sharma v. quasi linear hyperbolic systems, compressible flows, and waves (crc, 2010)(isbn...

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Library of Congress Cataloging‑in‑Publication Data

Sharma, Vishnu D.Quasilinear hyperbolic systems, compressible flows, and waves / Vishnu D. Sharma.

p. cm. -- (Monographs and surveys in pure and applied mathematics ; 142)Includes bibliographical references and index.ISBN 978-1-4398-3690-3 (hardcover : alk. paper)1. Wave equation--Numerical solutions. 2. Differential equations,

Hyperbolic--Numerical solutions. 3. Quasilinearization. I. Title.

QC174.26.W28S395 2010515’.3535--dc22 2010008125

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Contents

Preface ix

About the Author xiii

1 Hyperbolic Systems of Conservation Laws 1

1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Traffic flow . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 River flow and shallow water equations . . . . . . . . 3

1.2.3 Gasdynamic equations . . . . . . . . . . . . . . . . . . 4

1.2.4 Relaxing gas flow . . . . . . . . . . . . . . . . . . . . . 5

1.2.5 Magnetogasdynamic equations . . . . . . . . . . . . . 7

1.2.6 Hot electron plasma model . . . . . . . . . . . . . . . 10

1.2.7 Radiative gasdynamic equations . . . . . . . . . . . . 11

1.2.8 Relativistic gas model . . . . . . . . . . . . . . . . . . 11

1.2.9 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . 12

1.2.10 Dusty gases . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.11 Zero-pressure gasdynamic system . . . . . . . . . . . 13

2 Scalar Hyperbolic Equations in One Dimension 15

2.1 Breakdown of Smooth Solutions . . . . . . . . . . . . . . . . 15

2.1.1 Weak solutions and jump condition . . . . . . . . . . . 17

2.1.2 Entropy condition and shocks . . . . . . . . . . . . . . 21

2.1.3 Riemann problem . . . . . . . . . . . . . . . . . . . . 22

2.2 Entropy Conditions Revisited . . . . . . . . . . . . . . . . . 25

2.2.1 Admissibility criterion I (Oleinik) . . . . . . . . . . . . 25

2.2.2 Admissibility criterion II (Vanishing viscosity) . . . . 25

2.2.3 Admissibility criterion III (Viscous profile) . . . . . . 26

2.2.4 Admissibility criterion IV (Kruzkov) . . . . . . . . . . 28

2.2.5 Admissibility criterion V (Oleinik) . . . . . . . . . . . 29

2.3 Riemann Problem for Nonconvex Flux Function . . . . . . . 30

2.4 Irreversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . 34

v

vi

3 Hyperbolic Systems in One Space Dimension 393.1 Genuine Nonlinearity . . . . . . . . . . . . . . . . . . . . . . 393.2 Weak Solutions and Jump Condition . . . . . . . . . . . . . 403.3 Entropy Conditions . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.1 Admissibility criterion I (Entropy pair) . . . . . . . . 413.3.2 Admissibility criterion II (Lax) . . . . . . . . . . . . . 423.3.3 k-shock wave . . . . . . . . . . . . . . . . . . . . . . . 433.3.4 Contact discontinuity . . . . . . . . . . . . . . . . . . 43

3.4 Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . 443.4.1 Simple waves . . . . . . . . . . . . . . . . . . . . . . . 443.4.2 Riemann invariants . . . . . . . . . . . . . . . . . . . . 453.4.3 Rarefaction waves . . . . . . . . . . . . . . . . . . . . 463.4.4 Shock waves . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 Shallow Water Equations . . . . . . . . . . . . . . . . . . . . 543.5.1 Bores . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.5.2 Dilatation waves . . . . . . . . . . . . . . . . . . . . . 573.5.3 The Riemann problem . . . . . . . . . . . . . . . . . . 593.5.4 Numerical solution . . . . . . . . . . . . . . . . . . . . 613.5.5 Interaction of elementary waves . . . . . . . . . . . . . 653.5.6 Interaction of elementary waves from different families 663.5.7 Interaction of elementary waves from the same family 68

4 Evolution of Weak Waves in Hyperbolic Systems 754.1 Waves and Compatibility Conditions . . . . . . . . . . . . . 75

4.1.1 Bicharacteristic curves or rays . . . . . . . . . . . . . . 774.1.2 Transport equations for first order discontinuities . . . 784.1.3 Transport equations for higher order discontinuities . 814.1.4 Transport equations for mild discontinuities . . . . . . 82

4.2 Evolutionary Behavior of Acceleration Waves . . . . . . . . . 844.2.1 Local behavior . . . . . . . . . . . . . . . . . . . . . . 854.2.2 Global behavior: The main results . . . . . . . . . . . 864.2.3 Proofs of the main results . . . . . . . . . . . . . . . . 894.2.4 Some special cases . . . . . . . . . . . . . . . . . . . . 91

4.3 Interaction of Shock Waves with Weak Discontinuities . . . . 944.3.1 Evolution law for the amplitudes of C1 discontinuities 944.3.2 Reflected and transmitted amplitudes . . . . . . . . . 96

4.4 Weak Discontinuities in Radiative Gasdynamics . . . . . . . 1004.4.1 Radiation induced waves . . . . . . . . . . . . . . . . . 1014.4.2 Modified gasdynamic waves . . . . . . . . . . . . . . . 1034.4.3 Waves entering in a uniform region . . . . . . . . . . . 104

4.5 One-Dimensional Weak Discontinuity Waves . . . . . . . . . 1064.5.1 Characteristic approach . . . . . . . . . . . . . . . . . 1064.5.2 Semi-characteristic approach . . . . . . . . . . . . . . 1094.5.3 Singular surface approach . . . . . . . . . . . . . . . . 110

4.6 Weak Nonlinear Waves in an Ideal Plasma . . . . . . . . . . 112

vii

4.6.1 Centered rarefaction waves . . . . . . . . . . . . . . . 1164.6.2 Compression waves and shock front . . . . . . . . . . 118

4.7 Relatively Undistorted Waves . . . . . . . . . . . . . . . . . 1204.7.1 Finite amplitude disturbances . . . . . . . . . . . . . . 1224.7.2 Small amplitude waves . . . . . . . . . . . . . . . . . . 1234.7.3 Waves with amplitude not-so-small . . . . . . . . . . . 130

5 Asymptotic Waves for Quasilinear Systems 1335.1 Weakly Nonlinear Geometrical Optics . . . . . . . . . . . . . 133

5.1.1 High frequency processes . . . . . . . . . . . . . . . . 1345.1.2 Nonlinear geometrical acoustics solution in a relaxing

gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.2 Far Field Behavior . . . . . . . . . . . . . . . . . . . . . . . . 1375.3 Energy Dissipated across Shocks . . . . . . . . . . . . . . . . 140

5.3.1 Formula for energy dissipated at shocks . . . . . . . . 1405.3.2 Effect of distributional source terms . . . . . . . . . . 1425.3.3 Application to nonlinear geometrical optics . . . . . . 144

5.4 Evolution Equation Describing Mixed Nonlinearity . . . . . . 1465.4.1 Derivation of the transport equations . . . . . . . . . 1475.4.2 The ε-approximate equation and transport equation . 1495.4.3 Comparison with an alternative approach . . . . . . . 1525.4.4 Energy dissipated across shocks . . . . . . . . . . . . . 1525.4.5 Application . . . . . . . . . . . . . . . . . . . . . . . . 155

5.5 Singular Ray Expansions . . . . . . . . . . . . . . . . . . . . 1575.6 Resonantly Interacting Waves . . . . . . . . . . . . . . . . . 160

