Handbook of radiation and scattering of waves: Acoustic waves in fluids, elastic waves in solids, electromagnetic waves: with corrections

$¢affering of Waves:

Elastic Waves in Solids Electromagnetic Waves

Adrianus T. de Hoop

Professor of Electromagnetic Theory and Applied Mathematics Delft University of Technology

Delft

Netherlands

ACADEMIC PRESS

Harcourt Brace and Company Publishers

LONDON NWl 7DX

U.S. Edition Published by ACADEMIC PRESS INC. San Diego, CA 92101

This book is printed on acid free paper

Copyright © 1995 ACADEMIC PRESS LIMITED

All rights reserved

No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical including photocopying, recording, or any information storage and retrieval system without permission in writing from the publisher

A catalogue record for this book is available from the British Library

ISBN 0-12-208655-4

Typeset by Technical Typesetters (UK), Ashford, Kent

De Hoop A.T.,

Handbook of Radiation and Scattering of Waves, London, Academic Press, 1995

Section On In Replace with

Front matter

xxiii 2 each each of the

xxv 20 ν, ν ν, N

Part 1. Radiation and scattering of acoustic waves in fluids

4.5 page 77 line 20 q(s)ˆ q(x, s)ˆ

4.5 page 78 line 11 fˆ_k(s) fˆ_k(x, s)

5.11 page 123 Eq.(5.11-11) q[{x + c(τ − τ₀)θ, t₀] q[x + c(τ − τ₀)θ, t₀] 7.12 page 215 line 5 elementary contributions contributions

7.12 page 215 line 6 equivalent elementary equivalent

8.6 page 280 line 11 In this case In this case,

Part 2. Radiation and scattering of elastic waves in solids

12.7 page 378 line 5 s-domain complex frequency-domain

12.7 page 380 Eq.(12.7-10) hˆS hˆs

12.7 page 380 line 8 hˆS hˆs

12.7 page 380 Eq.(12.7-11) fˆ_kS fˆ_ks

12.7 page 380 line 11 fˆ_kS fˆ_ks

13.4 page 391 line 24 x_m/|x| (x_m− x_m)/|x − x_m| 15.8 page 471 line 26 (Equation (15.2-7)) Equation (15.2-7) 15.8 page 472 line 6 (Equation (15.2-7)) Equation (15.2-7) 15.8 page 472 line 19 (Equation (15.2-7)) Equation (15.2-7) 15.8 page 472 line 33 (Equation (15.2-7)) Equation (15.2-7) 15.8 page 473 line 11 (Equation (15.4-7)) Equation (15.4-7) 15.8 page 473 line 24 (Equation (15.4-7)) Equation (15.4-7) 15.8 page 473 line 37 (Equation (15.4-7)) Equation (15.4-7) 15.8 page 474 line 14 (Equation (15.4-7)) Equation (15.4-7) 16.2 page 522 line 22 Equations (16.2-1) see Equations (16.2-1) 16.2 page 522 line 23 Equations (16.2-14) see, Equations (16.2-14)

Handbook of Radiation and Scattering of Waves, London, Academic Press, 1995

Section On In Replace with

16.2 page 524 line 8 Equations (16.2-3) see, Equations (16.2-3) 16.2 page 524 line 9 Equations (16.2-16) see, Equations (16.2-16) 16.2 page 525 line 22 Equations (16.2-1) see, Equations (16.2-1) 16.2 page 525 line 23 Equations (16.2-16) see, Equations (16.2-16) 16.2 page 527 line 1 (−βS, αP, t − t− t)) (−βS, αP, t − t− t) 16.2 page 527 line 2 (βP, αS, t − t− t)) (−βP, αS, t − t− t) 16.2 page 530 line 10 Equations (16.2-54) see, Equations (16.2-54) 16.2 page 530 line 11 Equations (16.2-67) see, Equations (16.2-67) 16.2 page 531 line 14 Equations (16.2-56) see, Equations (16.2-56) 16.2 page 531 line 15 Equations (16.2-69) see, Equations (16.2-69) 16.2 page 532 line 18 Equations (16.2-54) see, Equations (16.2-54) 16.2 page 532 line 19 Equations (16.2-69) see, Equations (16.2-69) 16.3 page 539 line 11 Equations (16.3-1) see, Equations (16.3-1) 16.3 page 539 line 12 Equations (16.3.-14) see, Equations (16.3-14) 16.3 page 540 line 19 Equations (16.3-3) see, Equations (16.3-3) 16.3 page 540 line 20 Equations (16.3.-16) see, Equations (16.3-16) 16.3 page 542 line 15 Equations (16.3-1) see, Equations (16.3-1) 16.3 page 542 line 16 Equations (16.3.-16) see, Equations (16.3-16) 16.3 page 547 line 12 Equations (16.3-45) see, Equations (16.3-45) 16.3 page 547 line 13 Equations (16.3.-58) see, Equations (16.3-58) 16.3 page 548 line 18 Equations (16.3-47) see, Equations (16.3-47) 16.3 page 548 line 19 Equations (16.3.-60) see, Equations (16.3-60) 16.3 page 550 line 6 Equations (16.3-45) see, Equations (16.3-45) 16.3 page 550 line 7 Equations (16.3.-60) see, Equations (16.3-60)

De Hoop A.T.,

Handbook of Radiation and Scattering of Waves, London, Academic Press, 1995

Section On In Replace with

Part 3. Radiation and scattering of electromagnetic waves

19.5 page 635 Eq.(19.5-25) κ_c(x, t) κ_c(x, t)

21.3 page 663 Exercise 21.3-1 E_kε_k,rE_r E_kε_k,r∂_tE_r 21.3 page 663 Exercise 21.3-2 H_jμ_j,pH_p H_jμ_j,p∂_tH_p

24.1 page 695 line 2 χ_Tt(t) χ_T(t)

26.6 page 740 line 1 Equation (26.5-38) Equation (26.5-39) 27.1 page 775 Eq.(27.1-16) Zˆ_r,p = ˆη_r,k−1ε_k,m,pγˆ_m Zˆ_r,p =−ˆη−1_r,kε_k,m,pˆγ_m 27.1 page 776 Eq.(27.1-25) Zˆ_r,p = ˆη_r,k−1ε_k,m,pξ_mγˆ Zˆ_r,p =−ˆη_r,k−1ε_k,m,pξ_mγˆ 28.8 page 848 line 8 (Equation (28.2-7)) Equation (28.2-7) 28.8 page 848 line 20 (Equation (28.2-7)) Equation (28.2-7) 28.8 page 848 line 32 (Equation (28.2-7)) Equation (28.2-7) 28.8 page 849 line 11 (Equation (28.2-7)) Equation (28.2-7) 28.8 page 849 line 23 (Equation (28.4-7)) Equation (28.4-7) 28.8 page 849 line 35 (Equation (28.4-7)) Equation (28.4-7) 28.8 page 850 line 11 (Equation (28.4-7)) Equation (28.4-7) 28.8 page 850 line 24 (Equation (28.4-7)) Equation (28.4-7)

28.9 page 854 Eq.(28.9-14) C_t ∂_tC_t 28.9 page 854 Eq.(28.9-15) C_t ∂_tC_t 28.9 page 854 Eq.(28.9-16) C_t ∂_tC_t 28.9 page 854 Eq.(28.9-17) C_t ∂_tC_t 28.11 page 864 Eq.(28.11-1) C_t ∂_tC_t 28.11 page 864 Eq.(28.11-2) C_t ∂_tC_t 28.11 page 865 Eq.(28.11-9) DT Ds 28.12 page 871 Eq.(28.12-4) μA_p,j μB_p,j 29.6 page 937 line 22 { [{ 29.6 page 938 line 2 } [}

