The Physics of Vibrations and Waves

by Pain, H. John

• ISBN13: 9780470012963
• ISBN10: 047001296X
• Edition: 6th
• Format: Paperback
• Copyright: 2005-05-20
• Publisher: WILEY

Summary

The main theme of this highly successful book is that the transmission of energy by wave propogation is fundamental to almost every branch of physics. Therefore, besides giving students a thorough grounding in the theory of waves and vibrations, the book also demonstrates the pattern and unity of a large part of physics. This new edition has been thoroughly revised and has been redeisgned to meet the best contemporary standards. It includes new material on electron waves in solids using the Kronig-Penney model to show how their allowed energies are limited to Brillouin zones, The role of phonons is also discussed. An Optical Transform is used to demonstrate the modern method of lens testing. In the last two chapters the sections on chaos and solitons have been reduced but their essential contents remain. As with earlier editions, the book has a large number of problems together with hints on how to solve them. The Physics of Vibrations and Waves, 6th Edition will prove invaluable for students taking a first full course in the subject across a variety of disciplines particularly physics, engineering and mathematics.

Author Biography

H. J. Pain is a graduate of the united states Naval Aviation Academy, Florida, and of the Universities of St Andrews, London and Aix Marseilles. He taught physics at Imperial College, London where his research interests were in shock waves and magnetohydrodynamics.

