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Index

absolutely continuous 313adiabatic assumption 19algorithm

DC 65DC-DU 66,67DC-du 67,79DC-wp 65,66Durbin 171Levinson-Durbin 98, 120, 169Levinson's 97, 101, 121, 171Schur 74,121split-Levinson 106

almost everywhere 302analytic continuation 209analytic function 69, 203

singularity of 204Bargman's theorem 265, 320bound state 225, 262, 276Bruckstein and Kailath's equation 106capacitance 15cascade rule 37Cauchy-Riemann equations 211Cauchy sequence 290, 294Cauchy's residue theorem 205, 215, 226Cauchy's theorem 203causality 26, 59, 91, 120, 128causal solutions 48,59,83, 147characteristics 123, 127, 133, 140Cholesky factorization 85, 97Claerbout's equation 96, 97, 101closed contour 203closed set 293compact set 294complex variable

theory of functions of 67,201connected set 69conservation law 2

continuousabsolutely 313uniformly 313

convolution 51integral 49

deconvolution 56difference schemes 40

down-up 46for initial value problem 133for wave problems 40

differentiable 69downward continuation 65, 74Durbin Algorithm 171eigenvalue of matrix 195eigenvector 195

orthogonality of 196entire function 253equation

Bruckstein and Kailath's 106Claerbout's 96, 97, 101conservation 39constitutive 2evolution 4Gel'fand-Levitan 103,237,241Gel'fand-Levitan integral 172, 178Gopinath-Sondhi discrete 115Gopinath-Sondhi integral 182hyperbolic 124integral (see integral equation)Krein's 163Marchenko's 110, 242Maxwell's 15Newton's 2, 16of state 2, 20, 231-D wave 3,17,21,24SchrOdinger'lI 149, 190standardised wave 27

355

356 Index

3-D wave 20telegrapher's 15

equivalence class 292Fourier's problem 211Fourier transform 316

Dirichlet's conditions for 316in L 2 317

Fredholm alternative 165Fredholm integral equation

first kind 164second kind 165

frequency 6circular 6eigen 6resonance 6

functionabsolutely continuous 313analytic: 69differentiable 69entire 253fairly good 14generalised 10good 10Green's (see Green's function)Jost 223,233odd 8of bounded variation 314periodic 8smudge 14square integrable 261uniformly continuous 313

gas law 19Gel'fand-Levitan

discrete equation for finite-differenceSc:hrodinger equation 241

discrete equation for Jacobi matri­ces 237

discrete equation for layered media 103linear integral equation for layered me­

dia 172non-linear integral equation for layered

media 178integral equation for Sc:hrodinger's equa­

tion 326generalised functions 10Gopinath-Sondhi

discrete equation 115integral equation 182

Goupillaud medium 33, 40, 91Green's function 48

causal 49discrete causal 51for Fourier's problem 212non-causal 95of recurrence 227

grideven 41odd 41

Heine-Borel theorem 294Holder condition 208Hooke's Law 2

for elastic: medium 16hyperbolic equation 124ill-conditioned 56, 169impedance

local 29, 30of line 16reconstruction of 149

impulse 49response 50, 60

inductance 15initial-value-boundary-value problem 135initial-value problem 128integral (see also integral equation)

Darboux 301Lebesgue 306Riemann 300Stieltjes 286

integral equationFredholm 164, 165Gopinath-Sondhi 182Krein's 163linear Gel'fand-Levitan 172,326Volterra 252

interval of dependence 128Jacobi matrix 195Jost function 223, 233, 273

and phase shift 273integral representation of 280zeroes of 226, 227

Jost solution 223, 270Levin's representation for 328

Kreindiscrete equation for layered media 110integral equation 163

Kronecker delta 51

Lame constants 16layered media

Goupillaud 33, 119non-Goupillaud 86, 119, 123

layer-peeling methods 55Lebesgue integral 306Levinson-Durbin algorithm 98, 120, 169Levinson's algorithm 97, 101, 121, 171Levinson's theorem 236,278lossless 40Luzin's theorem 306mapping 69Marchenko

