CONTENTS

Cover

Title page

Acknowledgement

About the companion website

Preface

Introduction

Table of Constants

Table of Energy Storing Processes

1 Simple Harmonic Motion

1.1 Displacement in Simple Harmonic Motion

1.2 Velocity and Acceleration in Simple Harmonic Motion

1.3 Energy of a Simple Harmonic Oscillator

1.4 Simple Harmonic Oscillations in an Electrical System

1.5 Superposition of Two Simple Harmonic Vibrations in One Dimension

2 Damped Simple Harmonic Motion

Introduction

2.1 Complex Numbers

2.2 The Exponential Series

2.3 Methods of Describing the Damping of an Oscillator

3 The Fornced Oscillator

Introduction

3.1 The Operation of i upon a Vector

3.2 Vector Form of Ohm's Law

3.3 The Tuned LCR Circuit

3.4 Power Supplied to Oscillator by the Input Voltage

3.5 The Q-Value in Terms of the Resonance Absorption Bandwidth

3.6 The Forced Mechanical Oscillator

3.7 Behaviour of Velocity v in Magnitude and Phase versus Driving Force Frequency ω

3.8 Behaviour of Displacement x versus Driving Force Frequency ω

3.9 The Q-Value as an Amplification Factor

3.10 Significance of the Two Components of the Displacement Curve

3.11 Problem on Vibration Insulation

3.12 The Effect of the Transient Term

4 Coupled Oscillations

Introduction

4.1 Stiffness (or Capacitance) Coupled Oscillators

4.2 Normal Modes of Vibration, Normal Coordinates and Degrees of Freedom

4.3 Mass or Inductance Coupling

4.4 Coupled Oscillations of a Loaded String

4.5 The Wave Equation

5 Transverse Wave Motion (1)

Introduction

5.1 Partial Differentiation

5.2 Waves

5.3 Velocities in Wave Motion

5.4 The Wave Equation

5.5 Solution of the Wave Equation

5.6 Characteristic Impedance of a String (the String as a Forced Oscillator)

5.7 Reflection and Transmission of Waves on a String at a Boundary

5.8 Reflection and Transmission of Energy

5.9 The Reflected and Transmitted Intensity Coefficients

5.10 Matching of Impedances

5.11 Standing Waves on a String of Fixed Length

5.12 Standing Wave Ratio

5.13 Energy in Each Normal Mode of a Vibrating String

6 Transverse Wave Motion (2)

Introduction

6.1 Wave Groups, Group Velocity and Dispersion

6.2 Wave Group of Many Components. The Bandwidth Theorem

6.3 Heisenberg’s Uncertainty Principle

6.4 Transverse Waves in Periodic Structures (1) Waves in a Crystal

6.5 Linear Array of Two Kinds of Atoms in an Ionic Crystal

6.6 Transverse Waves in Periodic Structures (2) The Diffusion Equation, Energy Loss from Wave Systems

7 Longitudinal Waves

Introduction

7.1 Sound Waves in Gases

7.2 Energy Distribution in Sound Waves

7.3 Intensity of Sound Waves

7.4 Longitudinal Waves in a Solid

7.5 Application to Earthquakes

7.6 Reflection and Transmission of Sound Waves at Boundaries

7.7 Reflection and Transmission of Sound Intensity

7.8 Water Waves

7.9 Doppler Effect

8 Waves on Transmission Lines

Introduction

8.1 Ideal or Lossless Transmission Line

8.2 Coaxial Cables

8.3 Characteristic Impedance of a Transmission Line

8.4 Reflections from the End of a Transmission Line

8.5 Short Circuited Transmission Line (ZL = 0)

8.6 The Transmission Line as a Filter

8.7 Effect of Resistance in a Transmission Line

8.8 Characteristic Impedance of a Transmission Line with Resistance

8.9 Matching Impedances

9 Electromagnetic Waves

Introduction

9.1 Maxwell’s Equations

9.2 Electromagnetic Waves in a Medium having Finite Permeability μ and Permittivity ε but with Conductivity σ = 0

9.3 The Wave Equation for Electromagnetic Waves

9.4 Illustration of Poynting Vector

9.5 Impedance of a Dielectric to Electromagnetic Waves

9.6 Electromagnetic Waves in a Medium of Properties μ, ε and σ (where σ ≠ 0)

9.7 Skin Depth

9.8 Electromagnetic Wave Velocity in a Conductor and Anomalous Dispersion

9.9 When is a Medium a Conductor or a Dielectric?

9.10 Why will an Electromagnetic Wave not Propagate into a Conductor?

9.11 Impedance of a Conducting Medium to Electromagnetic Waves

9.12 Reflection and Transmission of Electromagnetic Waves at a Boundary

9.13 Reflection from a Conductor (Normal Incidence)

10 Waves in More Than One Dimension

Introduction

10.1 Plane Wave Representation in Two and Three Dimensions

10.2 Wave Equation in Two Dimensions

10.3 Wave Guides

10.4 Normal Modes and the Method of Separation of Variables

10.5 Two-Dimensional Case

