Nonlinear Hyperbolic Waves in Multidimensions: 1st Edition (Hardback) book cover

Nonlinear Hyperbolic Waves in Multidimensions

1st Edition

By Phoolan Prasad

Chapman and Hall/CRC

360 pages | 50 B/W Illus.

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pub: 2001-05-18
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Description

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.

Reviews

"…In summary [this book] includes numerical examples to illustrate theory, the quality of the graphics is good, and an adequate subject index is provided. The author has achieved his goal as a textbook for advanced undergraduate applied mathematics…This reviewer strongly recommends that scientists and engineers who are interested in these topics read this book…"

-Applied Mechanics Review, Vol. 55, no. 4, July 2002

Table of 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 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

About the Series

Monographs and Surveys in Pure and Applied Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
SCI040000
SCIENCE / Mathematical Physics