discrete equation for finite differenceSchrodinger equation 242

discrete equation for layered media 110integral equation for layered media 181integral equation for Schrodinger's equa-

tion 331matrix

chain scattering 37Jacobi 195orthogonal 39persymmetric 98positive definite 82scattering 40spectral function of 197Toeplitz 83, 97unitary 39

maximum modulus principle 72Maxwell's equations 15measure 219measurable

function 305, 307set 305

measure 219method of successive approximations 252method of variation of parameters 246,

251metric space 292

complete 294distance functions for 293

Newton's equations 2,16noise in data 121non-causal 35

solution 91, 92open

ball 293

Index 357

set 69, 72, 293operator

delay 34linear 3, 172self-adjoint 132time-reversal 92

orthogonality conditionfor eigenfunctions 254

. for regular solutions 229, 283for wave solutions 230, 283

orthogonal polynomials 191three term recurrence relation for 193zeroes of 194

persymmetric 98phase shift 273, 275Plemelj formulae 209pole 205positive definite 82Pozner-Levitan representation 321predictive decomposition 120principal minor 83principle of the argument 206,211,235problem

ill-conditioned 56initial value 128intital-value-boundary-value 135well posed 135

propagationof wavefront 60, 140of singularities 138

quadratic form 82quiescent 24, 128radius of convergence 69recurrence relation

three term 194Chebyshev 194, 221continuous spectrum of 224

reflection function 148integral equation for 152reconstruction of 150upward 75

regularpotential 269, 320, 337sequence 10solution 268

residue 205Riemann integral 300Riemann's representation 131

358 Index

Schrodinger's equation 149, 190finite difference form of 220, 230inverse scattering for 320on finite line 251on half-line 251, 261, 320

Schur algorithm 74, 79, 98, 120Schur's lemma 71Schwarz's lemma 73set

closed 293compact 294connected 69measurable 303of measure zero 302open 293

singularitiesof function 204propagation of 138

solutioncausal 48,59,83, 147Jost 224, 270non-causal 157regular 225, 268wave 224

spaceBanach 297function 295inner product 297linear 292metric 292normed 297

spectral function 197spectrum

continuous 224, 262discrete 263enumerable 264

split-Levinson algorithm 106square-integrable 261standardised wave equation 27Stieltje's integral 286Sturm's theorem 257, 320support 49telegrapher's equation 15theorem

Bargman's 265, 320Cauchy's 203Cauchy's residue 205, 215, 228Heine-Borel 294

Lebesgue's convergence 311Luzin's 306Paley-Wiener 321Sturm's 257, 320Titchmarsh's 329Wiener-Boaz 322

time-difference 111time reversal 92time-sum 111time-symmetric 93Titchmarsh's theorem 329Toeplitz matrix 83, 98, 121transform

Fourier 316z- 50,52

transmission line 15travel time 28, 29Volterra integral equation 151wave 1

compressional 18down 33elastic 27, 30electrical 1evolution of 35incident 26in non-uniform media 123longitudinal 1one-dimensional 2one-parameter 21plane 16,21reflection of 23, 33, 62shear 18sound 18speed of 4standing 5transmission of 23, 33, 62transmitted 26transverse 1travelling 4up 33,140

wavefront 60wavelength 7wave solution 224well-posed 45, 91Wiener-Boaz theorem 322Wronskian 212, 221Yule-Walker equations 169z-tranform 50,52