6 Self-Similar Solutions Involving Discontinuities 1656.1 Waves in Self-Similar Flows . . . . . . . . . . . . . . . . . . . 167

6.1.1 Self-similar solutions and their asymptotic behavior . 1686.1.2 Collision of a C1-wave with a blast wave . . . . . . . 173

6.2 Imploding Shocks in a Relaxing Gas . . . . . . . . . . . . . . 1766.2.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . 1776.2.2 Similarity analysis by invariance groups . . . . . . . . 1786.2.3 Self-similar solutions and constraints . . . . . . . . . . 1816.2.4 Imploding shocks . . . . . . . . . . . . . . . . . . . . . 1886.2.5 Numerical results and discussion . . . . . . . . . . . . 189

6.3 Exact Solutions of Euler Equations via Lie Group Analysis . 1966.3.1 Symmetry group analysis . . . . . . . . . . . . . . . . 1976.3.2 Euler equations of ideal gas dynamics . . . . . . . . . 1986.3.3 Solution with shocks . . . . . . . . . . . . . . . . . . . 202

7 Kinematics of a Shock of Arbitrary Strength 2057.1 Shock Wave through an Ideal Gas in 3-Space Dimensions . . 206

7.1.1 Wave propagation on the shock . . . . . . . . . . . . . 2107.1.2 Shock-shocks . . . . . . . . . . . . . . . . . . . . . . . 212

viii

7.1.3 Two-dimensional configuration . . . . . . . . . . . . . 2147.1.4 Transport equations for coupling terms . . . . . . . . 2157.1.5 The lowest order approximation . . . . . . . . . . . . 2187.1.6 First order approximation . . . . . . . . . . . . . . . . 220

7.2 An Alternative Approach Using the Theory of Distributions 2237.3 Kinematics of a Bore over a Sloping Beach . . . . . . . . . . 230

7.3.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . 2317.3.2 Lowest order approximation . . . . . . . . . . . . . . . 2347.3.3 Higher order approximations . . . . . . . . . . . . . . 2367.3.4 Results and discussion . . . . . . . . . . . . . . . . . . 2377.3.5 Appendices . . . . . . . . . . . . . . . . . . . . . . . . 243

Bibliography 249

Index 265

Preface

The material in this book has evolved partly from a set of lecture notes usedfor topic-courses and seminars at IIT Bombay and elsewhere during the lastfew years; much of the material is an outgrowth of the author’s collaborativeresearch with many individuals, whom the author would like to thank for theirscientific contributions.

The book provides a reasonably self-contained discussion of the quasi-linear hyperbolic equations and systems with applications. Entries from thebibliography are referenced in the body of the text, but these do not seri-ously affect the continuity of the text, and can be omitted at first reading.The aim of this book is to cover several important ideas and results on thesubject to emphasize nonlinear theory and to introduce some of the most ac-tive research areas in this field following a natural mathematical developmentthat is stimulated and illustrated by several examples. As the book has beenwritten with physical applications in mind, the author believes that the an-alytical approach followed in this book is quite appropriate, and is capableof providing a better starting point for a graduate student in this fascinatingfield of applied mathematics. In fact, the book should be particularly suitablefor physicists, applied mathematicians and engineers, and can be used as atext in either an advanced undergraduate course or a graduate level courseon the subject for one semester. Care has been taken to explain the mate-rial in a systematic manner starting from elementary applications, in orderto help the reader’s understanding, progressing gradually to areas of currentresearch. All necessary mathematical concepts are introduced in the first threechapters, which are intended to be an introduction both to wave propagationproblems in general, and to issues to be developed throughout the rest of thebook in particular. The remaining chapters of the book are devoted to someof the recent research work highlighting the applications of the characteris-tic approach, singular surface theory, asymptotic methods, self-similarity andgroup theoretic methods, and the theory of generalized functions to severalconcrete physical examples from gasdynamics, radiation gasdynamics, magne-togasdynamics, nonequilibrium flows, and shallow water theory. A few generalremarks have been included at the end of sections or chapters with the hopethat they will provide useful source material for ideas beyond the scope of thetext.

Chapter 1 provides a link between continuum mechanics and the quasi-linear partial differential equations (PDEs); it begins with a discussion of the

ix

x Preface

conditions necessary for such systems to be hyperbolic. Several examples areconsidered, that illustrate the ideas and show some of the peculiarities thatmay arise in the classification of systems.

Chapter 2 introduces the scalar conservation laws. The main nonlinearfeature is the breakdown of smooth solutions, leading to the notion of weaksolutions, the loss of uniqueness, entropy conditions, and shocks. These are allpresented in detail with the aid of examples.

Chapter 3 is devoted to hyperbolic systems in two independent variables;notions of genuine nonlinearity, k-shock, contact discontinuity, simple wavesand Riemann invariants are introduced. Special weak solutions, namely, therarefaction waves and shocks are discussed; these special solutions are partic-ularly useful in solving Riemann problems, which occur frequently in physicalapplications. All these ideas are applied to solve the Riemann problem forshallow water equations with arbitrary data.

Chapter 4 presents the evolutionary behavior of weak and mild disconti-nuities in a quasilinear hyperbolic system, using the method of characteristicsand the singular surface theory. Local and global behavior of the solution oftransport equation, describing the evolution of weak discontinuities, is studiedin detail, modifying several known results in the literature. Weak nonlinearwaves, namely, the rarefaction waves, compression waves, and shocks are stud-ied in the one-dimensional motion of an ideal plasma permeated by a trans-verse magnetic field. The method of relatively undistorted waves is introducedto study high frequency waves in a relaxing gas. The problem of interactionbetween a weak discontinuity wave and a shock, which gives rise to reflectedand transmitted waves, is studied in detail.

Chapter 5 deals with weakly nonlinear geometrical optics (WNGO). It isan asymptotic method whose objective is to understand the laws governingthe propagation and interaction of high frequency small amplitude waves inhyperbolic systems. The procedure is illustrated for nonequilibrium and strat-ified gas flows. Expressions for the energy dissipated across shocks, and theevolution equations describing mixed nonlinearity are derived. Singular rayexpansions and resonantly interacting waves in hyperbolic systems are brieflydiscussed.

Chapter 6 demonstrates the power, generality, and elegance of the self-similar and basic symmetry (group theoretic) methods for solving Euler equa-tions of gasdynamics involving shocks. An exact self-similar solution and theresults of interaction theory are used to study the interaction between a weakdiscontinuity wave and a blast wave in plane and radially symmetric flows.The method of Lie group invariance is used to determine the class of self-similar solutions to a relaxing gas flow problem involving shocks of arbitrarystrength. The method yields a general form of the relaxation rate for whichthe self-similar solutions are admitted. A particular case of collapse of an im-ploding shock is worked out in detail for radially symmetric flows. Numericalcalculations have been performed to describe the effects of relaxation and theambient density on the self-similar exponent and the flow pattern. With the

Preface xi

help of canonical variables, associated with the group generators that leavethe first order system of PDEs invariant, the system of PDEs is reduced toan autonomous system whose simple solutions provide nontrivial solutions ofthe original system. We remark that one of the special solutions of the Eu-ler system, discussed here, is precisely the blast wave solution known in theliterature.

