About the Author ix
Editors’ Preface to the Manchester Physics Series x
Preface to First Edition xi
Preface to Second Edition xiii
About the Companion Website xiv
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 4
1.2.3 Displacement, Velocity and Acceleration in SHM 5
1.2.4 General Solutions for SHM and the Phase Angle φ 7
1.2.5 The Energy of a Simple Harmonic Oscillator 9
1.2.6 The Physics of Small Vibrations 11
1.3 The Pendulum 16
1.3.1 The Simple Pendulum 16
1.3.2 The Energy of a Simple Pendulum 18
1.3.3 The Physical Pendulum 21
1.3.4 Numerical Solution of SHM 23
1.4 Oscillations in Electrical Circuits: Similarities in Physics 26
1.4.1 The LC Circuit 26
1.4.2 Similarities in Physics 27
Problems 1 29
2 The Damped Harmonic Oscillator 35
2.1 Physical Characteristics of Damped Harmonic Oscillators 35
2.2 The Equation of Motion for a Damped Harmonic Oscillator 36
2.2.1 Light Damping 37
2.2.2 Heavy Damping 40
2.2.3 Critical Damping 40
2.3 Rate of Energy Loss in a Damped Harmonic Oscillator 43
2.3.1 The Quality Factor Q of a Damped Harmonic Oscillator 44
2.4 Damped Electrical Oscillations 47
Problems 2 49
3 Forced Oscillations 53
3.1 Physical Characteristics of Forced Harmonic Motion 54
3.2 The Equation of Motion of a Forced Harmonic Oscillator 54
3.2.1 Undamped Forced Oscillations 54
3.2.2 Forced Oscillations with Damping 57
3.3 Power Absorbed During Forced Oscillations 63
3.4 Resonance in Electrical Circuits 68
3.5 Transient Phenomena 70
3.6 The Complex Representation of Oscillatory Motion 72
3.6.1 Complex Numbers 72
3.6.2 The Use of Complex Numbers to Represent Physical Quantities 75
3.6.3 Use of the Complex Representation for Forced Oscillations with Damping 76
Problems 3 78
4 Coupled Oscillators 85
4.1 Physical Characteristics of Coupled Oscillators 85
4.2 Normal Modes of Oscillation 86
4.3 Superposition of Normal Modes 89
4.4 Oscillating Masses Coupled by Springs 93
4.5 Forced Oscillations of Coupled Oscillators 101
4.6 Transverse Oscillations 104
Problems 4 108
5 Travelling Waves 115
5.1 Physical Characteristics of Waves 116
5.2 Travelling Waves 116
5.2.1 Travelling Sinusoidal Waves 119
5.3 The Wave Equation 122
5.4 The Equation of a Vibrating String 124
5.5 The Energy in a Wave 126
5.6 The Transport of Energy by a Wave 129
5.7 Waves at Discontinuities 130
5.8 Waves in Two and Three Dimensions 134
5.8.1 Waves of Circular or Spherical Symmetry 138
Problems 5 141
6 Standing Waves 147
6.1 Standing Waves on a String 147
6.2 Standing Waves as the Superposition of Two Travelling Waves 153
6.3 The Energy in a Standing Wave 155
6.4 Standing Waves as Normal Modes of a Vibrating String 157
6.4.1 The Superposition Principle 157
6.4.2 The Superposition of Normal Modes 158
6.4.3 The Amplitudes of Normal Modes and Fourier Analysis 161
6.4.4 The Energy of Vibration of a String 163
Problems 6 165
7 Interference and Diffraction of Waves 169
7.1 Interference and Huygens’ Principle 169
7.1.1 Young’s Double-Slit Experiment 172
7.1.2 Michelson Spectral Interferometer 178
7.2 Diffraction 180
7.2.1 Diffraction at a Single Slit 181
7.2.2 Circular Apertures and Angular Resolving Power 185
7.2.3 Double Slits of Finite Width 187
Problems 7 188
8 The Dispersion of Waves 193
8.1 The Superposition of Waves in Non-Dispersive Media 193
8.1.1 Beats 194
8.1.2 Amplitude Modulation of a Radio Wave 196
8.2 The Dispersion of Waves 197
8.2.1 Phase and Group Velocities 197
8.3 The Dispersion Relation 201
8.4 Wave Packets 204
8.4.1 Formation of a Wave Packet 205
Problems 8 209
Appendix: Solutions to Problems 215
Index 237