30.5 page 973 Eq.(30.5-18) I_max I_max

30.5 page 973 Eq.(30.5-18) H[t − t₀− t_w− (t_r+t_f)/2] H[t − t₀ − t_w− (t_r+t_f)/2]

30.5 page 973 line 25 configurtation configuration

Handbook of Radiation and Scattering of Waves, London, Academic Press, 1995

Section On In Replace with

Appendices

A.3 page 1000 Eq.(A.3-27) < x, x > < x, y > A.3 page 1000 Eq.(A.3-29) < x, y > < x, y >2

A.7 page 1018 line 17 a_p,n α_p,n

A.9 page 1024 Eq.(A.9-7) [∂_tx_m(t)∂_tx_m(t)]−1/2 [∂_tx_m(t)∂_tx_m(t)]1/2 A.10 page 1033 Eq.(A.10-12) _s(A)s(A) _s(A)s(B)

A.10 page 1035 Eq.(A.10-19) α[x(IK)] σ[x(IK)]

A.10 page 1038 Eq.(A.10-29) V_PN V_PN

A.10 page 1041 Eq.(A.10-55) AN_p (N) AN−1_p (N)

A.11 page 1045 Fig.(A.12-1) dA dA

Back matter

Index 1082 col. 2, line 20 Cauchy Cauchy’s

Revised: 2008 December 28

Suggestions for classroom use ... xxiii

Printing of symbols ... xxv

General introduction ... xxvii

Part I Radiation and scattering of acoustic waves in fluids

Basic

2.1

Number density, drift velocity, volume density of mass, and mass flow density of a collection of moving particles ... 7

Exercises ... 14

Conservation of the number of particles and its consequences ... 16

The equation of motion ... 20

The deformation rate equation ... 24

The constitutive relations ... 25

Exercises ... 28

The boundary conditions ... 29

Low-velocity linearisation: the equations of linear acoustics ...31

Exchange of acoustic energy ... 37

Exercises ... 40

The frictional-force/bulk-viscosity acoustic loss mechanism ...41

Exercises ... 43

Acoustic scalar and vector potentials in the theory of radiation from sources 44 Exercises ... 46

Point-source solutions; Green’s functions ... 47

Exercises ... 48

SI units of acoustic wave quantities ... 49

The principle of superposition and its application to acoustic wave fields in configurations wi|h geometrical symmetry ...

3.1 The principle of superposition ... 51

3.2 Symmetry with respect to a plane ... 52

Exercises ... 57

3.3 Symmetry with respect to a line ... 58

Exercises ... 62

3.4 Symmetry with respect to a point ... 63

Exercises ... 67

The acoustic wave equations, constitutive relations, and boundary conditions in the time Laplace-transform domain (complex frequency domain) ... 69

4.1 The complex frequency-domain acoustic wave equations ...70

Exercises ... 71

4.2 The complex frequency-domain constitutive relations; the Kramers-Kronig causality relations for a fluid with relaxation ...71

Exercises ... 74

4.3 The complex frequency-domain boundary conditions ...75

Exercises ... 75

4.4 The complex frequency-domain coupled acoustic wave equations ...76

4.5 Complex frequency-domain acoustic scalar and vector potentials ...77

Exercises ... 79

4.6 Complex frequency-domain point-source solutions and Green’s functions . 80 Exercises ... 81

References ... 81

Acoustic radiation from sources in an unbounded, homogeneous, isotropic fluid ... 83

5.1 The coupled acoustic wave equations and their solution in the angular wave-vector domain ... 83

5.2 The Green’s function of the scalar Helmholtz equation ... 86

Exercises ... 89

5.3 The complex frequency-domain source-type integral representations for the acoustic pressure and the particle velocity ... 89

Exercises ... 93

5.4 The time-domain source-type integral representations for the acoustic pressure and the particle velocity in a lossless fluid ... 93

Exercises ... 97

5.5 The Green’s function of the dissipative scalar wave equation ... 97

Exercises ... 103

5.6 Time-domain source-type integral representations for the acoustic pressure and the particle velocity in a fluid with frictional-force/bulk-viscosity losses . 104 5.7 The acoustic wave field emitted by a monopole transducer ... 106

5.9 5.10

5.11

Far-field radiation characteristics of extended sources

(complex frequency-domain analysis) ... 116

Far-field radiation characteristics of extended sources (time-domain analysis for a lossless fluid) ... 119

Exercises ... 122

The time evolution of an acoustic wave field. The initial-value problem (Cauchy problem) for a homogeneous, isotropic, lossless fluid ... 122

Exercises ... 124

References ... 125

Plane

6.1

Plane waves in the complex frequency domain ... 127

Exercises ... 130

6.2 Plane waves in lossless fluids; the slowness surface ... 130

Exercises ... 132

6.3 Plane waves in the real frequency domain; attenuation vector and phase vector ... 133

Exercises ... 139

6.4 Time-domain uniform plane waves in an isotropic, lossless fluid ... 140

Exercises ... 142

6.5 Structure of the plane wave motion near the planar boundary of an acoustically impenetrable object ... 144

7 Acoustic reciprocity theorems and their applications ... 149

7.9 7.10 7.1 The nature of the reciprocity theorems and the scope of their consequences 149 Exercises ... 156

7.2 The time-domain reciprocity theorem of the time convolution type .... 157

Exercises ... 160

7.3 The time-domain reciprocity theorem of the time correlation type ... 160

Exercises ... 164

7.4 The complex frequency-domain reciprocity theorem of the time convolution type ... 164

Exercises ... 167

7.5 The complex frequency-domain reciprocity theorem of the time correlation type ... 169

Exercises ... 172

7.6 Transmission/reception reciprocity properties of a pair of acoustic transducers ... 173

Exercises ... 176

7.7 Transmission/reception reciprocity properties of a single acoustic transducer ... 177

7.8 The direct (forward) source problem; point-source solutions and Green’s functions ... 181

Exercises ... 189

The direct (forward) scattering problem ... 193

7.11 7.12

The inverse scattering problem ... 205

Acoustic wave-field representations in a subdomain of the configuration space; equivalent surface sources; Huygens’ principle and the Ewald-Oseen extinction theorem ... 212

Exercises ... 219

References ... 220

Plane wave scattering by an object in an unbounded, homogeneous, isotropic, Iossless embedding ... 221

8.1 The scattering configuration, the incident plane wave and the far-field scattering amplitudes ... 221

Exercises ... 230

8.2 Far-field scattered wave amplitude reciprocity of the time convolution type 231 Exercises ... 239

8.3 Far-field scattered wave amplitude reciprocity of the time correlation type 240 Exercises ... 249

8.4 An energy theorem about the far-field forward scattered wave amplitude . 249 Exercises ... 253

8.5 The Neumann expansion in the integral equation formulation of the scattering by a penetrable object ... 254

8.6 Far-field plane wave scattering in the first-order Rayleigh-Gans-Born approximation; time-domain analysis and complex frequency-domain analysis for canonical geometries of the scattering object ... 259

Exercises ... 278

References ... 285

Part 2 Radiation and scattering of elastic waves in solids Introduction ... 289 Exercises ... 291 References ... 291

10

Basic

10.1

Number density, drift velocity, volume density of mass, and mass flow density of a collection of moving particles ... 293