Index of Notations

o convolution 149* time reversal operator 92. time-difference operator 111- time-sum operator 111• Fourier transform 198

reversal operator 82J Cauchy Principal Value 208

( , ) inner product 191, 238[ , J inner product 230( , ) inner product 297

II II norm 197,297A 125

A,Ao 237A(z) cross section of string 2

cross section of hom 22arbitrary function 252

A(z,O') 327Ag(t) 333

an 98,117as, a? 237

aj 221,246as, a? 238a;({) 142B(z) 252

B(z,r) 293B(z,e) 342

Bo(2z + e) 343bi, b? 237bi({) 143

C 292C 203

Co 268C1 265Cp 20Cv 19Cp 227C; 230

C(z) 15C(S) 305Co(z) 78

c 92c 3,17,30

en 100D 238

D, DO 81D,D+,D- 203

D(e,t) 31Dj 246Dn 333Doj 238

Do(z) 77d 18

d(z) 22d(z,y) 293d(e,t) 138dp(z) 229,241dp(~) 219,238,286

dtJ 63dt(z) 79

E 293E 190en 100F 239,245

F(t) 310F(w) 273F«() 227F(8) 223

Fpc 239F1«(),F1(8) 236

f 199,223/ 6i 321

i(~) 199/(w,z) 273

359

360 Notations

rj 225f,,(z) 73

G 72G(z) 253

G(z,y) 266G(y,z) 327G(e,,,) 212

G(w;z,y) 285

Gj."(') 227g(y) 253gee) 215

gj 218Hi 36

H(t) 236H(z) 13

H(z,y) 266h 110

h n 99h(z) 259h(z) 77h(t) 183

hj 77I(z,t) 15

J 40K,K o 238K(z,t) 323KU,t) 177K(z,t) 324K(z,t) 334Kt(e,t) 181

Kij 238k(e) 31

L(D), L(U) 81L(z) 15

U 311Ln(z) 192

M(n,z) 107M71

m(E) 304N number of zeros 207

number ofbound states 236P n 99

P gas pressure 19number of poles 207

P(w,z) 329Po 18

Pn(z) 192P 259

p(z) 50p(z,t) 17p(e,t) 123

PIc 50Pi,j 41Pi,j 112

Q 19q 81

q(e) 148qj 220

q" 50R field 292

matrix 82R n 293

R gas constant 19radius of convergence 71sum of residues 205

R(z) 337R(e,t) 152

Rm 284~j 84,154

~(z) 75r 110

rCA) 239r(e) 148

S 291S 246

SeC) 225s(9) 224

,,,(C) 248s,,(9) 248

T 98T tension in string 2

period 6absolute temperature 19

T(z) 339T(,) 234

Ti 61T[ 62

Tn(z) 193ti 61

tm 266t~ 62

U 197U, U O 81

Uo(z) 77Uj(z) 241Un(z) 193

Ui,;,UI,; 46u 18

u i 196,237u({) 212

u({, t) 138US 16

US(z) 79US'; 63

V 19V(:I:) 190,341

V(:I:,t) 1511 43

11 (1) 4911(2) 92

11(3),11(4) 93

11 (1) 51'oJ

17i,;, ii,; 111tI({) 212

tin 98W(:I:) 342

W(u,ll) 221W(u,tI) 212

W 251w(z) 51

w( :1:, t) elastic displacement 17volume velocity 23

w({,t) 123W(1) 50

Wlo 51Wi'; 41Wi'; 112X linear space 292

matrix 85z(i) 125

Zn 98y(i) 125

Z 16, 29, 30, 31Zi 24, 33, 86, 149

an 140am 194an 10413m 194ri 38

1 2011 36

1m 1941p 228~ 34

Notations 361

~i 88Li 33

Lie: 207Lii 86

6(:1:) 1161e,; 51

f(W) 275f(8) 224fi,; 16

( 222'1({) 31

'1i 35Bi 36

B({,t) 179A 6A 16

Ai 196Ai,A~ 237

jj 16'll"i 62p density of string 2

density of medium 16p(:I:) 191p(A) 197,237

11 39Ei 38tTi 36Tn 103

Ti,; 16f 148

f(w) 280~n(A) 196

q,(:I:) 252q,({) 140

q,(w,:I:) 252q,(n) 252

q,i 223q,.(A) 196

{ 29X(w,:I:) 322

~ 148~(w) 281,p(:I:) 190,251

,pi 220,pi({) 214

W 6W(A) 212

Wn 6

MechanicsSOUD MECHANICS AND ITS APPLICATIONS

Series Editor: G.M.L. Gladwell

Aims and Scope ofthe Series

The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of thisseries is to provide lucid accounts written by authoritative researchers giving vision and insight inanswering these questions on the subject of mechanics as it relates to solids. The scope of the seriescovers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics;variational formulations; computational mechanics; statics, kinematics and dynamics of rigid andelastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories ofelasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes;structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimentalmechanics; biomechanics and machine design.

