This book is based on an undergraduate course taught at the IAS/Park City Mathematics Institute (Utah) on linear and nonlinear waves. The first part of the text overviews the concept of a wave, describes one-dimensional waves using functions of two variables, provides an introduction to partial differential equations, and discusses computer-aided visualization techniques. The second part of the book discusses traveling waves, leading to a description of solitary waves and soliton solutions of the Klein-Gordon and Korteweg-deVries equations. The wave equation is derived to model the small vibrations of a taut string, and solutions are constructed via d'Alembert's formula and Fourier series. The last part of the book discusses waves arising from conservation laws. After deriving and discussing the scalar conservation law, its solution is described using the method of characteristics, leading to the formation of shock and rarefaction waves. Applications of these concepts are then given for models of traffic flow.
Author(s): Roger Knobel
Series: Student Mathematical Library 3
Publisher: American Mathematical Society
Year: 1999
Language: English
Pages: 196
Tags: mathematics;waves;vibrations;wave equation;differential equations
Cover ... 1
Title ... 6
Copyright ... 7
Contents ... 8
Foreword ... 12
Preface ... 14
Part 1. Introduction ... 16
Chapter 1. Introduction to Waves ... 18
§1.1. Wave phenomena ... 19
§1.2. Examples of waves ... 19
Chapter 2. A Mathematical Representation of Waves ... 22
§2.1. Representation of one-dimensional waves ... 22
§2.2. Methods for visualizing functions of two variables ... 24
Chapter 3. Partial Differential Equations ... 28
§3.1. Introduction and examples ... 28
§3.2. An intuitive view ... 30
§3.3. Terminology ... 32
Part 2. Traveling and Standing Waves ... 36
Chapter 4. Traveling Waves ... 38
§4.1. Traveling waves ... 38
§4.2. Wave fronts and pulses ... 41
§4.3. Wave trains and dispersion ... 42
Chapter 5. The Korteweg-de Vries Equation ... 46
§5.1. The KdV equation ... 47
§5.2. Solitary wave solutions ... 47
Chapter 6. The Sine-Gordon Equation ... 52
§6.1. A mechanical transmission line ... 52
§6.2. The Sine-Gordon equation ... 53
§6.3. Traveling wave solutions ... 57
Chapter 7. The Wave Equation ... 60
§7.1. Vibrating strings ... 60
§7.2. A derivation of the wave equation ... 61
§7.3. Solutions of the wave equation ... 65
Chapter 8. D'Alembert's Solution of the Wave Equation ... 68
§8.1. General solution of the wave equation ... 68
§8.2. The d'Alembert form of a solution ... 70
Chapter 9. Vibrations of a Semi-infinite String ... 74
§9.1. A semi-infinite string with fixed end ... 74
§9.2. A semi-infinite string with free end ... 79
Chapter 10. Characteristic Lines of the Wave Equation ... 82
§10.1. Domain of dependence and range of influence ... 82
§10.2. Characteristics and solutions of the wave equation ... 84
§10.3. Solutions of the semi-infinite problem ... 88
Chapter 11. Standing Wave Solutions of the Wave Equation ... 92
§11.1. Standing waves ... 92
§11.2. Standing wave solutions of the wave equation ... 93
§11.3. Standing waves of a finite string ... 95
§11.4. Modes of vibration ... 98
Chapter 12. Standing Waves of a Nonhomogeneous String ... 102
§12.1. The wave equation for a nonhomogeneous string ... 102
§12.2. Standing waves of a finite string ... 103
§12.3. Modes of vibration ... 105
§12.4. Numerical calculation of natural frequencies ... 106
Chapter 13. Superposition of Standing Waves ... 110
§13.1. Finite superposition ... 110
§13.2. Infinite superposition ... 113
Chapter 14. Fourier Series and the Wave Equation ... 116
§14.1. Fourier sine series ... 116
§14.2. Fourier series solution of the wave equation ... 121
Part 3. Waves in Conservation Laws ... 126
Chapter 15. Conservation Laws ... 128
§15.1. Derivation of a general scalar conservation law ... 128
§15.2. Constitutive equations ... 131
Chapter 16. Examples of Conservation Laws ... 134
§16.1. Plug flow chemical reactor ... 134
§16.2. Diffusion ... 136
§16.3. Traffic flow ... 138
Chapter 17. The Method of Characteristics ... 142
§17.1. Advection equation ... 142
§17.2. Nonhomogeneous advection equation ... 146
§17.3. General linear conservation laws ... 148
§17.4. Nonlinear conservation laws ... 149
Chapter 18. Gradient Catastrophes and Breaking Times ... 152
§18.1. Gradient catastrophe ... 153
§18.2. Breaking time ... 156
Chapter 19. Shock Waves ... 160
§19.1. Piecewise smooth solutions of a conservation law ... 160
§19.2. Shock wave solutions of a conservation law ... 162
Chapter 20. Shock Wave Example: Traffic at a Red Light ... 168
§20.1. An initial value problem ... 168
§20.2. Shock wave solution ... 169
Chapter 21. Shock Waves and the Viscosity Method ... 174
§21.1. Another model of traffic flow ... 174
§21.2. Traveling wave solutions of the new model ... 176
§21.3. Viscosity ... 178
Chapter 22. Rarefaction Waves ... 180
§22.1. An example of a rarefaction wave ... 180
§22.2. Stopped traffic at a green light ... 184
Chapter 23. An Example with Rarefaction and Shock Waves ... 188
Chapter 24. Nonunique Solutions and the Entropy Condition ... 196
§24.1. Nonuniqueness of piecewise smooth solutions ... 196
§24.2. The entropy condition ... 198
Chapter 25. Weak Solutions of Conservation Laws ... 202
§25.1. Classical solutions ... 202
§25.2. The weak form of a conservation law ... 203
Bibliography ... 208
Index ... 210
A ... 210
B ... 210
C ... 210
D ... 210
E ... 210
F ... 210
G ... 210
H ... 210
I ... 210
K ... 210
M ... 211
N ... 211
P ... 211
R ... 211
S ... 211
T ... 211
V ... 211
W ... 211
Back Cover ... 212
