HANDBOOK OF

MECHANICS

Edited by

W. FLÜGGE, Dr.-Ing. Professor of Engineering Mechanics

Stanford University

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New York Toronto London

McGRAW-HILL BOOK COMPANY, INC.

1962

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P A R T I . M A T H E M A T I C S

1. A l g e b r a , by F. R. Arnold D e t e r m i n a n t s , mat r ices , and l inear a lgebraic equa t ions 1-3; q u a d r a t i c equa t ions and general p roper t i es of in tegra l funct ions of one va r iab le 1-7; series 1-8

2. N u m e r i c a l A l g e b r a , by. M. G. Salvadori '•' Solut ion of algebraic equa t ions 2 - 1 ; solut ion of t r a n s c e n d e n t a l equa t ions 2 -5 ; solut ion

of s imul taneous l inear equa t ions 2-5 . ' ' . ,..'. , ,_

3 . P r o b a b i l i t y , by J. H. B. Kemperman Probab i l i ty 3 -1 ; r a n d o m var iab les 3-2; expec ta t ion va lues 3-2; m o m e n t s 3-3; correla­t ion 3-3; t h e no rma l d i s t r ibu t ion 3-4; corre la t ion con t inued 3-5; t h e centra l - l imit t h e o r e m 3-5; s tochas t ic processes 3-6; spect ra l analysis of a s t a t i ona ry process 3-7;

, , s ta t is t ical m e t h o d s 3-8; t e s t ing whe the r t w o r a n d o m var iab les h a v e t h e s a m e d.f, 3-8; ;,.;.. t e s t ing whe the r t w o r a n d o m var iables are i ndependen t 3-10

4. G e o m e t r y , by E. A. Ripperger '< Coord ina te sys tems 4 - 1 ; circle 4 - 3 ; pa rabo la 4-4; ellipse 4-5 ; hype rbo la 4-6; o ther curves

- 4-7; curvi l inear or thogonal coord ina tes 4-8;;solid, ana ly t i c geomet ry 4-8; differential . „geometry 4 - l l : . . , . . . ,

5. M o m e n t s of I n e r t i a , by H. V. Hahne

6. V e c t o r A n a l y s i s , by H. E. Newell Vector a lgebra 6 - 1 ; vec tor calculus 6-3; general coord ina tes 6-6; po ten t ia l t heo ry 6-8

7. T e n s o r A n a l y s i s , by, A. Schild I n t r o d u c t i o n 7 - 1 ; coordina tes , t r ans fo rma t ions 7-2; re la t ive tensors , examples 7-3 ; tensor a lgebra 7-5; me t r i c tensor , line e lement 7-7; cova r i an t and absolu te der iva t ives 7-10; ca r tes ian tensors 7-13; t ensor form of physical laws 7-17; o r thogona l coordina tes 7-17

8. C a l c u l u s , by A. Bronwell a n d R. H. Niemann Func t ion , l imit , con t inu i ty 8 -1 ; differentiation 8-2; in t eg ra t ion 8-4; funct ion of t w o var iables 8-6; i n d e t e r m i n a t e forms 8-7; m a x i m a a n d m i n i m a 8-9; T a y l o r ' s series and formula 8-10; imprope r in tegra ls 8-10; definite in tegra ls con ta in ing a rb i t r a ry p a r a m -

li eters 8-11; double in tegra ls 8-12; l ine i n t e g r a l 8-13

9. C o m p l e x V a r i a b l e s , by H. Lass I n t r o d u c t i o n 9 - 1 ; graphica l r ep resen ta t ion of complex n u m b e r s 9 -1 ; definition of a

hi .complex function, con t inu i ty 9-3; differentiabil i ty, t h e C a u c h y - R i e m a n n equa t ions 9-3; conformal m a p p i n g 9-6; t h e definite in tegra l 9-8; C a u c h y ' s in tegra l t heo rem 9-9; C a u c h y ' s in tegra l formula 9-11; T a y l o r series expansion 9-11 ; L a u r e n t ' s expansion 9-14; singular poin ts , residues 9-15; t h e res idue theorem, con tour in t eg ra t ion 9-16; t h e Schwartz-Christoffel t r ans fo rma t ion 9-17

