Riassunto
The propagation of curved, nonlinear wavefronts and shock fronts are very complex phenomena. Since the 1993 publication of his work Propagation of a Curved Shock and Nonlinear Ray Theory, author Phoolan Prasad and his research group have made significant advances in the underlying theory of these phenomena. This volume presents their results and provides a self-contained account and gradual development of mathematical methods for studying successive positions of these fronts. Nonlinear Hyperbolic Waves in Multidimensions includes all introductory material on nonlinear hyperbolic waves and the theory of shock waves. The author derives the ray theory for a nonlinear wavefront, discusses kink phenomena, and develops a new theory for plane and curved shock propagation. He also derives a full set of conservation laws for a front propagating in two space dimensions, and uses these laws to obtain successive positions of a front with kinks. The treatment includes examples of the theory applied to converging wavefronts in gas dynamics, a graphical presentation of the results of extensive numerical computations, and an extension of Fermat's principle. There is also a chapter containing approximate equations used to discuss stability of steady transonic flows. Full of new and original results, Nonlinear Hyperbolic Waves in Multidimensions is your only opportunity to explore a full treatment of these recent findings in book form. The material presented in this volume will prove useful not only for solving practical problems, but also in raising many difficult but important mathematical questions that remain open.
Contenuti
AN INTRODUCTION TO NONLINEAR HYPERBOLIC WAVES
A Wave Equation with Genuine Nonlinearity
Breakdown of a Genuine Solution
Conservation Law and Jump Condition
Stability Consideration, Entropy Condition, and Shocks
Some Examples
Shock Structure, Dissipation and Entropy Condition
The Persistence of a Shock
Nonlinear Wavefront and Shock Front
Hopf's Result on the General Solution
Equal Area Rule for Shock Fitting
HYPERBOLIC SYSTEM-SOME BASIC RESULTS
Hyperbolic System of First Order Equations in Two Independent Variables
The Wave Equation in m(>1) Space Dimensions
Hyperbolic System in More than Two Independent Variables
Bicharacteristic Curves, Rays, and Compatibility Condition
Propagation of Discontinuities of First Order Derivatives Along Rays
SIMPLE WAVE, HIGH FREQUENCY APPROXIMATION, AND RAY THEORY
Simple Wave
High-Frequency Approximation, Wavefront, Huygens' Method and Fermat's Principle
Kinematics of a Propagating Curve
Breakdown of the Continuity of a Solution of a Quasilinear System
Jump Conditions for a Curved Shock
WEAKLY NONLINEAR RAY THEORY (WNLRT) DERIVATION
A Historical Account
Derivation of CPW Theory
A Geometric Derivation of WNLRT
An Asymptotic Derivation of WNLRT
STABILITY OF SOLUTIONS NEAR A SINGULARITY OF SONIC TYPE
Introduction
One-Dimensional Weakly Nonlinear Wave Propagation
Waves in a Multi-Dimensional Steady Transonic Flow
WNLRT IN A POLYTROPIC GAS
Basic Equations
Geometrical Features of a Nonlinear Wavefront
Exact Solution of an Initial Value Problem
Conclusion and Validity of WNLRT
COMPATIBILITY CONDITIONS ON A SHOCK: SINGLE CONSERVATION LAW
Derivation of the Infinite System of Compatibility Conditions
Existence and Uniqueness of the Solution of the Infinite System
A New Theory of Shock Dynamics: Analytic Considerations
A New Theory of Shock Dynamics: Comparison of Numerical Results with the Exact Solution
Conclusion
ONE-DIMENSIONAL PISTON PROBLEM: AN APPLICATION OF NTSD
Formulation of the Problem
Dynamical Compatibility Conditions
Initial Conditions for the Piston Problem
Results and Discussions
COMPATIBILITY CONDITIONS ON A MULTI-DIMENSIONAL SHOCK
Shock Rays
Shock Manifold Equation for a Weak Shock
Geometrical and Kinematical Compatibility Conditions
Dynamical Compatibility Conditions
A Weak Shock Ray Theory
PROPAGATION OF A CURVED WEAK SHOCK
Governing Equations of the NTSD
Conservation Form of the Equations for a 2-Dimensional Shock Propagation
Initial Conditions, Results, and Discussion
Comparison with other Theories
Corrugational Stability and Persistence of a Kink
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