Selected Topics in Nonlinear Wave Mechanics

This book gives an overview ofthe current state of nonlinear wave mechanics with emphasis on strong discontinuities (shock waves) and localized self­ preserving shapes (solitons) in both elastic and fluid media. The exposition is intentionallyat a detailed mathematical and physical level, our expectation being that the reader will enjoy coming to grips in a concrete manner with advances in this fascinating subject. Historically, modern research in nonlinear wave mechanics began with the famous 1858 piston problem paper of Riemann on shock waves and con­ tinued into the early part of the last century with the work of Hadamard, Rankine, and Hugoniot. After WWII, research into nonlinear propagation of dispersive waves rapidly accelerated with the advent of computers. Works of particular importance in the immediate post-war years include those of von Neumann, Fermi, and Lax. Later, additional contributions were made by Lighthill, Glimm, Strauss, Wendroff, and Bishop. Dispersion alone leads to shock fronts of the propagating waves. That the nonlinearity can com­ pensate for the dispersion, leading to propagation with a stable wave having constant velocity and shape (solitons) came as a surprise. A solitary wave was first discussed by J. Scott Russell in 1845 in "Report of British Asso­ ciations for the Advancement of Science. " He had, while horseback riding, observed a solitary wave travelling along a water channel and followed its unbroken progress for over a mile.

Table of Contents:
1 Elastic Surface Waves, Resonances, and Inverse Acoustic Scattering.- 1.1 Introduction.- 1.2 Waves on elastic plates: Lamb-type and Scholte-Stoneley waves.- 1.3 Acoustic Interactions Wich Ocean-Floor Sediments.- 1.4 Conclusions.- 2 Amplitude Equation Models for the Interaction of Shocks with Nonlinear Dispersive Wave Envelopes.- 2.1 Introduction.- 2.2 The Weak Coupling Limit.- 2.3 The Riemann Problem.- 2.4 The Zero Diffusion Limit.- 2.5 Dynamics Prior to Shock Onset.- 2.6 The Incompressible Limit.- 2.7 Exact Solutions.- 2.8 The ? = 0 Problem.- 2.9 The Multi-Scale Expansions.- 2.10 Modulation Equations for the Dispersive Wave.- 2.11 Analysis of the Effective Shock Equation for the Nonlinear Limit.- 2.12 Analysis of the Effective Shock Equation in the Linear Limit.- 2.13 Numerical Experiments.- 3 Some Aspects of One-Dimensional Finite Amplitude Elastic Wave Propagation.- 3.1 First Order Equations.- 3.2 Systems of First Order Equations.- 3.3 Elastic Strings.- 3.4 Problems for membranes.- 3.5 Some problems for three dimensional elastic bodies.- 3.6 Thermodynamic considerations.- 4 Nonlinear Duality Between Elastic Waves and Quasi-particles.- 4.1 Introduction.- 4.2 Hyperbolicity and Conservation Laws.- 4.3 Elasticity as a field theory.- 4.4 Solitonic systems.- 4.5 Nearly integrable systems.- 4.6 Examples from Continuum Mechanics.- 4.7 Conclusions.- 5 Time-Harmonic Waves in Pre-Stressed Dissipative Materials.- 5.1 Introduction.- 5.2 Linearized Equations in Solids.- 5.3 Constitutive Equation for Solids.- 5.4 Linearized Equations and Restrictions for Fluids.- 5.5 Propagation Condition for Inhomogeneous Waves.- 5.6 Dissipation and Wave Decay.- 5.7 A New Approach to Wave Propagation.- 5.8 Rays in Pre-Stressed Solids.- 6 Dissipative Effects on the Evolution of Internal SolitaryWaves in a t Sheared, Stably Stratified Fluid Layer.- 6.1 Introduction.- 6.2 Model system.- 6.3 Discussion.- 6.4 Conclusions.- 7 Dissipative Nonlinear Strain Waves in Solids.- 7.1 Introduction.- 7.2 Modelling of nonlinear waves in an elastic rod.- 7.3 Mathematical tools for analysis of the governing equations.- 7.4 Amplification of a strain solitary wave in a narrowing rod.- 7.5 Selection of nonlinear strain waves in an elastic rod due to the influence of an external medium.- 7.6 Influence of macro- and micro-dissipation on the formation of dissipative solitary waves.- 7.7 Concluding remarks.