Table of Contents

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
Simple Harmonic Motion	1	(36)
Displacement in Simple Harmonic Motion	4	(2)
Velocity and Acceleration in Simple Harmonic Motion	6	(2)
Energy of a Simple Harmonic Oscillator	8	(2)
Simple Harmonic Oscillations in an Electrical System	10	(2)
Superposition of Two Simple Harmonic Vibrations in One Dimension	12	(3)
Superposition of Two Perpendicular Simple Harmonic Vibrations	15	(2)
*Polarization	17	(3)
Superposition of a Large Number n of Simple Harmonic Vibrations of Equal Amplitude a and Equal Successive Phase Difference δ	20	(2)
*Superposition of n Equal SHM Vectors of Length a with Random Phase	22	(3)
Some Useful Mathematics	25	(12)
Damped Simple Harmonic Motion	37	(16)
Methods of Describing the Damping of an Oscillator	43	(10)
The Forced Oscillator	53	(26)
The Operation of i upon a Vector	53	(1)
Vector form of Ohm's Law	54	(2)
The Impedance of a Mechanical Circuit	56	(1)
Behaviour of a Forced Oscillator	57	(3)
Behaviour of Velocity v in Magnitude and Phase versus Driving Force Frequency ω	60	(2)
Behaviour of Displacement versus Driving Force Frequency ω	62	(2)
Problem on Vibration Insulation	64	(2)
Significance of the Two Components of the Displacement Curve	66	(2)
Power Supplied to Oscillator by the Driving Force	68	(1)
Variation of Pav with ω. Absorption Resonance Curve	69	(1)
The Q-Value in Terms of the Resonance Absorption Bandwidth	70	(1)
The Q-Value as an Amplification Factor	71	(3)
The Effect of the Transient Term	74	(5)
Coupled Oscillations	79	(28)
Stiffness (or Capacitance) Coupled Oscillators	79	(2)
Normal Coordinates, Degrees of Freedom and Normal Modes of Vibration	81	(5)
The General Method for Finding Normal Mode Frequencies, Matrices, Eigenvectors and Eigenvalues	86	(1)
Mass or Inductance Coupling	87	(3)
Coupled Oscillations of a Loaded String	90	(5)
The Wave Equation	95	(12)
Transverse Wave Motion	107	(44)
Partial Differentiation	107	(1)
Waves	108	(1)
Velocities in Wave Motion	109	(1)
The Wave Equation	110	(2)
Solution of the Wave Equation	112	(3)
Characteristic Impedance of a String (the string as a forced oscillator)	115	(2)
Reflection and Transmission of Waves on a String at a Boundary	117	(3)
Reflection and Transmission of Energy	120	(1)
The Reflected and Transmitted Intensity Coefficients	120	(1)
The Matching of Impedances	121	(3)
Standing Waves on a String of Fixed Length	124	(2)
Energy of a Vibrating String	126	(1)
Energy in Each Normal Mode of a Vibrating String	127	(1)
Standing Wave Ratio	128	(1)
Wave Groups and Group Velocity	128	(4)
Wave Group of Many Components. The Bandwidth Theorem	132	(3)
Transverse Waves in a Periodic Structure	135	(3)
Linear Array of Two Kinds of Atoms in an Ionic Crystal	138	(2)
Absorption of Infrared Radiation by Ionic Crystals	140	(1)
Doppler Effect	141	(10)
Longitudinal Waves	151	(20)
Sound Waves in Gases	151	(4)
Energy Distribution in Sound Waves	155	(2)
Intensity of Sound Waves	157	(2)
Longitudinal Waves in a Solid	159	(2)
Application to Earthquakes	161	(1)
Longitudinal Waves in a Periodic Structure	162	(1)
Reflection and Transmission of Sound Waves at Boundaries	163	(1)
Reflection and Transmission of Sound Intensity	164	(7)
Waves on Transmission Lines	171	(28)
Ideal or Lossless Transmission Line	173	(1)
Coaxial Cables	174	(1)
Characteristic Impedance of a Transmission Line	175	(2)
Reflections from the End of a Transmission Line	177	(1)
Short Circuited Transmission Line (ZL = 0)	178	(1)
The Transmission Line as a Filter	179	(4)
Effect of Resistance in a Transmission Line	183	(3)
Characteristic Impedance of a Transmission Line with Resistance	186	(1)
The Diffusion Equation and Energy Absorption in Waves	187	(3)
Wave Equation with Diffusion Effects	190	(1)
Appendix	191	(8)
Electromagnetic Waves	199	(40)
Maxwell's Equations	199	(3)
Electromagnetic Waves in a Medium having Finite Permeability μ and Permittivity ε but with Conductivity σ = 0	202	(2)
The Wave Equation for Electromagnetic Waves	204	(2)
Illustration of Poynting Vector	206	(1)
Impedance of a Dielectric to Electromagnetic Waves	207	(1)
Electromagnetic Waves in a Medium of Properties μ, ε and σ (where σ ≠ 0)	208	(3)
Skin Depth	211	(1)
Electromagnetic Wave Velocity in a Conductor and Anomalous Dispersion	211	(1)
When is a Medium a Conductor or a Dielectric?	212	(2)
Why will an Electromagnetic Wave not Propagate into a Conductor?	214	(1)
Impedance of a Conducting Medium to Electromagnetic Waves	215	(2)
Reflection and Transmission of Electromagnetic Waves at a Boundary	217	(5)
Reflection from a Conductor (Normal Incidence)	222	(1)
Electromagnetic Waves in a Plasma	223	(4)
Electromagnetic Waves in the Ionosphere	227	(12)
Waves in More than One Dimension	239	(28)