10. O r d i n a r y d i f f e r e n t i a l E q u a t i o n s , by G, E, Latta \.\\ General r e m a r k s a b o u t o rd ina ry differential equa t ions 10 -1 ; e l e m e n t a r y m e t h o d s of in tegra t ion 10-2; l inear differential equa t ions 10-5; nonl inear equa t ions 10-19

11. Partial Differential Equations, by I. N. Sneddon Par t ia l differential equa t ions , the i r n a t u r e and occurrence l l - l ; first-order par t i a l differential equa t ions l l r 2 ; classification of second-order l inear equa t ions 11-5; m e t h -

" ods of solut ion of second-order pa r t i a l differential equa t ions 11-6; elliptic equa t ions

XV

XVI CONTENTS

11-13; hyperbolic equations 11-19; parabolic equations 11-23; equations of higher order than the second 11-24

12. Numerical Integration, by J. B. Scarborough Differences and interpolation 12-1; numerical differentiation and integration 12-4; numerical solution of ordinary differential equations 12-7

13. Relaxation Method, by D. N. de G. Allen

14. Numerical Solution of Hyperbolic Equations» 6^ G. G. O'Brien Introduction 14-1; differences 14-1; the numerical solution 14-2; error analysis 14-3; the truncation error 14-3; the numerical error 14-4; Von Neumann's method 14-4; sta­bility of the heat equation 14-4; stability of the wave equation 14-5; some comments on the stability criteria 14-5; an unstable difference equation 14-6; implicit difference equations 14-6

15. Special Functions, by F. Oberhettinger Trigonometric and hyperbolic functions 15-1; orthogonal polynomials 15-3; the gamma function 15-7; the hypergeometric function 15-9; the confluent hypergeometric func­tion 15-11; error functions, Fresnel's integrals, exponential integral, sine and cosine integral 15-12; Bessel functions 15-13; elliptic integrals 15-21; elliptic functions 15-24

16. Calculus of Variations, by P. W. Berg Functionals and the first variation 16-1; the Euler equation 16-2; the second variation and sufficient conditions 16-4; quadratic functionals 16-5; boundary conditions 16-7; subsidiary conditions 16-9; eigenvalue problems 16-10; approximation methods 16-12

17. Integral Equations, by M. A. Heaslet Introduction 17-1; integral equations of the second kind 17-5; integral equations of the first kind 17-11 , - -

18. Eigenvalue Problems, by L. Collatz Classification of matrix eigenvalue problems 18-1; classification of eigenvalue problems for ordinary differential equations 18-7; eigenvalue problems for partial differential equations and integral equations 18-12; extremal properties of the eigenvalues and further inclusion theorems 18-17; further methods for differential equations 18-23

19. Laplace Transformation, by E. J. Scott Definition and existence of the Laplace transform 19-1; the unit step function, the unit impulse, and the unit doublet 19-2; fundamental properties of Laplace transforms 19-4; the complex inversion theorem 19-7; ordinary differential equations 19-11; integrodifferential and integral equations 19-13; partial differential equations 19-14; difference equations 19-17

20. Tables Arcs, areas, volumes, centroids, moments of inertia 20-1; integrals 20-7; Fourier series 20-10

PART 2. MECHANICS OF RIGID BODIES

21. Statics, by D. H. Young Fundamentals of plane statics 21-3; equilibrium of coplanar forces 21-6; fundamentals of graphic statics 21-11; space statics 21-16; principle of virtual displacements 21-21

22. Kinematics, by F. H Raven Motion of a particle 22-1; motion of a rigid body 22-5; methods for motion analysis of mechanisms 22-7

23. Dynamics, by G. W. Housner Introduction 23-1; particle dynamics 23-5; systems of particles 23-9; dynamics of rigid bodies 23-13; alternate forms of the equations of motion 23-20

