+612 9045 4394
 
CHECKOUT
Nonlinear Waves in Inhomogeneous and Hereditary Media : Research Reports in Physics - Alexandr A. Lokshin

Nonlinear Waves in Inhomogeneous and Hereditary Media

Research Reports in Physics

Paperback

Ships: 15 business days
15 business days
$212.95
or 4 easy payments of $53.24 with Learn more

This monograph presents the study of one-dimensional waves in inhomogenous and hereditary media (media with memory). A new, mathematically rigorous factorization method is applied, which yields either an exact or an asymptotic decomposition of the nonlinear wave equations under consideration into first order multipliers. A central result presented in this book is a factorization theorem for the simple wave equation with constant coefficients, which permits a view of Riemann invariants and simple waves from a different angle, and which is used to go over to wave equations with variable coefficients and memory.

1. Nonlinear Waves in Homogeneous Media.- 1.1 Preliminaries.- 1.1.1 Equations of Motion of a Homogeneous Nonlinear Rod.- 1.1.2 Riemann Invariants and Characteristics.- 1.1.3 Simple Wave Equation.- 1.1.4 Conditions on the Strong Shock.- 1.1.5 Stability Condition for the Strong Shock.- 1.1.6 Weak Shocks.- 1.2 Nonlinear Hyperbolic Equations of the First Order.- 1.2.1 Conditions on the Shock.- 1.2.2 Constancy of the Integrals of Solutions.- 1.2.3 Solution of the Boundary Value Problem Method of Characteristics.- 1.2.4 Wave Breaking.- 1.2.5 Principle of Equal Areas.- 1.2.6 An Example.- 1.2.7 Ordinary Differential Equation for a Shock Propagating into an Undisturbed Domain.- 1.3 Exact Factorization of the Nonlinear Wave Equation with Constant Coefficients.- 1.3.1 Introductory Observations.- 1.3.2 Factorization Theorem for the Wave Equation for Stress.- 1.3.3 Difference Between Linear and Nonlinear Factorization.- 1.3.4 Factorization Theorem for the Deformation Wave Equation.- 1.3.5 Earnshaw's Theorem.- 1.3.6 Generalization of Earnshaw's Theorem.- 1.3.7 A Boundary Value Problem Posed in Terms of Displacements.- 1.4 Shock-Wave in a Simple System.- 1.4.1 Formulation of the Problem.- 1.4.2 Nonconformity of the Single-Wave Equation to the Shock Condition.- 1.4.3 Transformation of the Single-Wave Equation. Integral Equation for g(?) Generating the Transformation.- 1.4.4 Construction of the Function g(?).- 1.4.5 Discussion of the Results.- 1.5 The Shock-Wave in a Simple System (Continuation).- 1.5.1 Application of the Principle of Equal Areas.- 1.5.2 Application of Euler's Method.- 2. Nonlinear Short Waves of Finite Amplitude in Inhomogeneous Media.- 2.1 Asymptotic Factorization of the Nonlinear Wave Equation with a Variable Coefficient.- 2.1.1 Representation of the Nonlinear Wave Equation with a Variable Coefficient.- 2.1.2 Formulation of the Boundary Value Problem. Conditions of Asymptotic Factorization.- 2.1.3 Single-Wave Solution of the Boundary Value Problem.- 2.2 When is the Factorization Exact?.- 2.2.1 Nonlinear Case.- 2.2.2 Linear Case.- 2.3 Asymptotic Factorization of the General Nonlinear Wave Equation with Variable Coefficients.- 2.3.1 Preliminary Notes.- 2.3.2 Notation.- 2.3.3 Representation of the General Nonlinear Wave Equation with Variable Coefficients.- 2.3.4 Formulation of the Boundary Value Problem Conditions of Asymptotic Factorization.- 2.3.5 Linear Case.- 2.4 Evolution of Maximal Amplitude of the Stress Wave.- 2.4.1 Formulation of the Problem.