Nonlinear Hyperbolic Waves in Multidimensions

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.

Contents

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
• 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: PROPERTIES OF THE SOLUTION

• 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