10.6 Three-Dimensional Case

10.7 Normal Modes in Two Dimensions on a Rectangular Membrane

10.8 Normal Modes in Three Dimensions

10.9 3D Normal Frequency Modes and the de Broglie Wavelength

10.10 Frequency Distribution of Energy Radiated from a Hot Body. Planck’s Law

10.11 Debye Theory of Specific Heats

11 Fourier Methods

11.1 Fourier Series

11.2 Application of Fourier Sine Series to a Triangular Function

11.3 Application to the Energy in the Normal Modes of a Vibrating String

11.4 Fourier Series Analysis of a Rectangular Velocity Pulse on a String

11.5 Three-Phase Full Wave Rectification

11.6 The Spectrum of a Fourier Series

12 Waves in Optics (1) Interference

12.1 Light. Waves or Rays?

12.2 Fermat’s Principle

12.3 The Laws of Reflection

12.4 The Law of Refraction

12.5 Interference and Diffraction

12.6 Interference

12.7 Division of Amplitude

12.8 Newton’s Rings

12.9 Michelson’s Spectral Interferometer

12.10 The Structure of Spectral Lines

12.11 Fabry–Pérot Interferometer

12.12 Resolving Power of the Fabry–Pérot Interferometer

12.13 Division of Wavefront

12.14 Interference from Two Equal Sources of Separation f

12.15 Interference from Linear Array of N Equal Sources

13 Waves in Optics (2) Diffraction

13.1 Diffraction

13.2 Scale of the Intensity Distribution

13.3 Intensity Distribution for Interference with Diffraction from N Identical Slits

13.4 Fraunhofer Diffraction for Two Equal Slits (N = 2)

13.5 Transmission Diffraction Grating (N Large)

13.6 Resolving Power of Diffraction Grating

13.7 Resolving Power in Terms of the Bandwidth Theorem

13.8 Fraunhofer Diffraction from a Rectangular Aperture

13.9 Fraunhofer Diffraction from a Circular Aperture

13.10 The Airy Disc and Resolving Power

13.11 The Michelson Stellar Interferometer

13.12 Fresnel Diffraction

13.13 Zone Plate

13.14 Electron Diffraction and Brillouin Zones

14 Non-linear Oscillations

14.1 Free Vibrations of an Anharmonic Oscillator – Large Amplitude Motion of a Simple Pendulum

14.2 Forced Oscillations – Non-linear Restoring Force

14.3 Thermal Expansion of a Crystal

14.4 Non-linear Acoustic Waves and Shocks

14.5 Mach Number

Appendix 1: The Binomial Theorem

Appendix 2: Taylor’s and the Exponential Series

Appendix 3: Superposition of a Large Number n of Simple Harmonic Vibrations of Equal Amplitude a and Equal Successive Phase Difference δ

Appendix 4: Superposition of n Equal SHM Vectors of Length a with Random Phase ϕ

Appendix 5: Electromagnetic Wave Equations: Vector Method

Appendix 6: Planck’s Radiation Law

Appendix 7: Fraunhofer Diffraction from a Rectangular Aperture

Appendix 8: Reflection and Transmission Coefficients for a Wave Meeting a Boundary

Index

End User License Agreement

List of Tables

Chapter 13

Table 13.1 Relative intensities of subsidiary maxima in Figure 13.2 where I₀ is the principal maximum intensity, the slit width is d and a = n/λd sin θ. These are the origins of a, b and c in Figure 13.9.

Appendix 8

Table A1 Waves Incident Normally on a Plane Boundary Between Media of Characteristic Impedances Z₁ and Z₂

List of Illustrations

Chapter 1

Figure 1.1 Simple harmonic oscillators with their equations of motion and angular frequencies ω of oscillation. (a) A mass on a frictionless plane connected by a spring to a wall. (b) A simple pendulum. (c) A torsional pendulum. (d) A mass at the centre of a string under constant tension T. (e) A fixed length of non-viscous liquid in a U-tube of constant cross-section. (f) An acoustic Helmholtz resonator. (g) A hydrometer mass m in a liquid of density ρ. (h) An electrical L C resonant circuit.

Figure 1.2 Sinusoidal displacement of simple harmonic oscillator with time, showing variation of starting point in cycle in terms of phase angle ϕ.

Figure 1.3 Variation with time of displacement, velocity and acceleration in simple harmonic motion. Displacement lags velocity by π/2 rad and is π rad out of phase with the acceleration. The initial phase constant ϕ is taken as zero.

Figure 1.4 Parabolic representation of potential energy and kinetic energy of simple harmonic motion versus displacement. Inversion of one curve with respect to the other shows a 90° phase difference. At any displacement value the sum of the ordinates of the curves equals the total constant energy E.

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