Exercises ... 300

Conservation of the number of particles and its consequences ... 302

The equation of motion ... 305

Exercises ... 312

The deformation equation ... 313

Exercises ... 315

The constitutive relations ... 315

Exercises ... 321

The boundary conditions ... 322

10.8 10.9 10.10 10.11 10.12 10.13 10.14 Exercises ... 329

Exchange of elastodynamic energy ... ₃₃₀

Exercises ... 333

The frictional-force/viscosity elastodynamic loss mechanism ... 334

Exercises ... 336

Elastodynamic vector and tensor potentials in the theory of radiation from distributed sources ... 337

Exercises ... 339

Point-source solutions; Green’s functions ... 340

Exercises ... 341

The elastodynamic wave equation for the particle velocity in a lossless solid ... 341

The equivalent fluid model for dilatational waves in a solid ... 343

Exercises ... 346

SI units of elastic wave quantities ... 347

References ... 348

||

11.1 11.2 11.3 11.4 The principle of superposition ... 349

Symmetry with respect to a plane ... 350

Exercises ... 356

Symmetry with respect to a line ... 356

Exercises ... 361

Symmetry with respect to a point ... 361

Exercises ... 365

The elastic wave equations, constitutive relations, and boundary conditions in the time Laplace-transform domain (complex frequency domain) ... 367

12.1 12.2 12.3 12.4 12.5 12.6 12.7 The complex frequency-domain elastic wave equations ...368

Exercises ... 369

The complex frequency-domain constitutive relations; the Kramers-Kronig causality relations for a solid with relaxation ...369

The complex frequency-domain boundary conditions ... 372

Exercises ... 373

The complex frequency-domain coupled elastic wave equations ... 373

Complex frequency-domain elastodynamic vector and tensor potentials.. 374

Exercises ... 376

Complex frequency-domain point-source solutions; complex frequency-domain Green’s functions ... 376

Exercises ... 377

The complex frequency-domain elastic wave equations for dilatational waves (equivalent fluid model) ... ₃₇₈

Exercises ... ¯ ... 380

13

homogeneous, isotropic solid ... 381

13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 The coupled elastic wave equations in the angular wave-vector domain ... 381

The elastodynamic wave equation for the particle velocity and its solution in the angular wave-vector domain ... 384

Determination of Gp and Gs ... 385

Exercises ... 389

The complex frequency-domain source-type integral representations for the particle velocity and the dynamic stress ... 389

Exercises ... 393

The time-domain source-type integral representations for the particle velocity and the dynamic stress ... 394

Point-source solutions ... 396

Far-field radiation characteristics of extended sources (complex frequency-domain analysis) ... 398

Exercises ... 403

Far-field radiation characteristics of extended sources (time-domain analysis) ... 403

Exercises ... 407

The time evolution of an elastic wave field. The initial-value problem (Cauchy problem) for a homogeneous, isotropic, perfectly elastic solid . . 407

Exercises ... 410

14

Plane

14.1

elastic waves in homogeneous solids ...413

Plane waves in the complex frequency domain ... 413

Exercises ... 416

Plane waves in lossless solids; the slowness surface ... 416

Exercises ... 419

Plane waves in the real frequency domain; attenuation vector and phase vector ... 420

Exercises ... 422

Time-domain uniform plane waves in an isotropic, lossless solid ... 423

Exercises ... 426

15 Elastodynamic reciprocity theorems and their applications ... 429

15.1 15.2 15.3 15.4 The nature of the reciprocity theorems and the scope of their consequences ... 429

Exercises ... 436

The time-domain reciprocity theorem of the time convolution type .... 437

Exercises ... 440

The time-domain reciprocity theorem of the time correlation type ... 441

Exercises ... 444

15.5 15.6 15.7 15.8 15.9 15.10 15.11 15.12 Exercises ... 449

The complex frequency-domain reciprocity theorem of the time correlation type ... 450

Exercises ... 453

Transmission/reception reciprocity properties of a pair of elastodynamic transducers ... 455

Exercises ... 458

Transmission/reception reciprocity properties of a single elastodynamic transducer ... 459

The direct (forward) source problem. Point-source solutions and Green’s functions ... 463

Exercises ... 471

The direct (forward) scattering problem ... 475

The inverse source problem ... 481

The inverse scattering problem ... 487

Elastic wave-field representations in a subdomain of the configuration space; equivalent surface sources; Huygens’ principle and the Ewald-Oseen extinction theorem ... 494

Exercises ... 501

References ... 503

16

Plane wave scaltering by an object in an unbounded, homogeneous,

isotropic, Iossless embedding ... 505

16.1 16.2 16.3 16.4 16.5 16.6 The scattering configuration, the incident plane waves and the far-field scattering amplitudes ... 505

Exercises ... 516

Far-field scattered wave amplitudes reciprocity of the time convolution type ... 517

Exercises ... 533

Far-field scattered wave amplitudes reciprocity of the time correlation type ... 534

An energy theorem about the far-field forward scattered wave amplitudes 551 Exercises ... 559

The Neumann expansion in the integral equation formulation of the scattering by a penetrable object ... 560

Far-field plane wave scattering in the first-order Rayleigh-Gans-Born approximation; time-domain analysis and complex frequency-domain analysis for canonical geometries of the scattering object ... 565

Exercises ... 591

References ... 597

Part 3 Radiation and scaffering of electromagnetic waves

17

Exercises ... 604

18 The electromagnetic field equations ...605

Exercises ... 607

The electromagnetic field equations in vacuum ... 608

Exercises ... 609

The electromagnetic field equations in matter ... 610

Exercises ... 613

The electromagnetic field equations for time-independent fields (quasi-static field equations) ... 613

Exercises ... 614

SI units of the electromagnetic field quantities ... 615

References ... 616

19 The electromagnetic constitutive relations ... 617

19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 Conductivity, permittivity and permeability of an isotropic material .... 618

Conductivity, permittivity and permeability of an anisotropic material . . 619

Conductivity, permittivity and permeability of a material with relaxation . 620 Exercises ... 621

Electric current as a flow of electrically charged particles. The conservation of electric charge ... 622

Exercises ... 629

The conduction relaxation function of a metal ... 632

Exercises ... 639

The conduction relaxation function of an electron plasma ... 639

Exercises ... 641

The dielectric relaxation function of an isotropic dielectric ... 642

Exercises ... 643

SI units of the quantities associated with the electromagnetic constitutive behaviour of matter ... 644

References ... 645

20 The electromagnetic boundary conditions ... 647

20.1 20.2 20.3 Boundary conditions at the interface of two media ... 647

Exercises ... 649

Boundary condition at the surface of an electrically impenetrable object . 650 Exercises ... 650

Boundary condition at the surface of a magnetically impenetrable object . 651 Exercises ... 651

21 Exchange of energy in the electromagnetic field ... 653

21.1 21.2 21.3 Energy theorem for the electromagnetic field associated with the flow of a collection of electrically charged particles ... 653

Energy theorem for the electromagnetic field in stationary matter ... 657

21.4

Exercises ... 663

SI units of the quantities associated with the exchange of electromagnetic energy ... 666

Vector potentials, point-source solutions and Green’s functions in the theory of electromagnetic radiation from sources ... 667

22.1 Vector potentials in the theory of electromagnetic radiation from distributed sources ... 667

Exercises ... 669

22.2 Point-source solutions; Green’s functions ... 670

Exercises ... 671

23

23.1 23.2 23.3 23.4 The principle of superposition ... 673

Symmetry with respect to a plane ... 674

Exercises ... 680

Symmetry with respect to a line ... 681

Exercises ... 685

Symmetry with respect to a point ... 686

Exercises ... ... 690

24

24.1 24.2 24.3 24.4 The complex frequency-domain electromagnetic field equations ... 694

Exercises ... 695

The complex frequency-domain electromagnetic constitutive relations; Kramers-Kronig causality relations for a medium with relaxation ... 695

Exercises ... 706

The complex frequency-domain boundary conditions ... 710

Exercises ... 711

The complex frequency-domain coupled electromagnetic wave equations 711 Exercises ... 712

References ... 714

Complex frequency-domain vector potentials, point-source solutions and Green’s functions in the theory of electromagnetic radiation from sources ... 715

25.1 25.2 Complex frequency-domain vector potentials in the theory of electromagnetic radiation from distributed sources ... 715

Exercises ... 717

Complex frequency-domain point-source solutions; complex frequency-domain Green’s functions ... 717

26

Electromagnetic radiation from sources in an unbounded,

homogeneous, isotropic medium ...