- 2.4.2 Equation for Maximal Amplitudes.- 2.4.3 The Curve of Maximums as a Characteristic.- 2.5 Propagation of a Stress Wave in a Homogeneous Nonlinear Elastic Rod Located in the Gravity Field.- 2.5.1 Formulation of the Problem.- 2.5.2 Uselessness of Exact Factorization.- 2.5.3 Asymptotic Factorization.- 2.5.4 Single-Wave Solution of the Problem.- 3. Nonlinear Waves in Media with Memory.- 3.1 Hereditary Elasticity.- 3.1.1 Linear Equations.- 3.1.2 Nonlinear Equations.- 3.2 Small Quadratic Nonlinearity.- 3.2.1 Asymptotic Factorization of the Nonlinear Wave Equation with Memory.- 3.2.2 Why Can't the Factorization be Exact?.- 3.2.3 Single-Wave Equation.- 3.2.4 Condition on the Shock for the Stress Wave.- 3.2.5 New Notation.- 3.3 Continuous Stationary Profile Waves and Nonzero Solutions of Homogeneous Integral Volterra Equations.- 3.3.1 Waves Propagating in an Undisturbed Medium.- 3.3.2 Integral Equation for the Wave of Stationary Profile.- 3.3.3 Estimate of the Solution of the Integral Equation.- 3.3.4 Existence of Stationary Profile Waves. Special Case.- 3.3.5 Existence of the Wave of Stationary Profile. General Case.- 3.3.6 The Exponential Kernel.- 3.3.7 The Simplest Oscillatory Kernel.- 3.3.8 A More Complicated Oscillatory Kernel.- 3.3.9 Waves Propagating in a Prestressed Medium.- 3.3.10 The Exponential Kernel.- 3.4 Stationary Profile Shock-Waves and Self-Coordinated Integral Volterra Equations.- 3.4.1 Waves Propagating in an Undisturbed Medium.- 3.4.2 Integral Equation for Stationary Profile Waves.- 3.4.3 Estimate of the Solution of the Integral Equation.- 3.4.4 Existence of Stationary Profile Shock-Waves.- 3.4.5 The Power Kernel.- 3.4.6 The Exponential Kernel.- 3.4.7 Waves Propagating in a Prestressed Medium.- 3.5 Waves Tending to a Stationary Profile.- 3.5.1 Intuitive Approach.- 3.5.2 Rok's Method.- 3.6 Nonstationary Waves Analog of the Landau-Whitham Formula.- 3.6.1 Formulation of the Problem.- 3.6.2 Linear Case.- 3.6.3 Case of Small Quadratic Nonlinearity.- 3.6.4 Estimate of Quality of the Approximate Solution.- 3.6.5 Single-Wave Equation for Deformation.- 3.6.6 Single-Wave Equation for Displacement.- 3.6.7 A Boundary Value Problem Posed in Terms of Displacement.- 3.7 General Nonlinearity. Further Factorization Theorems for Nonlinear Wave Equations with Memory.- 3.7.1 Preliminary Notes.- 3.7.2 The Exact Factorization Theorem.- 3.7.3 The Asymptotic Factorization Theorem.- 3.7.4 Waves in Rods in the Presence of External Friction.- 3.8 Nonstationary Waves for an Exponential Memory Function.- 3.8.1 Formulation of the Problem.- 3.8.2 Derivation of a Single-Wave Differential Equation.- 3.8.3 The Analytic Solution in a Smoothness Domain.- 3.8.4 Wave Breaking.- 3.8.5 Case of Small Amplitudes. Asymptotic Analysis of the Shock-Wave.- 3.9 Reflection of a Wave from the Boundary Between Linear Elastic and Nonlinear Hereditary Media.- 3.9.1 Formulation of the Boundary Value Problem.- 3.9.2 Reduction of the Problem to an Integro-Functional Equation.- 3.9.3 Solution of the Integro-Functional Equation.- 3.10 The Exactly Factorizable Linear Wave Equation with Memory and a Variable Coefficient.- 3.10.1 Factorization Theorem.- 3.10.2 Solution of the Boundary Value Problem.- References.

ISBN: 9783540545361
ISBN-10: 3540545360
Series: Research Reports in Physics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 121
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 0.76
Weight (kg): 0.2