I. RT. Haftka, Z. Gilrdal and M.P. Kamat: Elements of Structural Optimization. 2nd rev.ed.,1990 ISBN 0-7923-0608-2

2. lJ. Kalker: Three-Dimensional Elastic Bodies in Rolling Contact. 1990ISBN 0-7923-0712-7

3. P. Karasudhi: Foundations ofSolid Mechanics. 1991 ISBN 0-7923-0772-04. N. Kikuchi: Computational Methods in Contact Mechanics. (forthcoming)

ISBN 0-7923-0773-95. Not published.6. J.F. Doyle: Static and Dynamic Analysis of Structures. With an Emphasis on Mechanics and

Computer Matrix Methods. 1991 ISBN 0-7923-1124-8; Pb 0-7923-1208-27. 0.0. Ochoa and J.N. Reddy: Finite Element Analysis ofComposite Laminates.

ISBN 0-7923-1125-68. M.H. Aliabadi and D.P. Rooke: Numerical Fracture Mechanics. ISBN 0-7923-1175-29. J. Angeles and e.S. L6pez-Cajl1n: Optimization ofCam Mechanisms. 1991

ISBN 0-7923-1355-010. D.E. Grierson, A. Franchi and P. Riva: Progress in Structural Engineering. 1991

ISBN 0-7923-1396-81I. RT. Haftka and Z. Gilrdal: Elements ofStructural Optimization. 3rd rev. and expo ed. 1992

ISBN 0-7923-1504-9; Pb 0-7923-1505-712. J.R. Barber: Elasticity. 1992 ISBN 0-7923-1609-6; Pb 0-7923-161O-X13. H.S. Tzou and G.L. Anderson (oos.): Intelligent Structural Systems. 1992

ISBN 0-7923-1920-614. E.E. Gdoutos: Fracture Mechanics. An Introduction. 1993 ISBN 0-7923-1932-X15. J.P. Ward: Solid Mechanics. An Introduction. 1992 ISBN 0-7923-1949-416. M. Farshad: Design and Analysis ofShell Structures. 1992 ISBN 0-7923-1950-817. H.S. Tzou and T. Fukuda (eds.): Precision Sensors, Actuators and Systems. 1992

ISBN 0-7923-2015-818. J.R Vinson: The Behavior ofShells Composed ofIsotropic and Composite Materials. 1993

ISBN 0-7923-2113-8

Kluwer Academic Publishers - Dordrecht / Boston / London

MechanicsSOUD MECHANICS AND ITS APPLICATIONS

Series Editor: G.M.L. Gladwell

19. H.S. Tzou: Piezoelectric Shells. Distributed Sensing and Control of Continua. 1993ISBN 0-7923-2186-3

20. W. Schiehlen: Advanced Multibody System Dynamics. Simulation and Software Tools. 1993ISBN 0-7923-2192-8

21. C.-W. Lee: Vibration Analysis ofRotors. 1993 ISBN 0-7923-2300-922. D.R. Smith: An Introduction to Continuum Mechanics. 1993 ISBN 0-7923-2454-423. G.M.L. Gladwell: Inverse Problems in Scattering. An Introduction. 1993 ISBN 0-7923-2478-124. G. Prathap: The Finite Element Method in Structural Mechanics. 1993 ISBN 0-7923-2492-725. 1. Herskovits (ed.): Structural Optimization '93. 1993 ISBN 0-7923-2510-926. M.A. Gonzalez-Palacios and 1. Angeles: Cam Synthesis. 1993 ISBN 0-7923-2536-2