719

26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9 26.10 26.11 26.12 26.13 The electromagnetic field equations and their solution in the angular wave-vector domain ... 719

The Green’s function of the scalar Helmholtz equation ... 723

Exercises ... 726

The complex frequency-domain source-type representations for the electric and the magnetic field strengths ... 726

Exercises ... 729

The time-domain source-type representations for the electric and the magnetic field strengths in a lossless medium ... 730

Exercises ... 733

The Green’s function of the dissipative scalar wave equation ... 734

Exercises ... 740

Time-domain source-type integral representations for the electric and the magnetic field strengths in a medium with conductive electric and linear hysteresis magnetic losses ... 740

The Green’s function of the scalar wave equation associated with plasma oscillations and superconductivity ... 743

Time-domain source-type integral representations for the electric and the magnetic field strengths in an electron plasma or a superconducting metal 749 The electromagnetic field emitted by a short segment of a thin, conducting, current-carrying wire ... 752

The electromagnetic field emitted by small, conducting, current-carrying loop ... 757

Exercises ... 762

Far-field radiation characteristics of extended sources (complex frequency-domain analysis) ... 762

Exercises ... 765

Far-field radiation characteristics of extended sources (time-domain analysis for a lossless medium) ... 765

Exercises ... 768

The time evolution of an electromagnetic wave field. The initial-value problem (Cauchy problem) for a homogeneous, isotropic, lossless medium 768 Exercises ... 770 References ... 771

27

Plane

27.1

Plane waves in the complex frequency domain ... 773

Exercises ... 778

Plane waves in lossless media; the slowness surface ... 780

Exercises ... 782

Plane waves in the real frequency domain; attenuation vector and phase vector ... 782

Exercises ... 800

Exercises ... 805

28

28.1 The nature of the reciprocity theorems and the scope of their consequences ... 807

Exercises ... 814

28.2 The time-domain reciprocity theorem of the time convolution type .... 814

Exercises ... 817

28.3 The time-domain reciprocity theorem of the time correlation type ... 818

Exercises ... 822

28.4 The complex frequency-domain reciprocity theorem of the time convolution type ... 822

Exercises ... 826

28.5 The complex frequency-domain reciprocity theorem of the time correlation type ... 827

Exercises ... 830

28.6 Transmission/reception reciprocity properties of a pair of electromagnetic antennas ... 832

Exercises ... 836

28.7 Transmission/reception reciprocity properties of a single electromagnetic antenna ... 837

28.8 The direct (forward) source problem. Point-source solutions and Green’s functions ... 840

Exercises ... 848

28.9 The direct (forward) scattering problem ... 851

28.10 The inverse source problem ... 857

28.11 The inverse scattering problem ... 863

28.12 Electromagnetic wave-field representations in a subdomain of the configuration space; equivalent surface sources; Huygens’ principle and the Ewald-Oseen extinction theorem ... 870

Exercises ... 877

References ... 878

29

29.1 29.2 29.3 29.4 The scattering configuration, the incident plane wave and the far-field scattering amplitudes ... 879

Exercises ... 887

Far-field scattered wave amplitude reciprocity of the time convolution type ... 888

Exercises ... 896

Far-field scattered wave amplitude reciprocity of the time correlation type ... 897

Exercises ... 906

30

The Neumann expansion in the integral equation formulation of the

scattering by a penetrable object ... 911

Far-field plane wave scattering in the first-order Rayleigh-Gans-Born approximation; time-domain analysis and complex frequency-domain analysis for canonical geometries of the scattering object ... 915

Exercises ... 935

References ... 941

Interference and shielding of electromagnetic systems accessible via low-frequency terminations. ElectroMagnetic Compatibility (EMC) . 943 30.1 30.2 30.3 30.4 30.5 30.6 30.7 The reciprocity surface interaction integral for a low-frequency multiport system ... 943

Exercises ... 945

The electromagnetic N-port system as a transmitting system (electromagnetic emission analysis) ... 947

Exercises ... 949

The electromagnetic N-port system as a receiving system (electromagnetic susceptibility analysis) ... 950

Exercises ... 957

Remote interaction between an M-port system and an N-port system . . . 959

Exercises ... 963

Electromagnetic interference ... 967

Exercises ... 975

The shielding effectiveness of a spherical shield for a radiating electric dipole placed at its centre (complex frequency-domain analysis) ... 979

The shielding effectiveness of a spherical shield for a radiating magnetic dipole placed at its centre (complex frequency-domain analysis) ... 984

References ... 988

Appendices

A.3 A.4

A.5

A.6

The summation convention ... 992

Exercises ... 992

Cartesian reference frames in affine space and in Euclidean space ... 993

Exercises ... 999

Definition of a Cartesian tensor ... 1001

Exercises ... 1003

Addition, subtraction and multiplication of tensors ... 1003

Exercises ... 1006

Symmetry properties ... 1008

Exercises ... 1009

A.8 A.9 A.10 A.11 A.12 Exercises ... 1017 Differentiaion of a tensor ... 1019 Exercises ... 1022

Geometrical objects of a particular shape in N-dimensional

Euclidean space ... 1023

Exercises ... 1031

Integration of a tensor ... 1032

Exercises ... 1042

The Taylor expansion ... 1043

Exercises ... 1044

Gauss’ integral theorem ... 1045

Exercises ... 1046

Appendix B Integral-transformation methods ... 1049

B.2

B.1 Laplace transformation of a causal time function ... 1049

Exercises ... 1057

Spatial Fourier transformation ... 1060

Exercises ... 1064

B.3 The Kramers-Kronig causality relations ... 1065

Exercises ... 1070

B.4 Fourier series and Poisson’s summation formula ... 1071

Exercises ... 1073

References ... 1074

and Radiation and Scattering of Electromagnetic Waves taught at the Delft University of Technology in the curricula of the Departments of Electrical Engineering, Engineering Mathematics, Mining and Petroleum Engineering, Applied Physics and Civil Engineering, and in the graduate courses provided by the Delft Centre for Technical Geoscience. A standard introductory knowledge of differential and integral calculus, as well as some undergraduate physics, is preassumed, while an introductory notion of Laplace and Fourier transformation methods is also helpful. Should the reader need to refresh his or her understanding of these concepts, the necessary mathematical prerequisites are recapitulated in the appendices that also include a number of results needed in the main text.

In view of the ever-increasing number of applications where wave problems are modelled for realistic three-dimensional configurations on large-capacity high-speed digital computers, the emphasis is mainly put on general principles and theorems that can serve to check numerical results rather than on highly specialized configurations for which more or less complicated analytical answers can be obtained. With this type of application in mind, the subscript notation for Cartesian vectors and tensors is consistently used as an interdisciplinary notational tool. This notation has the advantage that expressions and equations can be copied almost effortlessly to produce the corresponding statements in any of the high-level programming languages (for example, Fortran 77 or Fortran 90), while they are also directly amenable to symbolic

, TM

manipulation with programs like Mathematlca . Another advantage of the subscript notation is that the common structure of the wave and field equations in different branches of physics, and of the theorems resulting from them, becomes immediately clear. Moreover, the notation enables one to take into account anisotropy in the media in which the waves propagate.