24. Variational Principles of Mechanics, by C. Lanczos The nature of variational principles 24-1; the configuration space 24-2; the stationary value of V 24-3; the principle of virtual work 24-4; d'Alembert's principle 24-5; Ham­ilton's principle 24-5; the Euler-Lagrange equations 24-6; the geometry of the con­figuration space 24-8; elimination of algebraic variables 24-9; the Lagrangian multiplier method 24-9; physical significance of the Lagrangian. multiplier method 24^11; elimi­nation of kinosthenie variables 24-12; Jacobi's principle 24-13; the principle of Euler-Lagrange 24-15; the canonical equations of Hamilton 24-17; the phase space and the phase fluid 24-18; the extended phase space 24-19; canonical transformations 24-20; the partial differential equation of Hamilton-Jacobi 24-22

CONTENTS XVII

26. Gyroscopes, by H. Ziegler Basic concepts 25*1; free gyros 25-2; constrained precession 25-4; fast symmetric gyros 25-6; the heavy symmetric gyro 25-7; Cardan gimbals and friction 25-10; fast symmetric gyros on vehicles 25-11

PART 3. THEORY OF STRUCTURES

26. Statically Determinate Structures, by J. M. Gere Introduction 26-3; reactions 26-4; plane trusses 26-6; beams 26-11; deflections of structures 26-14; space trusses 26-25

27. Statically Indeterminate Structures, by J. M. Gere and C.-K. Wang Introduction 27-1; the general method 27-2; method of least work 27-5; three-moment equations 27-6; the slope-deflection method 27*8; the moment-distribution method 27-11

28. Influence Diagrams, by W. Flügge Definition 28-1; use of influence lines 28-1; calculation of influence functions of beams 28-2; indirect loading, trusses 28-6; connection with the kernel of integral equations 28-6; influence diagrams for plates 28-8; rotating load 28-11

29. Torsion-box Analysis, by P. Kuhn Introduction 29-1; idealized box of rectangular doubly symmetrical cross section 29-1; the real box of rectangular cross section 29-4; box tapered in width and depth 29-6; curvilinear double symmetrical cross sections 29-6; arbitrary four-flange sections 29-6; cutouts 29-8

30. Second-order Theory, by E. Chwalla General considerations 30-1; suspension bridges 30-10; wide-span arches 30-18

31. Beams on Elastic Foundation, by M. Eetkhyi Basic assumptions, differential equation, general solution 31-1; the infinitely long beam 31-2; the method of end conditioning 31-4; simultaneous axial and transverse forces 31-6; solution by trigonometric series 31-7; fields of application, foundations of various types 31-9

32. Table of Reactions, Bending Moments, and Deflection of Beams

PART 4. ELASTICITY

33. Basic Concepts, by K. Marguerre The concepts of stress and strain 33-3; strain and compatibility 33-4; stress and equilibrium 33-9; Mohr's circle 33-12; Hooke's law and the fundamental equations 33-14; energy principles 33-17; methods for solving the basic equations 33-22

34. Basic Equations in Tensor Notation, by H. J. Weiss The strain tensor 34-1; physical components of strain 34-4; the stress tensor 34-4; equations of equilibrium 34-5; physical components of stress 34-6; stress-strain relations 34-6

36. Bending of Beams, by H. V. Hahne Straight slender bars in pure tension or compression 35-1; bending of slender bars 35-3; strain energy 35-22

36. Torsion, by J. N. Goodier Introduction, bar of circular cross section 36-1; cylinder or prism of any cross section, Saint-Venant theory 36-2; the membrane analogy 36-8; approximate formulas for thin-walled open sections 36-9; stress concentration at notches, corners, and keyways 36-14; effect of initial tensile or bending stress 36-15; effect of pretwisted form 36-16; restrained warping, nonuniform torsion 36-18; torsion of nonuniform circular shafts, Michell theory 36-23

37. Two-dimensional Problems, by Ch. Massonnet General properties 37-1; plane problem in cartesian coordinates 37-5; plane problems in polar coordinates 37-11; numerical solution of edge-load problems by superposition of radial stress systems 37-28

38. Complex-variable Approach, by J. R. M. Radok Complex representation of stresses and displacements 38-1; transformation of coordi­nates, boundary conditions, the fundamental boundary-value problems 38-3; some elementary results, single-valuedness of displacements 38-4; Fourier series, the funda-

/

XV111 CONTENTS

mental problems for the unit circle 38-5; conformal mapping, curvilinear coordinates 38-7; solution for regions, mapped by polynomials 38-9; other methods of solution 38-10