Chapter 7 contains a discussion of the kinematics of a shock of arbitrarystrength in three dimensions. The dynamic coupling between the shock frontand the rearward flow is investigated by considering an infinite system oftransport equations that hold on the shock front. At the limit of vanishingshock strength, the first order truncation approximation leads to an exactdescription of acceleration waves. Asymptotic decay law for weak shocks andrearward precursor disturbances are obtained as special cases. At the strongshock limit, the first order approximation leads to a propagation law thathas a structural resemblance with Guderley’s exact similarity solution. Atten-tion is drawn to the connection between the transport equations along therays obtained here and the corresponding results obtained by an alternativemethod using the theory of distributions. Finally, the procedure is used todescribe the behavior of a bore of arbitrary strength as it approaches theshoreline on a sloping beach. The evolutionary behavior of weak and strongbores described by the lowest order truncation approximation is in excellentagreement with that predicted by the characteristic rule. Strangely enough, inthe characteristic rule (CCW approximation), the critical values of the borestrength for which the bore height and the bore speed attain extreme values,remain uninfluenced by the initial bore strength and the undisturbed waterdepth. The present method takes care of this situation in a natural manner,and the first few approximations yield results with reasonably good accuracy.It may be remarked that even the lowest order approximation describes theevolutionary behavior of a bore with good accuracy.

AcknowledgmentsI have been helped in the present venture by many colleagues and formerstudents. I am indebted to all of them; in particular, I mention Dr. R. Radha,Dr. G. K. Srinivasan, and Dr. T. Rajasekhar for their cooperation and help.I am thankful to C. L. Antony, Pradeep Kumar and Ashish Mishra for theirpersistent support, which was essential for the completion of this book. Duringthe work on the manuscript, I have benefited from contact with Prof. J. B.Keller, Prof. T. P. Liu, Prof. T. Ruggeri, Prof. D. Serre, and Prof. C. Dafermos,to whom I express my most sincere appreciation. I am greatly indebted to Prof.A. Jeffrey for being a constant source of inspiration and encouragement overthe years.

I acknowledge the continuous support from IIT Bombay and the Depart-ment of Mathematics, in particular, for giving me time and freedom to work

xii Preface

on this book. I also acknowledge the support of the Curriculum DevelopmentProgramme at IIT Bombay in writing the book. I express my sincere thanksto the personnel of the CRC Press, Dr. Sunil Nair and his editorial staff, fortheir kind cooperation, help, and encouragement in bringing out this book. Inparticular, I thank Karen Simon who painstakingly went through the draft ofthe entire manuscript.

I express my deep gratitude to my family members for providing me muchneeded moral support without which this work would have remained only inmy mind. Finally, I dedicate this book to my teacher Prof. Rishi Ram who notonly introduced me to this difficult area of mathematics but also guided methrough the tortuous paths of life; to him I owe more than words can express.

V. D. Sharma

About the Author

Dr. V.D. Sharma has been at the Indian Institute of Technology, Bombay(IITB) as a professor since 1988. Presently, he holds the position of InstituteChair Professor in the Department of Mathematics at IIT Bombay, and isPresident of the Indian Society of Theoretical and Applied Mechanics. He is aFellow of the National Academy of Sciences India, and of the Indian NationalScience Academy. Professor Sharma has published numerous research articlesin the area of hyperbolic systems of quasilinear partial differential equationsand the associated nonlinear wave phenomena. He is also a member of the ed-itorial board of the Indian Journal of Pure and Applied Mathematics. He hasvisited several universities and research institutions such as the University ofMaryland at College Park, the Mathematics Research Center at the Universityof Wisconsin, and Stanford University. He was the head of the Departmentof Mathematics at IITB from 1996 to 2000 and from 2003 to 2006. ProfessorSharma received IITB’s awards for excellence in research in basic sciences in2005, and for excellence in teaching in 1998 and in 2003. He received the C.L.Chandna Mathematics Award from the Canadian World Education Founda-tion for distinguished and outstanding contributions to mathematics researchand teaching in 1999. He also received the M.N. Saha Award for Research inTheoretical Sciences from the University Grants Commission, Government ofIndia in 2001.

xiii

Chapter 1

Hyperbolic Systems of Conservation

Laws

Any mathematical model of a continuum is given by a system of partial dif-ferential equations (PDEs). In continuum mechanics, the conservation laws ofmass, momentum and energy form a common starting point, and each mediumis then characterized by its constitutive laws. The conservation laws and con-stitutive equations for the field variables, under quite natural assumptions,reduce to field equations, i.e., partial differential equations, which, in general,are nonlinear and nonhomogeneous. For nonlinear problems, neither the meth-ods of their solutions nor the main characteristics of the motion are as wellunderstood as in the linear theory. Before we proceed to discuss mathemati-cal concepts and techniques to understand the phenomena from a theoreticalstandpoint and to solve the problems that arise, we introduce here the hyper-bolic systems of conservation laws in general (see Benzoni-Gavage and Serre[13]), and then present some specific examples, for application or motivation,which are of universal interest.

1.1 Preliminaries

A large number of physical phenomena are modeled by systems of quasi-linear first order partial differential equations that result from the balancelaws of continuum physics. These equations, expressed in terms of divergence,are commonly called conservation laws. The general form of the system ofconservation laws, in its differentiated form, is given by

∂u

∂t+

∂

∂xifi(u) = g(u,x, t), 1 ≤ i ≤ m, (1.1.1)

where u(x, t) = (u1, u2, . . . , un)tr is a vector of conserved quantities, depen-

dent on x = (x1, x2, . . . , xm)tr ∈ IRm and t ∈ [0,∞), and the superscripttr denotes transposition. The vector fields fi(u) = (f1i, f2i, . . . , fni)

tr andg = (g1, g2, . . . , gn)

tr represent, respectively, the flux and production densi-ties, which are assumed to be smooth functions of their arguments. Here, andthroughout, summation convention on repeated indices is automatic unlessotherwise stated.

System (1.1.1) arises in the study of nonlinear wave phenomena, whenweak dissipative effects such as chemical reaction, damping, stratification,relaxation etc., are taken into account. It expresses that the time variation

1

2 1. Hyperbolic Systems of Conservation Laws

of the total amount of the substance∫D

udx, contained in any domain D ofIRm, is equal to the flux of the vector fields fi across the boundary of D,plus contribution due to the sources or sinks distributed in the interior of D;system (1.1.1) with g = 0 is said to be in strict conservation from. When thedifferentiation in (1.1.1) is carried out, the following quasilinear system of firstorder results

u,t + Ai(u)u,i = g, (1.1.2)

where Ai = (∂fji/∂uk)1≤j,k≤n ≡ ∇fi is the n × n Jacobian matrix with∇ = (∂/∂u1, ∂/∂u2, . . . , ∂/∂un) as the gradient operator with respect to theelements of u, and a comma followed by t (respectively, an index i), denotespartial differentiation with respect to time t (respectively, the space variablexi).

The system (1.1.1) is called hyperbolic if for each x, t and u, and theunit vector ξ = (ξ1, ξ2, . . . , ξm)tr ∈ IRm, the n × n matrix ξjAj has n realeigenvalues λ1, λ2, . . . , λn with linearly independent eigenvectors r1, r2, . . . , rn;if, in addition, the eigenvalues are all distinct, the system (1.1.2), and hence(1.1.1), is called strictly hyperbolic. However, if the algebraic multiplicity ofan eigenvalue is greater than its geometrical multiplicity, the system is referredto as nonstrictly hyperbolic and cannot be diagonalized (see Li et al. [107] andZheng [215]).

It may be remarked that the eigenvalues of the system (1.1.2) are thesame as those of (1.1.1), and for a smooth solution the two forms (1.1.1) and(1.1.2) are equivalent. Working with (1.1.1) allows us to consider discontinuoussolutions as well, so that the equation is interpreted in some generalized sense.It is our goal here to study certain problems involving these equations. Beforeproceeding with the discussion of some of the consequences of such equations,it will be useful to have some specific physical examples which illustrate theiroccurrence.

