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 |  موضوع: كتاب The Physics of Vibrations and Waves الثلاثاء 21 مايو 2013, 10:03 am | |
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The Physics of Vibrations and Waves - H. J. Pain
Sixth Edition
H. J. Pain
Formerly of Department of Physics,
Imperial College of Science and Technology, London, UK
ويتناول الموضوعات الأتية :
Introduction to First Edition xi
Introduction to Second Edition xii
Introduction to Third Edition xiii
Introduction to Fourth Edition xiv
Introduction to Fifth Edition xv
Introduction to Sixth Edition xvi
1 Simple Harmonic Motion 1
Displacement in Simple Harmonic Motion 4
Velocity and Acceleration in Simple Harmonic Motion 6
Energy of a Simple Harmonic Oscillator 8
Simple Harmonic Oscillations in an Electrical System 10
Superposition of Two Simple Harmonic Vibrations in One Dimension 12
Superposition of Two Perpendicular Simple Harmonic Vibrations 15
Polarization 17
Superposition of a Large Number n of Simple Harmonic Vibrations of
Equal Amplitude a and Equal Successive Phase Difference d 20
Superposition of n Equal SHM Vectors of Length a with Random Phase 22
Some Useful Mathematics 25
2 Damped Simple Harmonic Motion 37
Methods of Describing the Damping of an Oscillator 43
3 The Forced Oscillator 53
The Operation of i upon a Vector 53
Vector form of Ohm’s Law 54
The Impedance of a Mechanical Circuit 56
Behaviour of a Forced Oscillator 57
Behaviour of Velocity v in Magnitude and Phase versus Driving Force Frequency x 60
Behaviour of Displacement versus Driving Force Frequency x 62
Problem on Vibration Insulation 64
Significance of the Two Components of the Displacement Curve 66
Power Supplied to Oscillator by the Driving Force 68
Variation of Pav with x. Absorption Resonance Curve 69
The Q-Value in Terms of the Resonance Absorption Bandwidth 70
The Q-Value as an Amplification Factor 71
The Effect of the Transient Term 74
4 Coupled Oscillations 79
Stiffness (or Capacitance) Coupled Oscillators 79
Normal Coordinates, Degrees of Freedom and Normal Modes of Vibration 81
The General Method for Finding Normal Mode Frequencies, Matrices,
Eigenvectors and Eigenvalues 86
Mass or Inductance Coupling 87
Coupled Oscillations of a Loaded String 90
The Wave Equation 95
5 Transverse Wave Motion 107
Partial Differentiation 107
Waves 108
Velocities in Wave Motion 109
The Wave Equation 110
Solution of the Wave Equation 112
Characteristic Impedance of a String (the string as a forced oscillator) 115
Reflection and Transmission of Waves on a String at a Boundary 117
Reflection and Transmission of Energy 120
The Reflected and Transmitted Intensity Coefficients 120
The Matching of Impedances 121
Standing Waves on a String of Fixed Length 124
Energy of a Vibrating String 126
Energy in Each Normal Mode of a Vibrating String 127
Standing Wave Ratio 128
Wave Groups and Group Velocity 128
Wave Group of Many Components. The Bandwidth Theorem 132
Transverse Waves in a Periodic Structure 135
Linear Array of Two Kinds of Atoms in an Ionic Crystal 138
Absorption of Infrared Radiation by Ionic Crystals 140
Doppler Effect 141
6 Longitudinal Waves 151
Sound Waves in Gases 151
vi Contents
Energy Distribution in Sound Waves 155
Intensity of Sound Waves 157
Longitudinal Waves in a Solid 159
Application to Earthquakes 161
Longitudinal Waves in a Periodic Structure 162
Reflection and Transmission of Sound Waves at Boundaries 163
Reflection and Transmission of Sound Intensity 164
7 Waves on Transmission Lines 171
Ideal or Lossless Transmission Line 173
Coaxial Cables 174
Characteristic Impedance of a Transmission Line 175
Reflections from the End of a Transmission Line 177
Short Circuited Transmission Line ðZL ¼ 0Þ 178
The Transmission Line as a Filter 179
Effect of Resistance in a Transmission Line 183
Characteristic Impedance of a Transmission Line with Resistance 186
The Diffusion Equation and Energy Absorption in Waves 187
Wave Equation with Diffusion Effects 190
Appendix 191
8 Electromagnetic Waves 199
Maxwell’s Equations 199
Electromagnetic Waves in a Medium having Finite Permeability l and
Permittivity e but with Conductivity r ¼ 0 202
The Wave Equation for Electromagnetic Waves 204
Illustration of Poynting Vector 206
Impedance of a Dielectric to Electromagnetic Waves 207
Electromagnetic Waves in a Medium of Properties l, e and r (where r 6¼ 0) 208
Skin Depth 211
Electromagnetic Wave Velocity in a Conductor and Anomalous Dispersion 211
When is a Medium a Conductor or a Dielectric? 212
Why will an Electromagnetic Wave not Propagate into a Conductor? 214
Impedance of a Conducting Medium to Electromagnetic Waves 215