The analysis is carried out in parallel in both the time domain and in the complex frequency domain where the time coordinate has been replaced by the complex Laplace transform parameter. The physical requirement of causality of the wave motion is conveniently accounted for through this procedure. Furthermore, the expressions pertaining to the steady-state quantities for a fixed frequency of oscillation result from the imaginary values of the time Laplace transform. The major steps in the time-domain and the complex frequency-domain analyses are presented in both these domains, thereby allowing a choice between either of the two in classroom use if the time allotted is curtailed.

problem, the direct (forward) scattering problem, the inverse source problem and the inverse scattering problem can be formulated in a natural manner with reciprocity as the point of departure, and the setup also reveals that all known algorithms for the computational analysis of these problems are consequences of a proper choice and interpretation of the two Field States that occur in the reciprocity theorems.

A set of carefully selected Exercises forms an integral part of the text. The exercises are hardly ever of the substitution-of-numbers type. They often illustrate the models that underlie the physics of the problem or cover additional aspects of the subject in question. They are placed at the end of each section and all answers are provided.

Acknowledgements

It is impossible to name all the Ph.D. students, colleagues and scientists who, through numerous discussions over many years, have helped me to mature my insight into the subject matter. It is virtually impossible to quantify how much I owe to them. I have enjoyed the privilege of having two unforgettable teachers: the late Professor J.P. Schouten and Professor C.J. Bouwkamp. The everyday discussions with my younger colleagues and friends, Professors H. Blok, P.M. van den Berg and J.T. Fokkema, and the senior staff members Dr D. Quak and Dr G. Mur of the Laboratory of Electromagnetic Research at Delft University of Technology are still an everyday pleasure. With gratitude I further mention the scientific opportunities that Schlumberger has granted me, both through the Visiting Scientist positions with Schlumberger-Doll Research, Ridgefield, CT, USA, from 1982 onward and with Schlumberger Cambridge Research Limited, Cambridge, England, from 1991 onward, and through the generous Research Grants from the Stichting Fund for Science, Technology and Research (a companion organization to the Schlumberger Foundation in the USA). Thanks are also due to Mr W.J.P. van Nimwegen for preparing the illustrations and again to Dr D. Quak for providing me with the numerical results. Above all, however, my special thanks go to Ms Janine van den Nouweland of the Secretarial Staff of the Laboratory of Electromagnetic Research, Delft University of Technology, for preparing the manuscript with such care, skill, efficiency and devotedness.

The material presented in the Handbook is straightforwardly suited for classroom use. For each three kinds of wave motion considered, the underlying physical concepts and the road leading to their governing equations are presented in detail. Furthermore, analytical results are derived for relatively simple configurations in the radiation and scattering of waves. Finally, each section concludes with a set of carefully selected Exercises (with all answers provided). Whenever the coherence in the presentation made this desirable, some arguments, reasonings and calculations have been repeated in Parts 1, 2 and/or 3.

A suggestion is made below for the material to be included in introductory courses that cover the fundamentals, but leave out a number of more advanced topics, in case time constrictions necessitate this.

Introductory course on Radiation and Scal~ering of Acoustic Waves in Fluids

General Introduction Appendix A:

tions only) Appendix B: and from Part 1:

Chapter 1 Chapter 2: Chapter 4: Chapter 5: Chapter 6: Chapter 7:

Sections A. 1-A.4, A.7 (only the Kronecker tensor), A. 10-A. 12 (the

Introductory course on Radiation and Scattering of Elastic Waves in Solids

General Introduction

Appendix A: Sections A. 1-A.4, A.7 (only the Kronecker tensor and the symmetrical unit tensor of rank four), A. 10-A. 12 (the definitions only)

Appendix B: Sections B.1-B.3 and from Part 2:

Chapter 9 Chapter 10: Sections 10.1-10.9, 10.12 Chapter 12: Sections 12.1-12.4 Chapter 13: Sections 13.1-13.8 Chapter 14: Sections 14.1-14.4 Chapter 15: Sections 15.1, 15.4

Introductory course on Radiation and Scattering of Electromagnetic Waves

General Introduction Appendix A:

Appendix B: and from Part 3:

Chapter 17 Chapter 18: Chapter 19: Chapter 20: Chapter 21: Chapter 24: Chapter 26: Chapter 27: Chapter 28: Chapter 30:

In accordance with international standards, symbols for physical quantities and symbols for numerical values are printed in italic (sloping) type in the Latin and Greek alphabets, while descriptive subscripts and numerical subscripts are printed in roman (upright) type. For reference, the two sets of symbols as used in the printing of this Handbook are reproduced below.

Note that in particular the difference between v, ~ and v is small.

Italic type Latin alphabet

a, A h, H o, 0 v, V b, B i, I p, P w, W c,C j,J q,Q x,X d,D k,K r,R y,Y e,E 1, L s,S z,Z fF m,M t,T g,G n,N u,U

Italic type Greek alphabet

ct, A alpha t, I iota p, P rho

fl, B beta ~, K kappa a, E sigma

~, F gamma 2, A lambda z-, T tau

6, A delta #, M mu v,/~ upsilon

e, E epsilon v, v nu q~, 45 phi

~, Z zeta ~, ~ xi ;(,, X chi

r/, H eta o, O omicron ~, g¢ psi

three wave phenomena, related in their mathematical structure, viz. Radiation and Scattering

of Acoustic Waves in Fluids (Part 1), Radiation and Scattering of Elastic Waves in Solids (Part

2) and Radiation and Scattering of Electromagnetic Waves (Part 3). Each of the three parts can be read and studied independently. Some general remarks applying to all three kinds of wave motion are given in this General Introduction. Some mathematical preliminaries are collected in Appendices A and B. Appendix A deals with the elementary properties of Cartesian vectors

and tensors, for which the subscript notation, supplemented with the summation convention, is used as an interdisciplinary notational tool. Appendix B deals with the description of signals in linear, time-invariant systems. For such systems, the time-domain characterisation and the complex frequency-domain characterisation yield complementary descriptions, that are related through the time Laplace transformation. In addition, the spatial Fourier transformation is discussed, which finds its application in spatially shift-invariant configurations (in one, two or three dimensions). This transformation interrelates the spatial-domain and spectral-domain

descriptions of wave phenomena.

The physical laws that underlie the properties of acoustic, elastic or electromagnetic waves are deduced from a series of basic standard experiments. To carry out these experiments, an observer must be able to register both the position in space and the instant at which an observation is made. To register, the position in space, the existence of a three-dimensional, isotropic, Euclidean, background space R3 is presumed. In this space, distance can be measured along three mutually perpendicular directions with the same position- and orientation-independent standard measuring rod. To register instants, the existence of a position- and orientation-independent standard clock is presumed.

The standard measuring rod is used to define, at a certain position that is denoted as the origin O, an orthogonal Cartesian reference frame consisting of three base vectors

{ i(1),i(2),i(3)} that are mutually perpendicularly oriented, are each of unit length, and form in

the indicated order a right-handed system (Figure 1). (The property that each base vector specifies geometrically a length and an orientation makes it a vectorial quantity, or a vector; notationally, vectors will, whenever appropriate, be represented by bold-face symbols.) Let

Figure | Orthogonal Cartesian reference frame with origin 0 and three mutually perpendicular base

vectors {i(1),i(2),i(3) } of unit length each, position vectorx = xli(1) + x2i(2) ÷ x3i(3), and time coordinate

t.

x : xli(1) + x2i(2) + x3i(3) 3

= ~ Xmi(m). (1)

m=l

The numbers {Xl,X2,X3 } are denoted as the right-handed, orthogonal Cartesian coordinates of the point of observation. The time coordinate is real and is denoted by t. In the notation of the theory of sets we write x~R3 and t~R.