1.2 Examples

1.2.1 Traffic flow

The simplest example of a nonlinear conservation law in one space dimen-sion is a first order partial differential equation

u,t + (f(u)),x = 0, (1.2.1)

where u is the density function and f(u), a given smooth nonlinear function,is the flux function. Such an equation appears in the formulation of trafficflow where u(x, t) denotes the density, the number of cars passing through theposition x at time t on a highway, and the function f(u) = uv denotes the flux

1.2 Examples 3

of cars with v being the average (local) velocity of the cars, which is assumedto be a given function of u. This assumption seems to be reasonable sincethe drivers are supposed to increase or decrease their speed as the densitydecreases or increases, respectively. The simplest model is the linear relationdescribed by the equation v = vmax(1 − (u/umax)), which shows that themaximum value of v occurs when u = 0, and when u is maximum, v = 0. Onsetting u = u/umax and x = x/vmax, the resulting normalized conservationlaw, after suppressing the overhead tilde sign, reduces to the form

u,t + (u(1 − u)),x = 0. (1.2.2)

Further details on traffic modeling and analyses may be found in Whitham[210], Goldstein [64], Sharma et al. [181], and Haberman [67].

1.2.2 River flow and shallow water equations

For river flows, a rectangular channel of constant breadth and inclination isconsidered, and it is assumed that the disturbance is roughly the same acrossthe breadth. If h(x, t) be the depth and u(x, t) the mean velocity of the fluidin the channel, then the governing equations for the river flow can be writtenin the conservation form (see Whitham [210] and Ockendon et al. [132])

u,t + f,x = b, (1.2.3)

whereu = (h, hu)tr , f = (hu, hu2 + (gh2 cosα)/2)tr , and b = (0, gh sinα− Cfu

2)tr ;

α is the angle of inclination of the surface of the river, g the acceleration dueto gravity, and Cf the friction coefficient that appears in the expression for thefriction force of the river bed. As the Jacobian matrix A = ∇f has distinct realeigenvalues u±

√gh cosα, the system (1.2.3) is strictly hyperbolic. Equations

(1.2.3) may be simplified to become the equivalent pair in nonconservativeform.

h,t + uh,x + hu,x = 0,u,t + uu,x + g cosα h,x = g sinα− C2

f (u2/h).

In the shallow water theory, where the height of the water surface above thebottom is small relative to the typical wave lengths, usually the slope andfriction terms are absent in (1.2.3), and so the governing system of equationsis in strict conservative form, and assumes on simplification the following form

c,t + uc,x + (cu,x/2) = 0,u,t + uu,x + 2cc,x = 0,

(1.2.4)

where c =√gh. It may be noticed that the above system (1.2.4) describes

the flow over a horizontal level surface; for a nonuniform bottom, there is anadditional term in the horizontal momentum equation due to the force acting


on the bottom surface, and so the corresponding shallow water equations canbe written in the following nonconservative form

c,t + uc,x + (cu,x/2) = 0,u,t + uu,x + 2cc,x = g(dh0/dx),

(1.2.5)

where the function −h0(x) describes the bottom surface relative to an originlocated on the equilibrium surface of the water along which lies the x-axis.

1.2.3 Gasdynamic equations

One of the simplest examples that appears to be fundamental in the studyof gasdynamics is provided by unsteady compressible inviscid gas flow; theEuler equations in the Cartesian coordinate system xi, 1 ≤ i ≤ 3, can bewritten in the following conservative form.

ρ,t + (ρui),i = 0,

(ρui),t + (ρuiuj + pδij),j = 0, i, j = 1, 2, 3 (1.2.6)

(ρE),t + ((ρE + p)uj),j = 0,

where ρ is the gas velocity, ui the ith component of gas velocity vec-tor u, p the gas pressure, E = e + (|u|2/2) the total energy per unitmass with e as the internal energy, and δij the Kronecker δ’s. The equa-tion of state can be taken in the form p = p(ρ, e). Identifying (1.2.6)with (1.1.1), we see that these equations are in conservative form withu = (ρ, ρu1, ρu2, ρu3, ρE)tr , f1 = (ρu1, p + ρu2

1, ρu1u2, ρu1u3, (ρE + p)u1)tr ,

f2 = (ρu2, ρu1u2, p + ρu22, ρu2u3, (ρE + p)u2)

tr , f3 = (ρu3, ρu1u3, ρu2u3,p+ ρu2

3, (ρE + p)u3)tr , and b = 0.

The eigenvalues of the matrix ξjAj , where Aj is the Jacobian matrix offj and |ξ| = 1, are easily found to be uiξi − a, uiξi + a and uiξi; the first twoeigenvalues are simple and the remaining one, i.e., uiξi, is of multiplicity threehaving three linearly independent eigenvectors associated with it. Thus, all thefive eigenvalues of ξjAj are real, but not distinct, and the eigenvectors spanthe space IR5, the system (1.2.6) is hyperbolic. Conservative forms (1.2.6)are obtained from the corresponding integral representation forms, and areneeded for the treatment of shocks. But for other purposes, the equations maybe simplified. The energy equation (1.2.6)3 can be written in various forms;using the other two equations (1.2.6)1 and (1.2.6)2, it takes an alternativeform

de/dt− (p/ρ2)(dρ/dt) = 0, (1.2.7)

where d/dt = ∂/∂t + ui∂/∂xi denotes the time derivative following anindividual particle. Further, using the thermodynamical relations TdS =de+pd(1/ρ) = dh− (1/ρ)dp, where T (p, ρ), S(p, ρ) and h(p, ρ) denote, respec-tively, the absolute temperature, the specific entropy and the specific enthalpy,(1.2.7) reduces to

T (dS/dt) = 0 or dh/dt− (1/ρ)dp/dt = 0. (1.2.8)

1.2 Examples 5

Since the expression for S in terms of p and ρ may be solved in principleas p = p(ρ, S), an equivalent form of (1.2.8)1 is dp/dt − a2dρ/dt = 0, wherea =

√(∂p/∂ρ)

Sis the local speed of sound.

Thus, the following alternative formulation of the Euler equations (1.2.6),though not in conservative form, will be convenient for future reference.

dρ/dt+ ρui,i = 0, dui/dt+ (1/ρ)p,i = 0, dS/dt = 0. (1.2.9)

For a polytropic gas, we have

e = p/(ρ(γ − 1)), S = Cv ln(p/ργ) + constant, a2 = γp/ρ, (1.2.10)

where γ > 1 is the specific heat ratio for the gas.If we assume that the flow has some symmetry, one can reduce the number

of space variables. For instance for a one-dimensional unsteady compressibleinviscid gas flow with plane, cylindrical or spherical symmetry, system (1.2.6)can be written [96]

(xmρ),t + (xmρu),x = 0,

(xmρu),t + (xm(p+ ρu2)),x = mpxm−1, (1.2.11)

(xmρ(e+u2

2)),t + (xmρu(e+

p

ρ+u2

2)),x = 0,

where u = u(x, t), ρ = ρ(x, t), p = p(x, t) and e = e(x, t) are, respectively,the gas velocity, density, pressure and internal energy with x as the spatialcoordinate being either axial in flows with planar (m = 0) geometry, or ra-dial in cylindrically symmetric (m = 1) and spherically symmetric (m = 2)configurations. System (1.2.11) can be written as

u,t + f,x = b, (1.2.12)

where u, f and b are column vectors, which can be read off by inspectionof (1.2.11); with the usual equation of state, the system (1.2.12) is strictlyhyperbolic as the Jacobian matrix of f has real and distinct eigenvalues u± aand u.

