Nonlinear Waves in Inhomogeneous and Hereditary Media

ISBN-13: 9783540545361 / Angielski / Miękka / 1992 / 121 str.

Alexandr A. Lokshin; Elena A. Sagomonyan; A. A. Lokshin

Opis

This booklet presents a study of one-dimensional waves in solids which can be modelled by nonlinear wave equations of different types. The factorization method is the main tool in this analysis. It allows for an exact or at least asymp- totic decomposition of the wave(s) under consideration in terms of first order multipliers. Chapter 1 provides a general introduction. It presents some well-known results on characteristics, Riemann invariants, simple waves, etc. The main result of Chap. 1 is Theorem 1.3.2. (Sect. 1.3.2) which establishes the possibility of exact factorization of the nonlinear wave equation EPa(a) 1 EPa _ 0 Ij(l-u- x2 with constant coefficients. This theorem permits one to construct further factor- izations of more complicated wave equations which the reader will meet in the following chapters. Chapter 2 is devoted to short wave processes in inhomogeneous media, the main result being the uniform asymptotic factorization of nonlinear wave equa- tions with variable coefficients and the description of corresponding single-wave processes without the usual assumption of a small wave amplitude.

Spis treści

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.