One of the purposes of the basic standard experiments is to define the units in terms of which the measured physical quantities are expressed¯ In accordance with international convention, the International System of Units (SystEme International d’Unitds) - abbreviated to SI - is employed. This system is based on seven basic quantities¯ The basic quantities, the basic units and the basic dimensions of SI are shown in Table 1 (see also, Symbols, Units, Nomenclature

and Fundamental Constants in Physics, 1987 Revision, Document IUPAP-25 (SUNAMCO

87-1) prepared by Richard Cohen and Pierre Giacomo, International Union of Pure and Applied Physics, SUNAMCO Commission, Sevres).

Basic quantities, units and dimensions of the International System of Units (SI)

Basic quantity Basic unit Basic dimension

Name Symbol Name Symbol Symbol

Length I metre m L

Mass m kilogramme kg M

Time t second s T

Electric current I ampere A I

Thermodynamic temperature T kelvin K 0

Amount of substance n mole mol N

Luminous intensity I candela cd J

(such as second (s), kilogramme (kg)), and an upper-case letter or an upper-case letter followed by one or more lower-case letters if the relevant unit is derived from a proper name (such as ampere (A), pascal (Pa)). Either simple or compound quantities that are derived from the basic quantities will occur in the sequel. The compound units are formed from the pertinent simple units by using the dot (o) as the multiplication sign and the solidus (/) as the division sign. Examples are: newton.second (N.s) for the unit of momentum and metre/second (m/s) for the unit of velocity. The only exception occurs if a simple unit would need a division sign; in this case, the notation ’to the power -1’ is used. An example of the latter is second-1 (s-1) for the unit ’per time’.

In SI the basic unit of length (m) is derived from the basic unit of time (s) via the speed of light (or electromagnetic wave speed) in vacuum: c0 = 299 792 458 mis (exactly).

The prefixes that should be used to indicate decimal multiples or submultiples of a unit are listed in Table 2.

When a prefix symbol is used with a unit symbol the combination should be considered as a single new symbol that can be raised to a positive or negative power without using parentheses;

3 3 3 9 3 3 3 3 9 3

for example, lmm =(10- m) =10- m,lkm =(10 m) =10 m.Compoundprefixes formed by the juxtaposition of two or more prefixes should not be used.

The mathematical framework by which the results from the standard basic experiments are cast into the macroscopic physical laws that govern the wave motion is furnished by tensor

calculus. In fact, a postulate to this effect has been used by Einstein to arrive at the theory of

relativity (Einstein, 1956). In addition to this, the corresponding subscript notation has the important advantage that expressions and equations can be copied almost effortlessly to produce the corresponding statements in any of the high-level programming languages (for example, Fortran 77 or Fortran 90), while they are also directly amenable to symbolic manipulation in programs like MathematicaTM. For this reason, Appendix A gives an introduction to the notation

and the properties of the Cartesian tensors that are needed in our further analysis.

Table 2 Prefixes for use with SI units

Prefix

Power of 10 Name Symbol

10-18 atto a 10-15 femto f 10-12 pico p 10-9 nano n 10-6 micro IX 10-3 milli m 10-2 centi c 10-1 deci d 101 deca da 102 hecto h 103 kilo k 106 mega M 109 giga G 1012 tera T 1015 peta P 1018 exa E Exercises Exercise 1

Do symbols for SI units include a full stop (period)? Answer: No.

Exercise 2

Are symbols for SI units altered in the plural?

Answer: No.

References

nd scatfe

coust’¢ w

uids

the elementary building blocks of matter are, by some cause, displaced out of their equilibrium position and try to return to this position under the influence of restoring forces. In this respect, acoustic waves differ from electromagnetic waves in that the latter can also be present in vacuo (i.e. in the absence of matter), whereas the former cannot. In the present part of the Handbook, the investigation of the properties of acoustic waves in fluids is our main concern; the properties of elastic waves in solids are discussed in Part 2.

When following an acoustic wave on its course, we start with its excitation by an acoustic source or transmitting device (the human voice, a musical instrument, a loudspeaker, a vibrating machine, an ultrasonic transducer). Once it has been generated, the wave propagates along a certain, more or less confined, path from the source to the receiver. Depending on the properties of the medium through which the wave passes, this propagation can lead to continuous

refraction by spatial and/or temporal changes in the medium (for example, the atmosphere), to reflection against and transmission across interfaces between different media, or to

discontinu-ous scattering or diffraction by objects whose acdiscontinu-oustic properties show a contrast with those of their surroundings. Finally, the wave motion is received by an acoustic receiving device (the human ear, a microphone, a hydrophone, a geophone, an electronic transducer). Figure 1.1 illustrates these different aspects.

Each of these aspects is the subject of extensive theoretical and experimental investigation. Usually, when the attention is focused on a particular detail, the remaining circumstances are

chosen as simply as possible. For example, when one wants to investigate the directional characteristics of an acoustical source, the surrounding medium will be taken to be of the utmost simplicity as far as its acoustic properties are concerned, and of infinite extent. When studying refraction phenomena during the propagation of an acoustic wave, the source will be taken to be a simple one (mostly a point source, i.e. a source whose dimensions are negligibly small

Figure 1.1 Acoustic wave phenomena on their course from source to receiver.

results of the partial model studies together in a judicious way in order to compose a judgement of the behaviour of acoustic waves and vibrations in the more complicated situations met in practice.

The practical applications of acoustic waves are widespread, and the number of fields in which they are used is ever increasing. In everyday life, acoustic waves are the carriers of sound, be it wanted (music, some speech) or unwanted (noise, other speech). In the field of medicine,

acoustic tomography, i.e. the imaging of an object (foetus, tumour) inside the body, is of

growing importance, the more so since acoustic radiation in the applied dosages is either non-hazardous or much less hazardous than the X-ray radiation used in X-ray tomography. The same idea of acoustic imaging underlies the use of acoustic waves in exploration geophysics, be it in surface seismics, vertical seismic profiling, cross-borehole seismics, or borehole acoustics. Here, acoustic waves are used to map the subsurface structure of the Earth in the search of fossil energy resources (coal, oil, natural gas). Furthermore, the non-destructive

evaluation of materials and of mechanical structures makes use of acoustic waves to a large

extent. The scattering of these waves by interior defects (inclusions, bubbles, cracks) makes the presence of these defects detectable at the surface of the structure, which surface is

accessible for carrying out the necessary measurements. For underwater locations, SONAR (SOund Navigation And Ranging) acoustic systems are installed on almost any vessel. Finally,

earthquake engineering, i.e. the design of earthquake-resistant structures, requires the

knowl-edge of the properties of acoustic waves in the Earth’s crust. These different applications are listed in Figure 1.2.

I~igure 1.2 Applications of acoustic waves.

involved and that the exchange of energy between these building blocks takes place through large numbers of energy quanta. Occasionally, we shall use microscopic considerations to elucidate the underlying physical picture of the macroscopic phenomena. Classical treatises on the subject are those by Lord Rayleigh (Strutt, 1945), Love (1959), Lamb (1952), Morse (1948), Morse and Ingard (1968), Mason (1964), and Friedlander (1958).

As in any kind of wave motion, the physical quantities that describe the acoustic wave motion depend on position and on time. Their time dependence in the domain where the source is acting is impressed by the excitation mechanism of the source. The subsequent dependence on position and time elsewhere in space is governed by the pertaining propagation and scattering laws, together with the principle of causality.