1.2.4 Relaxing gas flow

As a result of high temperatures attained by gases in motion, the effects ofnon equilibrium thermodynamics on the dynamics of gas motion can be im-portant. Assuming that the departure from equilibrium is due to vibrationalrelaxation, and the rotational and translational modes are in local thermo-dynamical equilibrium throughout, the governing system of equations for anunsteady flow in the absence of viscosity, heat conduction and body forces, isobtained by adjoining to the gasdynamic equations (1.2.6) the rate equation(see Vincenti and Kruger [207])

(ρσ),t + (ρσuj),j = ρΦ, (1.2.13)


where σ is the vibrational energy per unit mass, and Φ is the rate of changeof vibrational energy, which is assumed to be a known function of p, ρ andσ, given by Φ = (σ∗ − σ)/τ ; here, σ∗ is the equilibrium value of σ given byσ∗ = RΘv/(exp(Θv/T ) − 1) and τ−1 is the relaxation frequency given byτ−1 = k1p exp(−k2/T

1/3), where R is the gas constant, Θv the characteristictemperature of the molecular vibration, T = p/(ρR) the gas temperature,and k1, k2 the positive constants depending on the physical properties of thegas. Using the continuity equation (1.2.6)1, equation (1.2.13) can be put ina simpler form dσ/dt = Φ, where d/dt = ∂/∂t + ui∂/∂xi is the materialderivative.

To close the system of equations, we need to add an equation of statewhich can be written in the form

e = σ + (γ − 1)−1(p/ρ), (1.2.14)

where γ is the frozen specific heat ratio.Thus, in case of smooth flows, we may write the governing system of equa-

tions in nonconservative form

dρ/dt+ ρui,i = 0, dui/dt+ (1/ρ)p,i = 0,dp/dt+ ρa2

fui,i + (γ − 1)ρΦ = 0, dσ/dt = Φ,(1.2.15)

where a2f = γp/ρ is the frozen speed of sound.

System (1.2.15) may be written in the form u,t + Aiu,i = b, where u isthe six-dimensional column vector having components ρ,u, p and σ; the col-umn vector b and 6× 6 matrices Ai can be read-off by inspection of (1.2.15).Indeed the matrix Aiξi = (αIJ )1≤I,J≤6 with α11 = α22 = α33 = α44 = α55 =α66 = uiξi, α12 = ρξ1, α13 = ρξ2, α14 = ρξ3, α25 = ξ1/ρ, α35 = ξ2/ρ, α45 =ξ3/ρ, α52 = ρa2

f ξ1, α53 = ρa2f ξ2, α54 = ρa2

f ξ3, and the remaining entries beingzeros, has eigenvalues uiξi + af , uiξi − af and uiξi. The first two eigenvaluesare simple, whilst the third one, uiξi, is of multiplicity four, having four lin-early independent eigenvectors associated with it. Thus, the system (1.2.15)is hyperbolic.

For a one-dimensional motion with plane, (m = 0), cylindrical (m = 1) orspherical (m = 2) symmetry, the rate equation (1.2.13) can be written as

(xmρσ),t + (xmρσu),x = xmρΦ. (1.2.16)

Equations (1.2.11) together with (1.2.14) and the rate equation (1.2.16) de-scribe the one-dimensional motion of a vibrationally relaxing gas with plane,cylindrical or spherical symmetry; these equations may be simplified to assumethe following nonconservative form

ρ,t + uρ,x + ρu,x +mρu/x = 0, u,t + uu,x + (1/ρ)p,x = 0,p,t + up,x + ρa2

f (u,x +mux−1) + (γ − 1)ρΦ = 0, σ,t + uσ,x = Φ.(1.2.17)

System (1.2.17) is hyperbolic as the Jacobian matrix has four real eigenvalues,but not all distinct, and the eigenvectors span the space IR4.

1.2 Examples 7

It is sometimes of interest to examine the entropy production in nonequi-librium flows, where there is a transfer of energy from one internal mode of gasmolecules to another, and one needs to supplement the gasdynamic equationswith the rate equation (see Chu [36] and Clarke and McChesney [37])

(ρq),t + (ρqui),i = ρω(p, S, q), or dq/dt = ω, (1.2.18)

where q is the progress variable characterizing the extent of internal trans-formation in the fluid, and ω is the rate of internal transformation which isassumed to be a known function of p, S and q. Then by specifying the specificenthalpy h = h(p, S, q), through the equation of state and by invoking theGibbs relation, TdS = dh− (1/ρ)dp+ αdq, where T denotes the temperatureand α the affinity of internal transformation characterized by the variable q;all other variables such as ρ, T and α are known functions of p, S and q, givenby

ρ−1 = ∂h/∂p, T = ∂h/∂S, α = −∂h/∂q. (1.2.19)

When the internal transformation attains a state of equilibrium

ω(p, S, q) = 0 = α(p, S, q) ⇒ q = q∗(p, S), (1.2.20)

where q∗ is the equilibrium value of q evaluated at local p and S. When theGibbs relation is applied to the changes following a fluid element, and use ismade of the energy equation dh/dt = (1/ρ)dρ/dt along with the rate equation(1.2.18)2, we obtain dS/dt = ωα/T. This gives the rate of entropy productionfollowing a fluid element which, in general, is not zero in a nonequilibriumflow. When ρ = ρ(p, S, q) is substituted into the equation of continuity anduse is made of (1.2.18)2 along with the entropy equation dS/dt = ωα/T , weobtain an equivalent form of the continuity equation as

dp/dt+ ρa2fui.i = −ωa2

f (∂ρ/∂q + (α/T )∂ρ/∂S),

where af is the frozen speed of sound given by a2f = (∂p/∂ρ)S,q.

Thus, in the absence of external body forces, the equations governing anunsteady nonequilibrium flow of an inviscid and non-heat conducting gas,which has only one lagging internal mode, can be written in the followingnonconservative form

dp/dt+ ρa2fui,i = −ωa2

f (∂ρ/∂q + (α/T )∂ρ/∂S),

dui/dt+ (1/ρ)p,i = 0, dS/dt = αω/T, dq/dt = ω.(1.2.21)

One can easily check that the system (1.2.21) is hyperbolic; regarding someof the results concerning (1.2.21), the reader is referred to Singh and Sharma[186].

1.2.5 Magnetogasdynamic equations

Magnetogasdynamics is concerned with the study of interaction betweenmagnetic field and the gas flow; the governing equations, thus, consist of


Maxwell’s equations and the gasdynamic equations of motion. The effect ofmagnetic field on the conventional gasdynamic momentum equation is to adda term corresponding to the Lorentz force which, with the aid of Maxwell’sequations, may be written as the divergence of a tensor. For a perfectly con-ducting inviscid gas motion in a magnetic field H = (H1, H2, H3), the equa-tions of magnetogasdynamics can be written in the conservative form (seeJeffrey and Tanuiti [79] and Cabannes [27])

ρ,t + (ρui),i = 0;(ρui),t + (ρuiuj + pδij − µ(HiHj − 1

2 |H|2δij)),j = 0;(ρe+ 1

2ρ|u|2 + 12µ|H|2),t + (ρui(e+ p

ρ + 12 |u|2) + µ(|H|2ui − ujHjHi)),i = 0;