39. Plates, by S. Way Basic equations of laterally loaded plates 39-1; theory of rectangular plates 39-5; circular plates 39-24

40. Shells, by W. Flügge Definitions 40-1; shells of revolution, membrane theory 40-2; cylinders, membrane theory 40-9; general membrane theory 40-12; circular cylinder, bending theory 40-16; shells of revolution, bending theory 40-23

41. Bodies of Revolution, by Y. Y. Yu Basic equations 41-1; methods of solution by the use of stress and displacement func­tions 41-2; direct solution for a hollow sphere subjected to uniform internal and external pressures 41-6; axially symmetric stress distributions in finite-sized bodies 41-7; integral solutions for bodies extending to infinity in certain directions 41-12* problems involving concentrated forces, nuclei of strain 41-16; stress concentrations in bodies of revolution 41-18

42. Contact Problems by J. L. Lubkin The Hertz theory of elastic contact 42-1; extensions of the basic Hertz theory 42-6

43. Thermal Stresses, by H. Parkus General relations 43-1; stress-free temperature fields 43-4; straight or slightly curved bars 43-4; plane strain 43-5; plane stress 43-6; bending of plates 43-7; bodies of revolu­tion 43-8; thin shells of revolution 43-9; applications 43-10; change in rigidity and thermal buckling 43-16; quasi-static nonsteady problems 43-19; dynamic effects 43-19

44. Elastic Stability, by C. Libove Introduction 44-1; theoretical formulations of a classical buckling problem 44-3; illustrative solutions 44-14; numerical data 44-22

45. Nonlinear Problems, by D. G. Ashwell Causes of nonlinearity 45-1; beam on elastic foundation with local lifting off 45-1; the problem of the elastica 45-3; buckling of low arches and slightly curved beams 45-7; external and internal instability 45-9; torsion of thin sections 45-10; transition between beam and plate 45-11; deformation of plates into developable surfaces 45-12; snapping of a thin strip with transverse curvature 45-13; the Brazier effect 45-13; large deflec­tions of plates 45-14; circular plates with axial symmetry 45-15; relaxation method for laterally loaded membranes 45-15; typical results for plates and membranes 45-16; postbuckling behavior of plates and shells 45-17

PART 5. PLASTICITY AND VISCOELASTICITY

46. Basic Concepts, by D. C. Drucker Introduction 46-3; some fundamental aspects of inelastic behavior 46-3; stress-strain relations based on Ji, J\ 46-9; more general loading functions 46-11; a fundamental approach to plasticity and general inelasticity 46-13; idealizations and simplifications 46-14

47. Bending of Beams, by A. Phillips Basic assumptions 47-1; the general equations of bending 47-1; the positions of the neutral axis 47-2; symmetrical bending without axial force 47-2; superposition of perfectly plastic materials 47-6; the neutral axis has no fixed position, bending with axial force 47-7; deflections 47-9; shear stresses in bending of beams 47-11

48. Torsion, by W. F. Freiberger Fundamental equations 48-1; sand-hill analogy 48-2; method of characteristics, uni­form bar 48-3; example, the solid elliptical section 48-7; circular ring sectors and non­uniform circular shafts 48-9

4& Limit Analysis, by P. S. Symonds I Introduction 49-1; basic concepts 49-3; theorems of limit analysis 49-7; remarks on the theorems 49-8; applications of the general theorems 49-10; limit analysis of engi­neering structures 49-13; continuous beams and frames of ductile material 49-14

60. Two-dimensional Problems, by A. P. Green Introduction 50-1; the expansion of a cylindrical tube 50-2; theory of plane plastic strain 50-5; theory of plane stress 50-15

CONTENTS xix

61. Plates and Shells, by P. G. Hodge, Jr. Introduction 51-1; rigid—perfectly plastic circular plates 51-1; elastic—perfectly plastic circular plate 51-3; dynamic loading of circular plates 51-4; circular cylindrical shells 51-5; circular cylindrical shell problems 51-7

52. Inelastic Buckling, by J. E. Duberg Introduction 52-1; equilibrium of the deflected column 52-1; initially straight column 52-3; column with initial imperfections 52-5