Exercises

Exercise 1, 1

Friedlander, EG., 1958, Sound Pulses, Cambridge: Cambridge University Press. Lamb, H., 1952, Hydrodynamics, 6th edn, Cambridge: Cambridge University Press.

Love, A.E.H., 1959,A Treatise on the Mathematical Theory of Elasticity, 4th edn, Cambridge: Cambridge University Press.

Mason, W.R (Ed.), 1964, Physical Acoustics, New York: Academic Press.

quantities are introduced, through an appropriate spatial averaging procedure: number density of particles, drift velocity, volume density of mass, and mass flow density. The conservation of mass during the flow of particles, together with its consequences, are investigated. Next, the equation of motion of the fluid, the equation of deformation rate, the constitutive relations, the boundary conditions at the interface of two different fluids, and the transport of acoustical energy are discussed. Finally, the low-velocity linearised versions of the equations for acoustic waves in a fluid at rest are presented. As an introduction to the theory of acoustic radiation from sources, the acoustic scalar and vector potentials and the associated point-source solutions (Green’s functions) are introduced.

2.1 Number density, drift velocity, volume density of mass, and mass flow density of a collection of moving padicles

Our analysis starts by considering a collection of identifiable particles whose geometrical dimensions are negligibly small. The collection is present in some domain D in three-dimensional space 9(3. Each particle carries a label by which it can be distinguished from all the other particles, and the particle with the label p occupies, at the instant t, the position x(P)(t). If x(p) changes with time, the (instantaneous) velocity Wr(p) of the particle is given by Wr(p) = dtxr(p), where dt indicates the change of position with time that an observer registers when moving along with the particle. We now select a standard, shift- and time-invariant subdomain De of

D, a so-called representative elementary domain, whose maximum diameter is small compared with the geometrical dimensions of the macroscopic system we are analysing and small compared with the scale on which the macroscopic quantities we are going to introduce show spatial changes, but that nevertheless contains so large a number of particles that it can be considered as an elementary part of a continuum on the macroscopic scale (Figures 2.1-1 and 2.1-2).

Figure 2.1-1 Domain ~9 in which a collection of moving particles is present; ~ge is a time- and

shift-invariant representative elementary domain with centre x.

Figure 9. |-2 Representative elementary domain ~ge with a collection of moving particles.

assumed to vary piecewise continuously with position. (This is the so-called continuum

hypothesis.)

Number density

Let x be the position of the (bary)centre of ~ge(x), and let Ne = Ne(x, t) be the number of particles present in ~De(x). Then, the macroscopic number density n = n(x,t) of the collection of particles attributed to the position x and taken at time t is defined as

where

(2.1-2)

is the volume of De. (The shift invariance of De implies that ifx’~De(x), then ~’~De(0), where

Note: The position of the barycentre of De(x) is defined as

~x

x = V[1 "~

The continuum hypothesis states that n = n(x,t) is a piecewise continuous function of x. The total number of particles N - N(t) present in some bounded domain D - D(t) then follows as the sum of the numbers of particles present in the representative elementary subdomains that belong to D(t), i.e.

N(t) = ~ n(x,t) dV. a x~ (t)

(2.1-4)

Drift velocity

Next, the average velocity, transport velocity, or drift velocity vr of the particles is introduced

as

Ne(x,t)

Vr(X,t) = (w,.)(x,t) = [Ne(x,t)] -1 Z wr(P)(t) ’ (2.1-5)

p=l

where (...) denotes the arithmetic mean of the quantity in angular brackets. It is noted that the chaotic part of the motion of the particles, which determines the thermodynamic notion of their temperature, averages out in Equation (2.1-5) and does not contribute to the right-hand side.

Conservation of particles

Upon following, for a short while At, the collection of particles present in D(t) on its course, a conservation law is arrived at. Let the number of particles present in D(t) at the instant t be

N(t) = ~ n(x,t) dV ,

x~(t)

(2.1-6)

and let these particles occupy at the instant t + At the domain D(t + At). Then, the number of particles N(t + At) present in D(t + At) is given by

N(t + At) = ~ n(x,t + At) dV , a x~ (t+at)

(2.1-7)

particles have somehow been created at the overall rate dtNcr or annihilated at the overall rate

dtNann. The consideration that no other processes than the ones that are mentioned are involved, leads to the balance equation

N(t + At) = N(t) + [dtNcr - d/Nann] At + o(At), as At--~0. (2.1-8) Now, assuming that n = n(x,t) is, throughout !D(t), continuously differentiable with respect to t, we have the Taylor expansion

n(x,t + At) = n(x,t) + ~tn(x,t)At + o(At) , as At---~0. (2.1-9)

Note: For the definition of Landau’s order symbols o and O, see Appendix A, Equations

(A.8-1)-(A.8-6).

Furthermore, the geometry of the domain ~9(t) as it changes with t entails the following (see Figure 2.1-3, where ~D(t + At) is decomposed into the part that it has in common with ~D(t), the part that has been left behind, and the part that has been acquired):

~x

~D ( t+ A t) ,l X~D ( t )

+ f n(x,t)Vr(X,t)At dAr q- o(At) as AI---~0, (2.1-10) x~O D (t)

where O~D(t) is the boundary surface of ~9(t) and vr is the local drift velocity with which the particles on ~!D(t) move, while

Otn(x,t)dV=fx Otn(x,t) dV+o(1) as At---~0. (2.1-11)

x~D (t+At) ~D ( t)

Combining Equations (2.1-9)-(2.1-11), the result

f x n(x,t + At) dV= f n(x,t) dV + f ~tn(x,t)At dV + o(At) ~ (t+At) .t x~D (t+At) d x~D (t+At)

= f rt(x,,) dV + f rt(X,t)Vr(X,t)At dAr x~D (t) ,1 x~D (t)

+ f ~tn(x,t)At dV+ o(At) as At---)0 (2.1-12)

d x~D (t)

is obtained. From Equations (2.1-12) and (2.1-6)-(2.1-8) it follows, by dividing by At and taking the limit At---~0, that

~x

~D (t) ,/xc-x3 ~D (t)

Equation (2.1-13) is known as the conservation law of particle flow.

Introducing the volume densities of the rates of particle creation hcr and particle annihilation

r~ann similar to Equation (2.1-1), we can write

Figure 2.1-3 Conservation law for particles occupying the domain ~9 with boundary surface O~D and

moving with a drift velocity v.

dtNann(t) = dtI

nam~(X,t) dV =I

x~D(t) x~D(t)

(2.1-15)

(The dot over a symbol is a standard notation in physics to indicate the rate of change with time.) Using these expressions in Equation (2.1-13) and applying Gauss’ integral theorem to the second integral on the left-hand side of Equation (2.1-13) under the assumption that nvr is continuously differentiable throughout ~D(t), we obtain

~x

(2.1-16)

Since Equation (2.1-16) has to hold for any domain, and the integrands are assumed to be continuous functions of position, we arrive at (for the justification of this step, see Exercise 2.1-2)

Otn + ~r(nVr) = hcr - r~mm.

(2,1-17) Equation (2.1-17) is known as the continuity equation of particle flow.

