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Nonlinear Waves in Elastic Media explores the theoretical results of one-dimensional nonlinear waves, including shock waves, in elastic media. It is the first book to provide an in-depth and comprehensive presentation of the nonlinear wave theory while taking anisotropy effects into account. The theory is completely worked out and draws on 15 years of research by the authors, one of whom also wrote the 1965 classic Magnetohydrodynamics. Nonlinear Waves in Elastic Media emphasizes the behavior of quasitransverse waves and analyzes arbitrary discontinuity disintegration problems, illustrating that the solution can be non-unique - a surprising result. The solution is shown to be especially interesting when anisotropy and nonlinearity effects interact, even in small-amplitude waves. In addition, the text contains an independent mathematical chapter describing general methods to study hyperbolic systems expressing the conservation laws. The theoretical results described in Nonlinear Waves in Elastic Media allow, for the first time, discovery and interpretation of many new peculiarities inherent to the general problem of discontinuous solutions and so provide a valuable resource for advanced students and researchers involved with continuum mechanics and partial differential equations.
Table of Contents
Mathematical Introduction Conservation Laws and Related Differential Equations. Hyperbolic Systems. Linear and Linearized Equations. Riemann Invariants. Boundary Conditions and Evolutionary Properties. Riemann Waves. Discontinuities and Relations on Them. Shock Adiabat. Evolutionary Conditions for Discontinuity. Low Intensity Discontinuities. Shock Adiabat Behavior in a Vicinity of the Jouget Point. Conservation Law in the Godunov Form. Entropy. Entropy Production and Entropy Density Change on a Discontinuity. Solutions with Discontinuities as a Limit of Continuous Solutions to Equations of a Complicated Model. Small Perturbations in Dissipative Media. Shock Wave Structure. On Plane Wave Problems in Elastic Media Elastic Medium Model. Governing Equations. Plane Wave Equations. Conditions on a Discontinuity. Shock Adiabat. Entropy Change Along the Shock Adiabat. Wave Isotropy and Anisotropy. Internal Energy of a Medium with Weak Wave Anisotropy. Elastic Potential for a Weakly Nonlinear Medium. Nonlinear Wave Propagation Through Media Interacting with Electromagnetic Fields. Riemann Waves Small Perturbations. Linear Waves. Equations for Riemann Waves. Quasilongitudinal Waves. Quasitransverse Riemann Wave. Parameter Variations in Quasitransverse Waves. Evolution of Quasitransverse Riemann Waves. Riemann Waves in the Case of Wave Isotropy. Shock Waves Relationships on a Weak Shock Wave. Quasilongitudinal Shock Waves. Quasitransverse Waves. Shock Adiabat. Entropy Nondecreasing Condition. Evolutionary Conditions on Shocks. Velocities in Quasitranverse Waves. The Number of Evolutionary Shock Waves and Their Types. Locations of Evolutionary Segments on the Shock Adiabat. Shock Transitions into a Given State. Special Forms of Initial Deformations. Quasitransverse Shock Waves for G/R2
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CHOICE – Outstanding Academic Title – Award Winner
CHOICE – 2018 Outstanding Academic Title – Award Winner
Shingo Research and Professional Publication Award Winner
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