63. Viscoelasticity, by E. H. Lee Introduction 53-1; stress-strain-time relations of linear viscoelasticity 53-4; stress analysis 53-11; discussion 53-20

64. Viscoelastic Buckling, by J. Kempner Introduction 54-1; deflection-time characteristics of linearly viscoelastic columns 54-2; deflection-time characteristics of nonlinearly viscoelastic columns 54-8

PART 6. VIBRATIONS

65. Kinematics of Vibrations, by R. S. Ayre Definitions 55-3; vectorial representation 55-3; combination of harmonic vibrations of the same frequency 55-4; nonharmonic periodic motions 55-4; beating 55-5; com­bination of two harmonic motions occurring at right angles to each other, Lissajous figures 55-6; phase-plane representation of vibration 55-7; amplitude modulation and frequency modulation 55-9; complex numbers and exponentials 55-10

56. Systems of One Degree of Freedom, by W. T. Thomson Undamped free vibration 56-1; damped free vibration 56-3; forced vibration with harmonic excitation 56-6; forced vibration with nonperiodic excitation 56-16

67. Systems of Several Degrees of Freedom, by F. B. E. Crossley Number of degrees of freedom 57-1; from physical to mathematical form 57-1; damp­ing 57-3; Lagrangian method 57-3; general equations and solution, free vibration of small amplitude 57-5; obtaining the roots of a polynomial 57-6; modes of free vibration of small amplitude 57-6; wandering of the energy 57-9; normal coordinates 57-9; the stationary property of the normal modes 57-12; orthogonality of the normal modes 57-13; a matrix iteration method 57-15; systems of chain form 57-18; damped free oscillations of systems 57-18; the Routh-Hurwitz stability criteria 57-19; Lagrange's equation for the nonconservative case 57-20; response of a doubly free system to harmonic stimulus 57-21; the dynamic absorber 57-23; extension of fixed-point theorem to multiple-coordinate systems 57-26; general equations of forced. oscillations of a linear system 57-26; transient motions and shock isolation 57-29; vibration isolation in multiple-mode systems 57-31

68. Rotating and Reciprocating Machines, by S. H. Crandall Introduction 58-1; balancing of rotors 58-1; balancing of reciprocating machines 58-3; torsional vibrations 58-7; whirling of shafts 58-13

59. Servomechanisms, Automatic Control, by G. A. Smith Introduction 59-1; mathematical description of a servo system 59-2; frequency-response approach to servo design 59-4; root-locus method 59-11; component transfer functions 59-14

60. Surge Tanks, by C. Jaeger Introduction 60-1; the basic equations of mass oscillations in the simple surge tank 60-2; solution neglecting tunnel friction 60-3; calculation of water-level oscillations including tunnel friction 60-4; introduction of ratios into the calculation of surge tanks 60-4; the graphical method of Schoklitsch 60-5; the stability of single and multiple surge tanks 60-7

61. Continuous Systems, by D. Young Definitions, general procedures 61-1; longitudinal vibrations of uniform bars 61-2; strings, shafts, helical springs, fluid columns 61-5; lateral vibrations of beams, classical theory 61-6; lateral vibrations of beams, Timoshenko theory 61-14; circular rings 61-18; membranes 61-20; plates 61-21; thin shells 61-25; special methods 61-27

62. Dynamic Buckling, by E. Mettler Introduction 62-1; dynamic stability of a beam under plane motion 62-1; additional topics 62-9

X X CONTENTS

63. Flutter, by Y. C. Fung Introduction, relationship of flutter to other aeroelastie problems 63-1; mathematical formulation of the flutter problem 63-3; example 63-6; methods of solution 63-7; dis­crete mass approximation 63-9; structural damping 63-11; modal approach, Galerkin's method of solution 63-11; electric analog methods 63-12; concluding remarks 63-13

64. Propagation of Elastic Waves, by E. E. Zajac Introduction 64-1; small-amplitude transverse waves in a flexible string 64-2; longi­tudinal waves in a bar or string 64-4; torsional waves in a rod of circular cross section 64-6; waves in dispersive, media, flexural waves in a beam (Bernoulli-Euler theory) 64-6; flexural waves in a beam (Timoshenko theory) 64-9; waves in an unbounded isotropic medium 64-11; reflection of plane waves at a plane boundary 64-13; Rayleigh surface waves 64-15; waves in an infinite slab 64-16; waves in circular cylinders, the Poehhammer-Chree waves 64-18

