Knobel R. An Introduction to the Mathematical Theory of Waves

Providence: American Mathematical Society, 1999. —212 p.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.ForewordIntroduction to Waves
Wave phenomena
Examples of waves
A Mathematical Representation of Waves
Representation of one-dimensional waves
Methods for visualizing functions of two variables
ial Differential Equations
Introduction and examples
An intuitive view
Terminology
Traveling and Standing Waves
Traveling Waves
Traveling waves
Wave fronts and pulses
Wave trains and dispersion
The Korteweg-de Vries Equation
The KdV equation
Solitary wave solutions
The Sine-Gordon Equation
A mechanical transmission line
The Sine-Gordon equation
Traveling wave solutions
The Wave Equation
Vibrating strings
A derivation of the wave equation
Solutions of the wave equation
D'Alembert's Solution of the Wave Equation
General solution of the wave equation
The d'Alembert form of a solution
Vibrations of a Semi-infinite String
A semi-infinite string with fixed end
A semi-infinite string with free end
Characteristic Lines of the Wave Equation
Domain of dependence and range of influence
Characteristics and solutions of the wave equation
Solutions of the semi-infinite problem
Standing Wave Solutions of the Wave Equation
Standing waves
Standing wave solutions of the wave equation
Standing waves of a finite string
Modes of vibration
Standing Waves of a Nonhomogeneous String
The wave equation for a nonhomogeneous string
Standing waves of a finite string
Modes of vibration
Numerical calculation of natural frequencies
Superposition of Standing Waves
Finite superposition
Infinite superposition
Fourier Series and the Wave Equation
Fourier sine series
Fourier series solution of the wave equation
Waves in Conservation Laws
Conservation Laws
Derivation of a general scalar conservation law
Constitutive equations
Examples of Conservation Laws
Plug flow chemical reactor
Diffusion
Traffic flow
The Method of Characteristics
Advection equation
Nonhomogeneous advection equation
General linear conservation laws
Nonlinear conservation laws
Gradient Catastrophes and Breaking Times
Gradient catastrophe
Breaking time
Shock Waves
Piecewise smooth solutions of a conservation law
Shock wave solutions of a conservation law
Shock Wave Example: Traffic at a Red Light
An initial value problem
Shock wave solution
Shock Waves and the Viscosity Method
Another model of traffic flow
Traveling wave solutions of the new model
Viscosity
Rarefaction Waves
An example of a rarefaction wave
Stopped traffic at a green light
An Example with Rarefaction and Shock Waves
Nonunique Solutions and the Entropy Condition
Nonuniqueness of piecewise smooth solutions
The entropy condition
Weak Solutions of Conservation Laws
Classical solutions
The weak form of a conservation law