Reflection and Transmission of Electromagnetic Waves at a Boundary 217
Reflection from a Conductor (Normal Incidence) 222
Electromagnetic Waves in a Plasma 223
Electromagnetic Waves in the Ionosphere 227
9 Waves in More than One Dimension 239
Plane Wave Representation in Two and Three Dimensions 239
Wave Equation in Two Dimensions 240
Contents vii
Wave Guides 242
Normal Modes and the Method of Separation of Variables 245
Two-Dimensional Case 246
Three-Dimensional Case 247
Normal Modes in Two Dimensions on a Rectangular Membrane 247
Normal Modes in Three Dimensions 250
Frequency Distribution of Energy Radiated from a Hot Body. Planck’s Law 251
Debye Theory of Specific Heats 253
Reflection and Transmission of a Three-Dimensional Wave at a
Plane Boundary 254
Total Internal Reflection and Evanescent Waves 256
10 Fourier Methods 267
Fourier Series 267
Application of Fourier Sine Series to a Triangular Function 274
Application to the Energy in the Normal Modes of a Vibrating String 275
Fourier Series Analysis of a Rectangular Velocity Pulse on a String 278
The Spectrum of a Fourier Series 281
Fourier Integral 283
Fourier Transforms 285
Examples of Fourier Transforms 286
The Slit Function 286
The Fourier Transform Applied to Optical Diffraction from a Single Slit 287
The Gaussian Curve 289
The Dirac Delta Function, its Sifting Property and its Fourier Transform 292
Convolution 292
The Convolution Theorem 297
11 Waves in Optical Systems 305
Light. Waves or Rays? 305
Fermat’s Principle 307
The Laws of Reflection 307
The Law of Refraction 309
Rays and Wavefronts 310
Ray Optics and Optical Systems 313
Power of a Spherical Surface 314
Magnification by the Spherical Surface 316
Power of Two Optically Refracting Surfaces 317
Power of a Thin Lens in Air (Figure 11.12) 318
Principal Planes and Newton’s Equation 320
Optical Helmholtz Equation for a Conjugate Plane at Infinity 321
The Deviation Method for (a) Two Lenses and (b) a Thick Lens 322
The Matrix Method 325
viii Contents
12 Interference and Diffraction 333
Interference 333
Division of Amplitude 334
Newton’s Rings 337
Michelson’s Spectral Interferometer 338
The Structure of Spectral Lines 340
Fabry -- Perot Interferometer 341
Resolving Power of the Fabry -- Perot Interferometer 343
Division of Wavefront 355
Interference from Two Equal Sources of Separation f 357
Interference from Linear Array of N Equal Sources 363
Diffraction 366
Scale of the Intensity Distribution 369
Intensity Distribution for Interference with Diffraction from N Identical Slits 370
Fraunhofer Diffraction for Two Equal Slits ðN ¼ 2Þ 372
Transmission Diffraction Grating (N Large) 373
Resolving Power of Diffraction Grating 374
Resolving Power in Terms of the Bandwidth Theorem 376
Fraunhofer Diffraction from a Rectangular Aperture 377
Fraunhofer Diffraction from a Circular Aperture 379
Fraunhofer Far Field Diffraction 383
The Michelson Stellar Interferometer 386
The Convolution Array Theorem 388
The Optical Transfer Function 391
Fresnel Diffraction 395
Holography 403
13 Wave Mechanics 411
Origins of Modern Quantum Theory 411
Heisenberg’s Uncertainty Principle 414
Schro¨dinger’s Wave Equation 417
One-dimensional Infinite Potential Well 419
Significance of the Amplitude wnðxÞ of the Wave Function 422
Particle in a Three-dimensional Box 424
Number of Energy States in Interval E to E þ dE 425
The Potential Step 426
The Square Potential Well 434
The Harmonic Oscillator 438
Electron Waves in a Solid 441
Phonons 450
14 Non-linear Oscillations and Chaos 459
Free Vibrations of an Anharmonic Oscillator -- Large Amplitude Motion of
a Simple Pendulum 459
Contents ix
Forced Oscillations – Non-linear Restoring Force 460
Thermal Expansion of a Crystal 463
Non-linear Effects in Electrical Devices 465
Electrical Relaxation Oscillators 467
Chaos in Population Biology 469
Chaos in a Non-linear Electrical Oscillator 477
Phase Space 481
Repellor and Limit Cycle 485
The Torus in Three-dimensional ð_x; x; t) Phase Space 485
Chaotic Response of a Forced Non-linear Mechanical Oscillator 487
A Brief Review 488
Chaos in Fluids 494
Recommended Further Reading 504
References 504
15 Non-linear Waves, Shocks and Solitons 505
Non-linear Effects in Acoustic Waves 505
Shock Front Thickness 508
Equations of Conservation 509
Mach Number 510
Ratios of Gas Properties Across a Shock Front 511
Strong Shocks 512
Solitons 513
Bibliography 531
References 531
Appendix 1: Normal Modes, Phase Space and Statistical Physics 533
Mathematical Derivation of the Statistical Distributions 542
Appendix 2: Kirchhoff’s Integral Theorem 547
Appendix 3: Non-Linear Schro¨dinger Equation 551
Index
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