bessel Interpolations for the Field u(1)					125
							s,h
6.2	Scattering at a Disk			126
		6.2.1	Physical Optics Approximation				127
		6.2.2	Field Generated by Nonuniform Scattering Sources							130
		6.2.3	Total Scattered Field			132

TEAM LinG

Contents vii


6.3 Scattering at Cones: Focal Field		134
6.3.1	Asymptotic Approximations for the Field		134
6.3.2	Numerical Analysis of Backscattering		138

6.4Bodies of Revolution with Nonzero Gaussian Curvature:

Backscattered Focal Fields			141
6.4.1	PO Approximation		143
6.4.2 Total Backscattered Focal Field: First-Order PTD
	Asymptotics	145
6.4.3	Backscattering from Paraboloids			145
6.4.4	Backscattering from Spherical Segments				151

6.5Bodies of Revolution with Nonzero Gaussian Curvature:

Axially Symmetric Bistatic Scattering	155
6.5.1 Ray Asymptotics for the PO Field	156

6.5.2Bessel Interpolations for the PO Field in the

Region π − ω ≤ ϑ ≤ π 159

6.5.3Bessel Interpolations for the PTD Field in the

Region π − ω ≤ ϑ ≤ π 160

6.5.4 Asymptotics for the PTD Field in the Region

2ω < ϑ ≤ π − ω away from the GO Boundary ϑ = 2ω 161

6.5.5Uniform Approximations for the PO Field in the Ray Region

			2ω ≤ ϑ ≤ π − ω Including the GO Boundary ϑ = 2ω						161
		6.5.6 Approximation for the PO Field in the Shadow
			Region for Reﬂected Rays		165
Problems		166
7. Elementary Acoustic and Electromagnetic Edge Waves									169

7.1	Elementary Strips on a Canonical Wedge					170
7.2	Integrals for js,h(1) on Elementary Strips				171
7.3	Triple Integrals for Elementary Edge Waves					175
7.4	Transformation of Triple Integrals into One-Dimensional Integrals								178
7.5	General Asymptotics for Elementary Edge Waves							183
7.6	Analytic Properties of Elementary Edge Waves						187
7.7	Numerical Calculations of Elementary Edge Waves							191
7.8	Electromagnetic Elementary Edge Waves					194
7.9	Improved Theory of Elementary Edge Waves						198
		7.9.1	Acoustic EEWs	199
		7.9.2	Electromagnetic EEWs		203
7.10	Some References Related to Elementary Edge Waves							209
Problems		210

TEAM LinG

viii Contents
8. Ray and Caustics Asymptotics for Edge Diffracted Waves											213

8.1	Ray Asymptotics			213
		8.1.1	Acoustic Waves		213
		8.1.2	Electromagnetic Waves			218
		8.1.3	Comments on Ray Asymptotics					219
8.2	Caustic Asymptotics			220
		8.2.1	Acoustic Waves		221
		8.2.2	Electromagnetic Waves			225
Problems		226
9. Multiple Diffraction of Edge Waves: Grazing Incidence and Slope
Diffraction											229

9.1	Statement of the Problem and Related References								229
9.2	Grazing Diffraction			230
		9.2.1	Acoustic Waves		230
		9.2.2	Electromagnetic Waves			234
9.3	Slope Diffraction in the Conﬁguration of Figure 9.1									236
		9.3.1	Acoustic Waves		236
		9.3.2	Electromagnetic Waves			238
9.4	Slope Diffraction: General Case					240
		9.4.1	Acoustic Waves		240
		9.4.2	Electromagnetic Waves			243
Problems		245
10. Diffraction Interaction of Neighboring Edges on a Ruled Surface											247

10.1	Diffraction at an Acoustically Hard Surface							248
10.2	Diffraction at an Acoustically Soft Surface							250
10.3	Diffraction of Electromagnetic Waves						252
Problems		254
11. Focusing of Multiple Acoustic Edge Waves Diffracted at a Convex
Body of Revolution with a Flat Base											255

11.1	Statement of the Problem and its Characteristic Features									255
11.2	Multiple Hard Diffraction				256
11.3	Multiple Soft Diffraction				258
Problems		260

TEAM LinG

						Contents	ix
12.	Focusing of Multiple Edge Waves Diffracted at a Disk						261

	12.1	Multiple Hard Diffraction			261
	12.2	Multiple Soft Diffraction			264
	12.3	Multiple Diffraction of Electromagnetic Waves				267
	Problems		268
13.	Backscattering at a Finite-Length Cylinder						269

	13.1	Acoustic Waves		269
			13.1.1 PO Approximation		269

13.1.2Backscattering Produced by the Nonuniform

Component j(1)	273
13.1.3 Total Backscattered Field		277
13.2 Electromagnetic Waves	279

13.2.1E-Polarization 279

13.2.2H-Polarization 283

Problems 285


14. Bistatic Scattering at a Finite-Length Cylinder								287

14.1	Acoustic Waves		287
	14.1.1	PO Approximation			287
	14.1.2	Shadow Radiation as a Part of the Physical Optics Field						289
	14.1.3	PTD for Bistatic Scattering at a Hard Cylinder					290
	14.1.4	Beams and Rays of the Scattered Field				297
	14.1.5	Reﬁned Asymptotics for the Specular Beam					300
14.2	Electromagnetic Waves			304

14.2.1E-Polarization 304

14.2.2H-Polarization 306

14.2.3Reﬁned Asymptotics for the Specular Beam Reﬂected

	from the Lateral Surface	308
Problems	311
Conclusion		313

References		315

Index		323

TEAM LinG

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