(Hi),t + ujHi,j −Hjui,j +Hiuj,j = 0; Hi,i = 0,(1.2.22)

where µ is the magnetic permeability and other symbols have their usualmeaning; the suffixes i, j run over 1,2,3. The thermal pressure p is related tothe internal energy e through the gamma-law equation of state p = ρe(γ− 1).Equation (1.2.22)4 which is a consequence of Maxwell’s equations and Ohm’slaw, is called the induction equation or the field equation. The magnetic fieldvector H, by virtue of the induction equation and initial conditions, satisfiesthe equation (1.2.22)5, which is a statement of the experimental fact thatthe magnetic charges do not exist in nature; in fact, it may be regarded as arestriction on H, and may be conveniently used in place of one of the projec-tions of the induction equation. Although the electric field E and the currentdensity J do not explicitly appear in equations (1.2.22), they may be obtainedthrough the magnetic field and flow velocity: E = µH×u and J = µ curl Hunder the approximation of ideal magnetogasdynamics. The theory of non-linear system (1.2.22) is basically similar to that of ordinary compressiblegasdynamics; nevertheless, in detail, the magnetogasdynamic phenomena aredecidedly more complex than the gasdynamic ones. The momentum and en-ergy equations in (1.2.22) can be simplified with the help of the remainingequations, and can be written in the following nonconservative form

ui,t + ujui,j + (1/ρ)p,i + (µ/ρ)Hj(Hj,i −Hi,j) = 0,p,t + uip,i + ρa2ui,i = 0,

(1.2.23)

where a =√

(γp/ρ) is the sound speed in the medium. The system of equa-tions described by (1.2.22)1, (1.2.22)4 and (1.2.23) is hyperbolic (see Singhand Sharma [185] and Sharma ([178], [176])).

When the flow and field variables depend only on the Cartesian coor-dinate x and on time t,H1 turns out to be a constant, and the govern-ing system of equations, represented by (1.2.22)1, (1.2.22)4 and (1.2.23),for the one-dimensional motion becomes u,t + Au,x = 0, where u =(ρ, u1, u2, , u3, H2, H3, p)

tr and A = (AIJ )1≤I,J≤7 is a 7 × 7 matrix withA11 = A22 = A33 = A44 = A55 = A66 = A77 = u1, A12 =ρ,A25 = µH2/ρ,A26 = µH3/ρ,A27 = 1/ρ,A35 = −µH1/ρ = A46, A52 =H2, A53 = A64 = −H1, A62 = H3, A72 = ρa2 and the remaining en-

1.2 Examples 9

tries being zeros. When H1 6= 0, the eigenvalues of A are real and dis-tinct, namely, u1 ± cf , u1 ± cs, u1 ± ca and u, where cf , cs and ca are re-spectively the fast magnetoacoustic speed, slow magnetoacoustic speed, and

Alfven speed given by c2f = 12

a2 + b2 +

√(a2 + b2)2 − 4a2c2a

1/2

, c2s =

12

a2 + b2 −

√(a2 + b2)2 − 4a2c2a

1/2

, and c2a = µH21/ρ with b2 = µ|H|2/ρ.

The corresponding eigenvectors span IR7 and so the system is strictly hyper-bolic.

For a one-dimensional planar and cylindrically symmetric motion of aplasma, which is assumed to be an ideal gas with infinite electrical conductiv-ity and to be permeated by a magnetic field orthogonal to the trajectories ofgas particles, the governing equations can be easily derived from the equations(1.2.22)1, (1.2.22)4 and (1.2.23). We assume that the vector H is directed per-pendicular to the trajectories of gas particles. In the cylindrical case, the fieldvector H may, in particular, be directed along the axis of symmetry, alongtangents to concentric circles with center on the symmetry axis, or in the gen-eral case, it may create a screw field with components Hφ and Hz. Thus, thebasic equations can be written down in the following nonconservative form(see Korobeinikov [96])

ρ,t + (ρu),r +mρu/r = 0,

ρ(u,t + uu,r) + (p+ (µ/2)H2),r + (nµ/r)H2 = 0,

H,t + (uH),r +m(1 − n)uH/r = 0, (1.2.24)

p,t + up,r + ρa2(u,r +mu/r) = 0,

where H is the transverse magnetic field, which in a cylindrically symmetric(m = 1) motion is either axial (n = 0) with H = Hz or annular (n = 1) withH = Hφ; for a planer (m = 0) motion, H = Hz and the constant n assumesthe value zero.

For a screw field, the vectors H and u have components (0, Hφ, Hz) and(u, 0, 0) along the axes associated with the cylindrical coordinates (r, φ, z),and the basic equations become

ρ,t + (ρu),r + ρu/r = 0,

ρ(u,t + uu,r) + (p+ (µ/2)(H2φ +H2

z )),r + (µ/r)H2φ = 0,

Hz,t+ (uHz),r + uHz/r = 0, (1.2.25)

Hφ,t+ (uHφ),r = 0,

p,t + up,r + ρa2(u,r + u/r) = 0.

Both the systems described above by (1.2.24) and (1.2.25) are hyperbolic,but not strictly hyperbolic; for further details regarding some of the resultsconcerning (1.2.24), the reader is referred to Menon and Sharma [122].


1.2.6 Hot electron plasma model

We consider a homogeneous unbounded plasma consisting of mobile elec-trons in a neutralizing background of stationary ions of number density no.Assuming that the processes taking place in a plasma are such that the heatconduction, viscosity and energy interchange due to collisional interactionsare all negligible, the evolution of the electron density n and mean electronvelocity u for a one component warm electron fluid, in a one-dimensionalconfiguration, are then given by

n,t + un,x + nu,x = 0,u,t + uu,x + (nm)−1p,x = −eE/m, (1.2.26)

where m is the electron mass, −e is the electron charge, E is the electric field,and p is the electron pressure, the evolution of which is determined from theapproximate moment of the Vlasov equation; in the one-dimensional case, theevolution of p is governed by the following equation (see Davidson [46])

p,t + up,x + 3pu,x = 0, (1.2.27)

which, in view of the continuity equation (1.2.26)1 yields the familiar adiabaticgamma-law (for γ = 3)

p = (const)n3. (1.2.28)

The electric field E in (1.2.26)2, in view of Poisson’s equation E,x =−4πe(n− no), and the Maxwell equation in the electrostatic approximation,E,t = 4πenu, is determined from the equation

E,t + uE,x = 4πeuno. (1.2.29)

It may be recalled that it is possible to heat a plasma due to collisional ef-fects by periodically compressing and decompressing it. This compression canbe one, two, or three-dimensional depending on the geometrical arrangement.The thermodynamic state of charged particles subject to adiabatic relation,pnγ = const, where γ = 1 + (2/δ) with δ being the number of degrees of free-dom affected by the compression. For a one-dimensional compression, δ = 1,we have γ = 3, and hence the adiabatic relation (1.2.28) (see Sharma etal. [180]). Equations (1.2.26), (1.2.27) and (1.2.29), thus constitute a closedone-dimensional description of a warm electron fluid in the electrostatic ap-proximation; it may be noticed that the description of this model is equivalentto the single-water bag model of a Vlasov plasma (see Bertrand and Feix [14]).This system can be expressed in the form u,t + Au,x = b, where the matrixA has eigenvalues u ± (3p/n)1/2 and u (multiplicity two) with four linearlyindependent associated eigenvectors; hence it is hyperbolic. For further detailsregarding some of the results concerning this system, the reader is referred to[180].

