Vibrations and Waves

This introductory text emphasises physical principles, rather than the mathematics. Each topic begins with a discussion of the physical characteristics of the motion or system. The mathematics is kept as clear as possible, and includes elegant mathematical descriptions where possible. Designed to provide a logical development of the subject, the book is divided into two sections, vibrations followed by waves. A particular feature is the inclusion of many examples, frequently drawn from everyday life, along with more cutting-edge ones. Each chapter includes problems ranging in difficulty from simple to challenging and includes hints for solving problems. Numerous worked examples included throughout the book.

Auteur
Professor George C. King, Department of Physics & Astronomy, University of Manchester, Manchester, UK.

Texte du rabat
The Manchester Physics Series
General Editors: F.K. Loebinger; F. Mandl; D.J. Sandiford, School of Physics and Astronomy, The University of Manchester Properties of Matter: B.H. Flowers and E Mendoza

Statistical Physics, Second Edition: F. Mandl

Electromagnetism, Second Edition: I.S. Grant and W.R.Phillips

Statistics, R.J. Barlow

Solid State Physics, Second Edition: J.R. Hook and H.E. Hall

Quantum Mechanics, F. Mandl

Computing for Scientists, R.J. Barlow and A.R. Barnett

The Physics of Stars, Second Edition, A.C. Phillips

Nuclear Physics, J.S. Lilley

Introduction to Quantum Mechanics, A.C. Phillips

Particle Physics, Third Edition: B.R. Martin and G. Shaw

Dynamics and Relativity, J.R. Forshaw and A.G. Smith

Vibrations and Waves, G.C. King

Vibrations and Waves is based on an introductory course given regularly by the author. The text provides the student with a thorough grounding in the theory of vibrations and waves.

Throughout the book, the fundamental principles of vibrations and waves are emphasised so that these principles can be applied to a wide range of oscillating systems and to different kinds of waves.

The text, which is divided into two sections, vibrations followed by waves, follows a logical progression from the simple harmonic oscillator to waves in continuous media.

Vibrations and Waves includes:

• Vibrations and waves beautifully and concisely described in terms of the mathematical equations used throughout the book
• Worked examples throughout
• Problems ranging in difficulty from simple to challenging
  Solutions and hints to the problems at the end of the book

Contenu

Editors' Preface to the Manchester Physics Series xi

Author's Preface xiii

1 SIMPLE HARMONIC MOTION 1

1.1 Physical Characteristics of Simple Harmonic Oscillators 1

1.2 A Mass on a Spring 2

1.2.1 A mass on a horizontal spring 2

1.2.2 A mass on a vertical spring 5

1.2.3 Displacement, velocity and acceleration in simple harmonic motion 5

1.2.4 General solutions for simple harmonic motion and the phase angle 7

1.2.5 The energy of a simple harmonic oscillator 10

1.2.6 The physics of small vibrations 12

1.3 The Pendulum 17

1.3.1 The simple pendulum 17

1.3.2 The energy of a simple pendulum 19

1.3.3 The physical pendulum 22

1.3.4 Numerical solution of simple harmonic motion3 24

1.4 Oscillations in Electrical Circuits: Similarities in Physics 27

1.4.1 The LC circuit 27

1.4.2 Similarities in physics 29

PROBLEMS 1 29

2 THE DAMPED HARMONIC OSCILLATOR 33

2.1 Physical Characteristics of the Damped Harmonic Oscillator 33

2.2 The Equation of Motion for a Damped Harmonic Oscillator 34

2.2.1 Light damping 35

2.2.2 Heavy damping 37

2.2.3 Critical damping 38

2.3 Rate of Energy Loss in a Damped Harmonic Oscillator 41

2.3.1 The quality factor Q of a damped harmonic oscillator 43

2.4 Damped Electrical Oscillations 46

PROBLEMS 2 47

3 FORCED OSCILLATIONS 49

3.1 Physical Characteristics of Forced Harmonic Motion 50

3.2 The Equation of Motion of a Forced Harmonic Oscillator 50

3.2.1 Undamped forced oscillations 50

3.2.2 Forced oscillations with damping 54

3.3 Power Absorbed During Forced Oscillations 60

3.4 Resonance in Electrical Circuits 64

3.5 Transient Phenomena 66

3.6 The Complex Representation of Oscillatory Motion 68

3.6.1 Complex numbers 68

3.6.2 The use of complex numbers to represent physical quantities 71

3.6.3 Use of the complex representation for forced oscillations with damping 74

PROBLEMS 3 74

4 COUPLED OSCILLATORS 77

4.1 Physical Characteristics of Coupled Oscillators 77

4.2 Normal Modes of Oscillation 78

4.3 Superposition of Normal Modes 81

4.4 Oscillating Masses Coupled by Springs 87

4.5 Forced Oscillations of Coupled Oscillators 93

4.6 Transverse Oscillations 96

PROBLEMS 4 99

5 TRAVELLING WAVES 105

5.1 Physical Characteristics of Waves 106

5.2 Travelling Waves 106

5.2.1 Travelling sinusoidal waves 109

5.3 The Wave Equation 112

5.4 The Equation of a Vibrating String 114

5.5 The Energy in a Wave 116

5.6 The Transport of Energy by a Wave 119

5.7 Waves at Discontinuities 121

5.8 Waves in Two and Three Dimensions 126

5.8.1 Waves of circular or spherical symmetry 130

PROBLEMS 5 133

6 STANDING WAVES 137

6.1 Standing Waves on a String 137

6.2 Standing Waves as the Superposition of Two Travelling Waves 144

6.3 The Energy in a Standing Wave 147

6.4 Standing Waves as Normal Modes of a Vibrating String 149

6.4.1 The superposition principle 149

6.4.2 The superposition of normal modes 150

6.4.3 The amplitudes of normal modes and Fourier analysis 153

6.4.4 The energy of vibration of a string 156

PROBLEMS 6 158

7 INTERFERENCE AND DIFFRACTION OF WAVES 161

7.1 Interference and Huygen's Principle 161

7.1.1 Young's double-slit experiment 163

7.1.2 Michelson spectral interferometer 170

7.2 Diffraction 172

7.2.1 Diffraction at a single slit 172

7.2.2 Circular apertures and angular resolving power 177

7.2.3 Double slits of finite width 179

PROBLEMS 7 181 **8 ...