Kluwer Academic Publishers - Dordrecht / Boston / London

MechanicsFLUID MECHANICS AND ITS APPLICATIONS

Series Editor: R. Moreau

Aims and Scope ofthe Series

The purpose of this series is to focus on subjects in which fluid mechanics plays a fundamentalrole. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transferetc., books will be published dealing with topics which are currently in a state of rapid develop­ment, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows andnumerical modelling techniques. It is a widely held view that it is the interdisciplinary subjects thatwill receive intense scientific attention, bringing them to the forefront of technological advance­ment. Fluids have the ability to transport matter and its properties as well as transmit force,therefore fluid mechanics is a subject that is particularly open to cross fertilisation with othersciences and disciplines of engineering. The subject of fluid mechanics will be highly relevant indomains such as chemical, metallurgical, biological and ecological engineering. This series isparticularly open to such new multidisciplinary domains.

1. M. Lesieur: Turbulence in Fluids. 2nd rev. ed., 1990 ISBN 0-7923-0645-72. O. Metais and M. Lesieur (eds.): Turbulence and Coherent Structures. 1991

ISBN 0-7923-0646-53. R. Moreau: Magnetohydrodynamics. 1990 ISBN 0-7923-0937-54. E. Coustols (ed.): Turbulence Control by Passive Means. 1990 ISBN 0-7923-1020-95. A.A. Borissov (ed.): Dynamic Structure ofDetonation in Gaseous and Dispersed Media. 1991

ISBN 0-7923-1340-26. K.-S. Choi (ed.): Recent Developments in Turbulence Management. 1991

ISBN 0-7923-1477-87. E.P. Evans and B. Coulbeck (eds.): Pipeline Systems. 1992 ISBN 0-7923-1668-18. B. Nau (ed.): Fluid Sealing. 1992 ISBN 0-7923-1669-X9. T.K.S. Murthy (ed.): Computational Methods in Hypersonic Aerodynamics. 1992

ISBN 0-7923-1673-810. R. King (ed.): Fluid Mechanics of Mixing. Modelling, Operations and Experimental Tech-

niques. 1992 ISBN 0-7923-1720-3II. Z. Han and X. Yin: Shock Dynamics. 1993 ISBN 0-7923-1746-712. L. Svarovsky and M.T. Thew (eds.): Hydroclones. Analysis and Applications. 1992

ISBN 0-7923-1876-513. A. Lichtarowicz (ed.): Jet Cutting Technology. 1992 ISBN 0-7923-1979-614. F.T.M. Nieuwstadt (ed.): Flow Visualization and Image Analysis. 1993 ISBN 0-7923-1994-X15. AJ. Saul (ed.): Floods and Flood Management. 1992 ISBN 0-7923-2078-616. D.E. Ashpis, T.B. Gatski and R. Hirsh (eds.): Instabilities and Turbulence in Engineering

Flows. 1993 ISBN 0-7923-2161-817. R.S. Azad: The Atmospheric Boundary Layer for Engineers. 1993 ISBN 0-7923-2187-118. F.T.M. Nieuwstadt (00.): Advances in Turbulence IV. 1993 ISBN 0-7923-2282-719. K.K. Prasad (ed.): Further Developments in Turbulence Management. 1993

ISBN 0-7923-2291-620. Y.A. Tatarchenko: Shaped Crystal Growth. 1993 ISBN 0-7923-2419-6

Kluwer Academic Publishers - Dordrecht / Boston / London

MechanicsFLUID MECHANICS AND ITS APPLICATIONS

Series Editor: R. Moreau

21. J.P. Bonnet and M.N. Glauser (OOs.): Eddy Structure Identification is Free Turbulent ShearFlows. 1993 ISBN 0-7923-2449-8

Kluwer Academic Publishers - Dordrecht / Boston / London