1.2 Examples 11

1.2.7 Radiative gasdynamic equations

In radiative gasdynamics, the basic equations governing the flow form asystem of coupled integro-differential equations of considerable complexity.The consequence of these complexities has been to stimulate a search for ap-proximate formulation of the equation of radiative transfer which leads merelyto a system of nonlinear differential equations. Thus, using a general differen-tial approximation for the equation of radiative transfer, which is applicableover the entire range from the transparent limit to the optically thick limit,the set of fundamental equations governing an unsteady flow of a thermallyradiating inviscid gas can be written in the form (see Vincenti and Kruger[207])

ρ,t + uiρ,i + ρui,i = 0; ρui,t + ρujui,j + p,i + (1/3)ER,i = 0;ρT (S,t + uiS,i) + (4/3)ERui,i + uiER,i − kp(cER − 4σT 4) = 0;ER,t + qi,i = −kp(cER − 4σT 4); (1/c)qi,t + (c/3)ER,i = −kpqi,

(1.2.30)

where 1 ≤ i, j ≤ 3; ui are the gas velocity components, ρ the density,p = p(ρ, S) the gas pressure with S being the entropy, ER is the radiativeenergy density, T = T (ρ, S) the temperature, c the velocity of the light, kpthe Plank’s absorption coefficient, σ the Stefan’s constant, and qi the compo-nents of the radiative flux vector. The system of equations can be written inthe form u,t+Aiu,i = b, where u is the column vector with nine componentsρ, u1, u2, u3, S, ER, and q; the column vector b can be read off by inspec-tion, and ξiAi = (αIJ)1≤I,J≤9 is a 9 × 9 matrix with α11 = α22 = α33 =α44 = α55 = uiξi, α12 = ρξ1, α13 = ρξ2, α21 = a2ξ1/ρ, α25 = p,Sξ1/ρ, α26 =ξ1/(3ρ), α31 = a2ξ2/ρ, α35 = p,Sξ2/ρ, α36 = ξ2/(3ρ), α41 = a2ξ3/ρ, α45 =p,Sξ3/ρ, α46 = ξ3/(3ρ), α52 = 4ERξ1/(3ρT ), α53 = 4ERξ2/(3ρT ), α54 =4ERξ3/(3ρT ), α56 = uiξi/(ρT ), α67 = ξ1, α68 = ξ2, α69 = ξ3, α76 =c2ξ1/3, α86 = c2ξ2/3 and α96 = c2ξ3/3 and the remaining entries being allzeros. The eigenvalues of ξiAi are real: ±c/

√3, uiξi (multiplicity three),

uiξi±(∂p/∂ρ+(4ER/3ρ2T )∂p/∂S)1/2 and zero (multiplicity two), and the cor-

responding eigenvectors span IR9; hence the system is hyperbolic. For furtherdetails regarding some of the consequences of (1.2.30), we refer to Helliwell[70], Sharma and Shyam [167], and Sharma [182].

1.2.8 Relativistic gas model

A model for one-dimensional, isentropic, barotropic fluid in special rela-tivity is given by the following conservation laws of mass and momentum (seeSmoller and Temple [188])

∂

∂t

(ρ+

(p+ ρc2)v2

c2(c2 − v2)

)+

∂

∂x

((p+ ρc2)v2

c2 − v2

)= 0,

∂

∂t

((p+ ρc2)v

c2 − v2

)+

∂

∂x

(p+

(p+ ρc2)v2

c2(c2 − v2)

)= 0,

(1.2.31)


where v is the particle speed, c the velocity of the light, ρ the density andp(ρ) the pressure which is a smooth function of ρ. For more general models,the reader is referred to Taub [195]. The above system is strictly hyperbolicas the eigenvalues of the above system are real and distinct, i.e.,

λ1 = c2(v − a)/(c2 − av) and λ2 = c2(v + a)/(c2 + av),

where a =√dp/dρ is the sound speed.

1.2.9 Viscoelasticity

Within the framework of the extended thermodynamics, a set of balancelaws describing viscoelastic materials has been proposed in Lebon et al. [104]by introducing the stress tensor and the heat flux vector among the statevariables; this model includes, as particular cases, a number of classical modelssuch as the ones of Maxwell, Kelvin-Voigt and Poynting-Thomson. Assumingslab symmetry, the balance equations for a one-dimensional case become

ε,t + vε,x − εv,x = 0,v,t + vv,x + (ε/ρo)σ,x = 0,e,t + ve,x + (σε/ρo)v,x = 0,σ,t + vσ,x = (φ1/τ)(ε,t + vε,x) + (φ2/τ)(T,t + vT,x) + φ3,

(1.2.32)

where ε is the strain, v is the velocity, ρo is the reference density, T denotesthe absolute temperature, σ is the stress, e = e(ε, T, σ) represents the inter-nal energy, τ is the relaxation time and φ1, φ2, φ3 are the material responsefunctions dependent upon ε, T and σ.

The group invariance of the above system gives rise to a complete charac-terization of the material response functions (see Fusco and Palumbo [57]and Jena and Sharma [86]). Using an invariance analysis for τ = const,the Maxwell model of viscoelastic media is characterized by the responsefunctions φ1 = ψT−2α/γ , φ2 = 0 and φ3 = −ΛσT−1/γ, where ψ,Λ, α andγ are constants with Λ/ψ ≤ 0; the invariant system of basic equationscan be written in the form u,t + Au,x = b where u = (ε, v, T, σ)tr ,b =(0, 0,Λσe,σT

−1/γ/(τe,T ),−ΛσT−1/γ/τ)tr , and A = (AIJ )1≤I,J≤4 is a 4 × 4matrix with A11 = A22 = A33 = A44 = v, A12 = −ε, A24 = ε/ρo, A42 =−ψT−2α/γε/τ , A32 = (ε/e,T )(e,ε+ρ−1

o σ+ψτ−1T−2α/γe,σ) and the remainingentries being all zeros. Matrix A admits eigenvalues v± εΓT−α/γ and v (mul-tiplicity two) with Γ = (−ψ/(ρoτ))1/2; hyperbolicity of the system is assuredby the condition ψ < 0.

1.2.10 Dusty gases

The study of a two-phase flow of gas and dust particles has been of greatinterest because of many applications to different engineering problems. Gasflows, which carry an appreciable amount of solid particles, may exhibit sig-nificant relaxation effects as a result of particles being unable to follow rapid

1.2 Examples 13

changes of the velocity and temperature of the gas. When the mass concen-tration of the particles is comparable with that of the gas, the flow propertiesbecome significantly different from that of a pure gas. Here, we consider amixture of a perfect gas and a large number of small dust particles of uniformspherical shape. The viscosity and heat conductivity of the gas are neglectedexcept for the interaction with the dust particles, which do not interact amongthemselves. The thermal motion of dust particles is also assumed negligible.Let p, ρ, T and u be respectively the pressure, density, temperature and ve-locity of the gas, and k, θ, v the mass concentration, temperature and velocityof dust particles. Then, assuming slab symmetry, the equations of mass, mo-mentum and energy, for a two-phase flow can be written in the following form(see Marble [118], Rudinger [156], Pai et al. [139], and Sharma et al. [177])

ρ,t + (ρu),x = 0; ρ(u,t + uu,x) + k(v,t + vv,x) + p,x = 0,

(ρ(u2

2+ cvT ) + k(

v2

2+ cmθ)),t + (ρu(

u2

2+ cpT ) + kv(

v2

2+ cmθ)),x = 0,

k,t + (kv),x = 0; v,t + vv,x = (u− v)/τv, (1.2.33)

θ,t + vθ,x = (T − θ)/τT ,

where cp, cv and cm are specific heats of the gas and dust particles; τv and τTare the velocity and temperature equilibrium times, which are assumed to beconstant. The equation of state of a perfect gas is taken to be p = ρRT withR as the gas constant. The above system, after simplification, can be writtenas u,t + Au,x = b, where u = (ρ, u, T, k, v, θ)tr ,

b =

(0,−k

ρ

u− v

τv,k

ρcv((u− v)2

τv− cm(T − θ)

τv), 0,

(u− v)

τv,(T − θ)

τT

)tr,

and A = (AIJ )1≤I,J≤6 is a 6 × 6 matrix with A11 = A22 = A33 = u,A44 =A55 = A66 = v,A12 = ρ,A21 = RT/ρ,A23 = R,A32 = (γ − 1)T,A45 = k andthe remaining entries being all zeros. The eigenvalues of A are real : u±a, u, v(multiplicity three) with a = (γp/ρ)1/2, but the corresponding eigenvectorsspan IR5; thus the system is nonstrictly hyperbolic.