Plane Wave Representation in Two and Three Dimensions	239	(1)
Wave Equation in Two Dimensions	240	(2)
Wave Guides	242	(3)
Normal Modes and the Method of Separation of Variables	245	(1)
Two-Dimensional Case	246	(1)
Three-Dimensional Case	247	(1)
Normal Modes in Two Dimensions on a Rectangular Membrane	247	(3)
Normal Modes in Three Dimensions	250	(1)
Frequency Distribution of Energy Radiated from a Hot Body. Planck's Law	251	(2)
Debye Theory of Specific Heats	253	(1)
Reflection and Transmission of a Three-Dimensional Wave at a Plane Boundary	254	(2)
Total Internal Reflection and Evanescent Waves	256	(11)
Fourier Methods	267	(38)
Fourier Series	267	(7)
Application of Fourier Sine Series to a Triangular Function	274	(1)
Application to the Energy in the Normal Modes of a Vibrating String	275	(3)
Fourier Series Analysis of a Rectangular Velocity Pulse on a String	278	(3)
The Spectrum of a Fourier Series	281	(2)
Fourier Integral	283	(2)
Fourier Transforms	285	(1)
Examples of Fourier Transforms	286	(1)
The Slit Function	286	(1)
The Fourier Transform Applied to Optical Diffraction from a Single Slit	287	(2)
The Gaussian Curve	289	(3)
The Dirac Delta Function, its Sifting Property and its Fourier Transform	292	(1)
Convolution	292	(5)
The Convolution Theorem	297	(8)
Waves in Optical Systems	305	(28)
Light, Waves or Rays?	305	(2)
Fermat's Principle	307	(1)
The Laws of Reflection	307	(2)
The Law of Refraction	309	(1)
Rays and Wavefronts	310	(3)
Ray Optics and Optical Systems	313	(1)
Power of a Spherical Surface	314	(2)
Magnification by the Spherical Surface	316	(1)
Power of Two Optically Refracting Surfaces	317	(1)
Power of a Thin Lens in Air (Figure 11.12)	318	(2)
Principal Planes and Newton's Equation	320	(1)
Optical Helmholtz Equation for a Conjugate Plane at Infinity	321	(1)
The Deviation Method for (a) Two Lenses and (b) a Thick Lens	322	(3)
The Matrix Method	325	(8)
Interference and Diffraction	333	(78)
Interference	333	(1)
Division of Amplitude	334	(3)
Newton's Rings	337	(1)
Michelson's Spectral Interferometer	338	(2)
The Structure of Spectral Lines	340	(1)
Fabry - Perot Interferometer	341	(2)
Resolving Power of the Fabry - Perot Interferometer	343	(12)
Division of Wavefront	355	(2)
Interference from Two Equal Sources of Separation f	357	(6)
Interference from Linear Array of N Equal Sources	363	(3)
Diffraction	366	(3)
Scale of the Intensity Distribution	369	(1)
Intensity Distribution for Interference with Diffraction from N Identical Slits	370	(2)
Fraunhofer Diffraction for Two Equal Slits (N = 2)	372	(1)
Transmission Diffraction Grating (N Large)	373	(1)
Resolving Power of Diffraction Grating	374	(2)
Resolving Power in Terms of the Bandwidth Theorem	376	(1)
Fraunhofer Diffraction from a Rectangular Aperture	377	(2)
Fraunhofer Diffraction from a Circular Aperture	379	(4)
Fraunhofer Far Field Diffraction	383	(3)
The Michelson Stellar Interferometer	386	(2)
The Convolution Array Theorem	388	(3)
The Optical Transfer Function	391	(4)
Fresnel Diffraction	395	(8)
Holography	403	(8)
Wave Mechanics	411	(48)
Origins of Modern Quantum Theory	411	(3)
Heisenberg's Uncertainty Principle	414	(3)
Schrodinger's Wave Equation	417	(2)
One-dimensional Infinite Potential Well	419	(3)
Significance of the Amplitude Ψ n(x) of the Wave Function	422	(2)
Particle in a Three-dimensional Box	424	(1)
Number of Energy States in Interval E to E + dE	425	(1)
The Potential Step	426	(8)
The Square Potential Well	434	(4)
The Harmonic Oscillator	438	(3)
Electron Waves in a Solid	441	(9)
Phonons	450	(9)
Non-linear Oscillations and Chaos	459	(46)
Free Vibrations of an Anharmonic Oscillator - Large Amplitude Motion of a Simple Pendulum	459	(1)
Forced Oscillations - Non-linear Restoring Force	460	(3)
Thermal Expansion of a Crystal	463	(2)
Non-linear Effects in Electrical Devices	465	(2)
Electrical Relaxation Oscillators	467	(2)
Chaos in Population Biology	469	(8)
Chaos in a Non-linear Electrical Oscillator	477	(4)
Phase Space	481	(4)
Repellor and Limit Cycle	485	(1)
The Torus in Three-dimensional (x,x,t) Phase Space	485	(2)
Chaotic Response of a Forced Non-linear Mechanical Oscillator	487	(1)
A Brief Review	488	(6)
Chaos in Fluids	494	(10)
Recommended Further Reading	504	(1)
References	504	(1)
Non-linear Waves, Shocks and Solitons	505	(28)
Non-linear Effects in Acoustic Waves	505	(3)
Shock Front Thickness	508	(1)
Equations of Conservation	509	(1)
Mach Number	510	(1)
Ratios of Gas Properties Across a Shock Front	511	(1)
Strong Shocks	512	(1)
Solitons	513	(18)
Bibliography	531	(1)
References	531	(2)
Appendix 1: Normal Modes, Phase Space and Statistical Physics	533	(14)
Mathematical Derivation of the Statistical Distributions	542	(5)
Appendix 2: Kirchhoff's Integral Theorem	547	(4)
Appendix 3: Non-Linear Schrodinger Equation	551	(1)
Index	552