66. Nonlinear Vibrations, by K. Klotter Free oscillations in systems not capable of auto-oscillations 65-1; self-sustained oscil­lations (auto-oscillations) 65-11; forced oscillations 65-18; stability considerations 65-27; special methods 65-31

66. Stochastic Loads, by A. C. Eringen Introduction 66-1; differential equations of distribution and correlation functions 66-6; correlation analysis of viscoelastic systems 66-9; random loads on infinite and semi-infinite media 66-11; Brownian motion of bars and plates 66-13; discrete models for continuous systems 66-16

67. Acoustics, by 0. K, Mawardi The sound field 67-1; mechanisms of sound generation 67-6

PART 7. FLUID MECHANICS

68. Basic Concepts, by C. E. Brown

Properties of fluids 68-3; dynamics of fluids 68-4; boundary conditions 68-12

69. Dimensionless Parameters, oy D. C. Baxter

70. Two-dimensional Ideal Fluid Flow, by V. L. Streeter Definitions 70-1; Euler's equations 70-1; continuity equation 70-2; boundary conditions 70-3; irrotational flow, velocity potential 70-3; integration of Euler's equations, Bernoulli equation 70-4; Laplace equation, equipotential lines 70-5; kinetic energy theorem 70-5; uniqueness theorems 70-6; stream function 70-6; circulation 70-7; complex variable, conformal mapping 70-8; linear operations 70-9; simple conformal transformations 70-9; inverse transformations, elliptic-hyperbolic net 70-14; Blasius' theorem 70-21; flow around circular cylinders 70-21; principles of free-streamline flow 70-23

71. Three-dimensional Ideal Fluid Flow, by I. Flügge-Lotz Basic equations 71-1; simple flow fields 71-4; ellipsoids of revolution 71-8; flow along arbitrary bodies of revolution 71-10; kinetic energy of the fluid flow, apparent mass 71-13; three-dimensional flow with vorticity in limited regions 71-16

72. Airfoil Theory, by A. Robinson Introductory remarks 72-1; two-dimensional wing theory 72-2; Joukowski airfoils 72-7; thin airfoil theory 72-10; Theodorsen's method 72-13; Lighthill's method 72-15; lifting-line theory 72-16; small-aspeet-ratio theory 72-19; general three-dimensional theory 72-21; other problems and methods in three-dimensional theory 72-23

73. General Thermodynamics, by K. Oswatitsch Fundamental concepts 73-1; equation of state 73-1; the first law of thermodynamics 73-3; specific heats, heats of transition 73-7; kinetic theory of gases 73-10; the second law of thermodynamics 73-12; some consequences of the second law of thermodynamics 73-15

74. Thermodynamics of Gas Flow, by A. Boshko Comparison with the static case 74-1; the energy equation 74-1; area-velocity relations, compressibility 74-2; procedure for a general equation of state 74-5; equations for a perfect gas 74-5; the constants of the energy equation 74-7; the area relation for a perfect gas 74-7; shock-wave equations 74-8; shock waves in a perfect gas 74-10; thermodynamics of a flow field 74-12

CONTENTS XXI

76. Subsonic Flow, by N. Bott Introduction 75-1; linearized theory of subsonic flow 75-4; higher approximations 75-12; the hodograph method 74-18; conclusions and discussion of the critical Mach number 75-23 • . : J - , J

76. Transonic Flow, by J. R. Spreiter Introduction 76-1; fundamental equations 76-1; similarity rules, wings 76-3; similarity rules, bodies of revolution 76-3; Mach number freeze at M„ = 1 76-5; transonic equivalence rule 76-5; self-similar solutions for Mx = 1 76-7; hodograph method 76-8; method of successive approximation 76-10; integral equation method 76-10;

' method of local linearization 76-11 .