Volume density of mass and mass flow density

Let us now concentrate on the mechanical properties of the particles. Consider again the representative elementary domain ~De(x) and let Ne(x,t) be the number of particles present in it. In addition, let m(p) be the mass of the particle with the label p, then the volume density of mass p is defined as

p(x,t) = VZ1 Z re(p)’ p=l

(2.1-18)

Ne(x,t)

¢k(x,t) = Ve-1 ~ m(P)w~)(t). (2.1-19)

p=l

Using Equation (2.1-1), Equation (2.1-18) can be rewritten as N~ P : (Ne/Ve)NfX E m(p) p=1 :n(m>, _(2.1-20) and Equation (2.1-19) as

(N/VON2

In terms of ~e volume density of mass, ~e total mass M = M(0 of the p~icles in some domain O(t) is given by

In what follows, it will be necess~y to distinguish between ~e different types of p~icles as far as their mass is concerned. Let the subscript B be the label that indicates the value of the mass of p~icles of type B. (In as far as ~e mechanical prope~ies ~e concerned, the subscript B indicates the different mechanicN substances out of which the collection of p~icles is composed.) Fu~e~ore, let ~e superscript p denote the label of an individual p~icle wi~in the collection of p~icles of a ce~Nn type. For all p~icles of type B we obviously have

m~) =mB. (2.1-23)

Now, let, N~,B = Ne,B(X,O denote the number of p~icles of type B present in the representative elementary domain ~@). The number density of p~icles of type B is then given by

nB(x,t) = Ne,B(X,t)/Ve,

their volume density of mass by

Ne,B(X,t)

PB(X’t) : V~-I E m~B)

p=l

: [Ne, B(X,t)]Ve] mB

= nB(x,t) mB,

and their mass flow density by

N~,B(x,t)

w E

p=l

(2.1-24)

Ve- 1roB Z (P)

w~;g(t)

p=l Ne,~(x,t)

: INs, B(x,t)/Ve] [Ne,B(X,t)]-ImB Z WB~’,2(t)

p=l

: nB(x,t)mBVB;k(x,t)

= DB(X,t)VB;k(X,t). (2.1-26)

Taking all types of particles together, the contributions from the different substances add up to the total volume density of mass

P=ZPB

B

and the total mass flow density

(2.1-27)

Ck= ~ CB;k. (2.1-28)

B

Conservation of mass

A relationship between PB and CB;k is obtained when the conservation law (Equation (2.1-13)) is applied to the particles of type B. Multiplication of Equation (2.1-13) by mB leads to

fx~D(t)~tpBdV+fd x~OD (t)¢B;~A~=~x ~D (t) (/~B’cr --/~B’ann ) dV’ (2.1-29)

where/5B,cr and/~B,ann denote the volume densitites of the rates at which mass is created and annihilated, respectively, through particles of type B. In the same way, multiplication of Equation (2.1-17) by mB leads to

~tPB + ~kOSB ;k =/~B,cr --/~B,ann ’ (2.1-30)

Now, in the majority of physical processes there is, on a macroscopic scale, no net creation or annihilation of mass. In such a case, summing over all types of particles leads to

~/SB,cr =0 and ~/SB,ann =0. (2.1-31)

B B

Under this condition, summing over all types of particles in Equation (2.1-29) yields

,I xf~D(t)Otp dV +d fx~ofD (t)¢k dAk = O,

(2.1-32) while summing over all types of particles in Equation (2.1-30) yields

~tP + ~kq)k = 0. (2.1-33)

Equation (2.1-32) is known as the conservation law of mass; Equation (2.1-33) is known as the

Stationary flow of particles

A flow of particles is called stationary, or steady, when n, vr, ticr and tiann are independent of time. As the mass of a particle is also independent of time, the quantities p, 05k, ~6cr and/Sann are,

for a stationary mass flow, independent of time.

Static distribution of particles

A distribution of particles is called static if no macroscopic transport of particles takes place; hence, Vr = 0 for a static distribution of particles. Correspondingly, for a static distribution of particles 45~: = 0.

Exercises

Exercise 2.1-1

In a collection of particles with number density n we consider a domain in the shape of a cube with edge length a. (a) What is the value of a if the cube is to contain, on average, a single particle? (b) What is the value of a if n = 26.86763 x 1024 m-3 (Loschmidt constant, i.e. number

density of particles of an ideal gas at a temperature of 273.15 K and at a pressure of 101325 Pa)?

Answers: (a) a = n-1/5; (b) a = 0.33388 x 10-8 m.

Exercise 2. 1-2

Show that iff=f(x,t) is a continuous function ofx and t and

f x f (x,t) d V = O

~D ( t)

for any D(t), thenf(x,t) : 0 for all x~D (t). (Hint: The proof follows by reductio ad absurduln. Assume that f(xo,t) > O, then, on account of the assumed continuity, f(x,t) > 0 in some neighbourhood B0 ofx0. By taking D to be this neighbourhood it follows that

fx

which is contrary to what is given. Repeat the same argument for f(xo,t) < 0 and draw the conclusion.)

Exercise 2. 1-3

three-dimensional rectangle D = {x’~3; xm -- AXm/2 < x£ < xm + Z~Cm]2}. Assume that nvr is continuously differentiable and use for nvr everywhere on 0~9 the first-order Taylor expansion

[nVr](X;t) = [nv,.](x,t) + (xr~ -Xm)Om[nVr](X,t) + o(Ix’-xl) as Ix’-xl-->0. Divide the resulting

expression by AXlZXX2AX3 and take the limit ZhXl--+0, Ax2-->0, Ax3-->0.

Exercise 2. 1-4

Show that for a stationary flow of particles the conservation law of particle flow (Equation (2.1-13)) reduces to

fx~O~(x)v"(x)dAr=fx~D[hcr(X)- tiann(X)] dV.

(2.1-34)

Exercise 2. 1-5

Show that for a stationary mass flow the conservation law of mass (Equation (2.1-32)) reduces to

Ix ~k(x) dAk =O "

(2.1-35)

Exorciso 2.

Show that for a stationary flow of particles the continuity equation of particle flow (Equation (2.1-17)) reduces to

Or(nVr) = ticr- ~ann" (2.1-36)

Exercise 2. 1-7

Show that for a stationary mass flow the continuity equation for mass flow (Equation (2.1-33)), reduces to

0/~45k = 0. (2.1-37)

Exercise 2. 1-8

Prove that for any continuously differentiable function 45 = 45(x,t) the following properties hold:

(a) Ot[VZlf,~x’~D~(x)45(x:t)dV] V~-I f ,,~x~D,(x)0t45(x;t)dV;

(b)

OP[ f ’O dUI= W -I fo

,~ x ~D~ (x) a x’~OD~ (x)

and applying Gauss’ integral theorem to the last integral.)

Exercise 2. 1-9

Show, by using the method that leads from Equations (2.1-6)-(2.1-8) to Equation (2.1-13), that for any function gt= gt(x,t) that is associated with the conservative flow of particles with drift velocity vr we have

dt fx gt(x,t) dV= f x OtlIt(x,t) dV + f gt(x,t)vr(x,t) dAr. _(2.1-38)

~D ( t) _{~ ( t) ,1 x~O~D ( t)}

(This result is known as Reynolds’ transport theorem.)

Exercise 2, 1-10

Show, from the result of Exercise 2.1-9, that for any continuously differentiable function gt =

N(x,t) that is associated with the conservative flow of particles with continuously differentiable

drift velocity Vr = Vr(X,t) we have

dtd x f~(t)gt(x’t) dV=d xf~D(t){ Otgj(x’t) + or [Vr(X’t) gt(x’t) ] } dV .

(Hint: Apply Gauss’ integral theorem to the boundary integral in Equation (2.1-38).) Upon

rewriting the left-hand side as

dt f gt(x,t) dV= f _{~t(x,t) dV,} (2.1-39)

,~ x~ (t) a x~ (t) we also conclude that

~(x,t) = Otgt(x,t) + Or [Vr(X,t)gt(x,t)] . (2.1-40)

2.2 Conservation of the number of particles and its consequences