77. Supersonic Flow, by M. D. Van Dyke Introduction 77-1; oblique shock waves 77-2; Prandtl-Meyer expansion 77-5; wedge and circular cone 77-6; airfoil section theory 77-8; linearized theory 77-12; thin wings 77-16; bodies 77-22

78. Hypersonic Flow, by A. J. Eggers and C. A. Syvertson Introduction 78-1; exact solutions 78-1; approximate solutions 78-1; real-gas and viscous effects 78-9

79. Slender-body Theory, by A. E. Bryson, Jr. Introduction 79-1; differential equation for the velocity potential and the pressure relation, relationship to acoustic theory 79-2; lift of flat wings and bodies of revolution 79-4; the vortex wake behind slender configurations 79-5; modified slender-body theory 79-9; quasi-steady aerodynamic forces on slender airplanes and missiles 79-10

80. Flutter, by I. E. Garrick Introductory remarks 80-1; nonsteady potential flow 80-2; dynamics and aerodynam­ics of idealized two-dimensional case, translation and rotation of a harmonically oscillating wing section 80-3; aerodynamic coefficients for the harmonic case 80-4; flutter determinant 80-5; pressure formula for general harmonic motion (two-dimen­sional flow at M = 0) 80-6; generalized aerodynamic forces 80-7; illustrative example for cantilever wing 80-8; supersonic flow (two-dimensional, nonsteady) 80-10; high Mach-number flows 80-11; supersonic flow (three-dimensional, nonsteady) 80-12; oscillating airfoil in subsonic compressible flow (two-dimensional and three-dimensional methods) 80-13; remarks on intermediate methods, small-aspect-ratio wings, and swept-baek wings 80-14; indicial functions associated with transient flows (growth of lift and gust functions), the superposition principle 80-15; concluding remarks 80-18

81. Flow at Low Reynolds Numbers, by P. A. Lagerstrom and / . D. Chang Introduction 81-1; some simple exact solutions of the Navier-Stokes equations 81-5; Stokes equations 81-11; Oseen equations 81-15; discussion of theoretical results, higher-order approximations 81-27

82. Incompressible Boundary Layers, by J. Kestin Outline of fluid motion with friction, real and perfect fluids 82-1; the physical concept of a boundary layer 82-1; the mathematical concept of a boundary layer 82-5; trans­formation of the boundary-layer equations, steady, two-dimensional flows 82-11; exact solutions of boundary-layer equations 82-13; approximate methods for the solution of the boundary-layer equations 82-19; origin of turbulence 82-23; turbulent flow along a flat plate 82-27

83. Compressible Boundary Layers, by K. Stewartson Introduction 83-1; equations 83-1; properties of Pr and p 83-4; range of application of the equations 83-4; nondimensional form of the equations 83-5; alternative forms of the boundary-layer equations 83-6; integral relations 83-6; the boundary layer on a flat plate in a uniform stream 83-7; boundary layers with pressure gradient 83-10; stability 83-12; interaction between shock waves and boundary layers 83-13 ; hypersonic boundary layer 83-15

84. Turbulence, by S. I. Pai Introduction 84-1; Reynolds' experiments and other facts about turbulent flows 84-1; the turbulent field and mean flow 84-2; Reynolds equations and Reynolds stresses 84-2; semiempirical theories of turbulence 84-3; the logarithmic velocity profile and general resistance law 84-4; turbulent jet-mixing regions and wakes 84-6; fundamentals of the statistical theory of turbulence 84-7; turbulence in compressible fluid flow 84-7

85. Lubrication, by H. Poritsky Slider bearing 85-1; journal bearing 85-3; negative pressures 85-5; end leakage, Reyn-

XX11 CONTENTS

olds' equation 85-6; application of Reynolds' equation 85-7; hydrostatic bearings 85-9; dynamically loaded bearing 85-10; lubrication of gears and rollers 85-12

86. Surface Waves, by T. Y. Wu General formulation of a surf ace-wave problem 86-1; two-dimensional problems 86-3; three-dimensional problems 86-10

87. Cavitation, by M. S. Plesset and B. Perry Introduction 87-1; incipient cavitation 87-2; cavity flow 87-7

88. Flow through Porous Media, by J. 8. Aronofsky Introduction 88-1; fluids 88-1; porous medium 88-1; permeability, Darcy's law 88-1; fluid-flow equations 88-3; typical solutions 88-5

I N D E X FOLLOWS CHAPTER 88. ..

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