1.2.11 Zero-pressure gasdynamic system

The following two-dimensional convective system, known as the pressure-less gasdynamic system, arises in the study of adhesion particle dynamics todescribe the motion of free particles, which stick under collision, and explainsthe formation of the large scale structures in the universe. (see Weinan et al.[209])

ρ,t + (ρui),i = 0, (ρuj),t + (ρuiuj),i = 0,

where i, j take values 1 and 2. This system has a repeated eigenvalueλ = ξ1u1+ξ2u2, in the direction of unit vector ξ = (ξ1, ξ2) with an incomplete


set of right eigenvectors; its one-dimensional version takes the form

ρ,t + (ρu),x = 0, (ρu),t + (ρu2),x = 0,

which again has a single eigenvalue whose geometrical multiplicity is less thanits algebraic multiplicity. Again, each of these systems is not hyperbolic, andis referred to as nonstrictly hyperbolic.

Remarks 1.2.1:

(i) There is no general theory to solve globally in time the initial valueproblem for the system of PDEs (1.1.1); indeed, it is usual to considerdiscontinuous weak solutions satisfying admissibility criteria, but notmuch is known about their existence (see [45] and [164]).

(ii) The study of nonstrictly hyberbolic systems is far from complete withrespect to the existence and uniqueness of their solutions that may com-prise waves such as undercompressive shocks, overcompressive shocks,delta shocks, and oscillations (see Li et al. [107], Zheng [215], and Dafer-mos [45]).

Chapter 2

Scalar Hyperbolic Equations in One

Dimension

It is well known that the Cauchy problem for the system (1.1.1), satisfyingthe initial condition u(x, 0) = uo(x),x ∈ IRm does not have, in general, asmooth solution beyond some finite time interval, even when u0 is sufficientlysmooth. For this reason, we study the weak solutions containing the discon-tinuities; these weak solutions are, in general, not unique, and one needs anadmissibility criterion to select physically relevant solutions. Here, we studythe one-dimensional scalar case to highlight some of these fundamental issuesin the theory of hyperbolic equations of conservation laws.

2.1 Breakdown of Smooth Solutions

Let us set m = n = 1,g = 0, and write (1.1.1) as

u,t + (f(u)),x = 0, x ∈ IR, t > 0 (2.1.1)

satisfying the initial condition

u(x, 0) = u0(x). (2.1.2)

We write (2.1.1) as u,t + a(u)u,x = 0, where a(u) = f ′(u); the basic ideaunderlying the hyperbolicity of an equation like (2.1.1) is that the Cauchyproblem, in which appropriate data are prescribed at some initial time, shouldbe well posed for it. Since (2.1.1) may be interpreted as an ordinary differentialequation (ODE) along any member of the characteristic family, determinedby the solution of dx/dt = a(u), the notion of hyperbolicity for (2.1.1) issynonymous with the existence of real characteristic curves.

Let us assume that a′(u) 6= 0 for all u; this means that f ′′(u) 6= 0, i.e.,the flux function f(u) is either convex (f ′′(u) > 0) or concave (f ′′(u) < 0).The convexity condition a(u) 6= 0 corresponds to the notion of genuine non-linearity introduced by Lax [102]. For convenience, we may take a′(u) > 0; itthen immediately follows that along the characteristic curves dx/dt = a(u), uremains constant, and so the characteristics are straight lines. The solution

15

16 2. Scalar Hyperbolic Equations in One Dimension

u = u(x, t) of (2.1.1) is then implicitly given by

u = u0(x− ta(u)). (2.1.3)

The above solution (2.1.3) represents a wave, with shape determined by thefunction u0(x), moving to the right with speed a(u); the dependence of a onu produces a typical nonlinear distortion of the wave as it propagates. If u0

is differentiable and u′0(x) ≥ 0, it follows from (2.1.1) on using the implicitfunction theorem that u,t and u,x stay bounded for all t > 0, and the solutionu exists for all time. On the other hand, if u′0 < 0 at some point, both u,tand u,x become unbounded as 1+u′0a

′0(u0)t approaches zero; this corresponds

to the situation when the characteristics, emanating from (x1, 0) and (x2, 0)with x1 < x2 and f ′(u0(x1)) > f ′(u0(x2)), intersect at a point (xc, tc) in thedomain t > 0, where a continuous solution is overdetermined. Thus, the initialvalue problem (2.1.1), (2.1.2) has a continuous single valued solution, calleda simple wave if u0(x) is nondecreasing (see Courant and Friedrichs [41]). Asimple wave of the kind in which the characteristics emanating from the initialline fan out in the positive t-direction is called an expansion (or rarefactionwave); see Figure 2.1.1(a). However, if the characteristics of different slopes fanout from a single point, the wave is called a centered simple wave; see Figure2.1.1(b). But, if u0(x) is monotonically decreasing so that the characteristicsemanating from the initial line converge in the positive t-direction, and thesimple wave region tends to be narrowed (see the region: t ≤ x ≤ α, 0 ≤ t < αin Figure 2.1.2(a)), then the simple wave is called a compression wave (or,a condensation wave). In fact, any compressive part of the wave, where thepropagation velocity is a decreasing function of x, ultimately breaks to give atriple-valued solution for u(x, t); the time t = tc > 0 of first breaking is givenby

tc = −(minξ∈IR

φ′(ξ))−1; φ(ξ) = a(u0(ξ)). (2.1.4)

Example 2.1.1: Consider the following initial value problem (IVP) for theinviscid Burgers equation

u,t + uu,x = 0, x ∈ IR, t > 0u(x, 0) = 1/(1 + x2).

(2.1.5)

Here, the characteristic speed a(u) = u and u′0(x) = −2x/(1+x2)2, we expectthat the wave will break for x > 0, and the solution u(x, t) will becomemultivalued. The breaking time tc in (2.1.4) requires finding the most negativevalue of φ′(ξ) = u′0(ξ) = −2ξ/(1 + ξ2)2, which is attained at ξ = 1/

√3. The

breaking time tc is, then, given by tc = 8/33/2.The differential equation (2.1.5)1 implies that along the characteristics,

x = ξ+ tu0(ξ), which meet the line t = 0 at x = ξ, the function u(x, t) has thevalue u0(ξ). For t < tc, these characteristics do not intersect, and the solutionof the Cauchy problem is given implicitly by

u(x, t) = u0(x− tu). (2.1.6)

2.1 Breakdown of Smooth Solutions 17

On the other hand, when t ≥ tc the characteristic lines begin to intersect;indeed, in this case the Jacobian of the transformation (ξ, t) → (x, t) is zero,i.e., (∂x/∂ξ)|t = 0, and (2.1.6) no longer defines a single valued solution tothe Cauchy problem.

The foregoing solution can be saved for all times t ≥ 0 by allowing discon-tinuities into the solution; this we shall discuss in the following subsection.

2.1.1 Weak solutions and jump condition

Let us consider the conservation law (2.1.1). The function u(x, t) is calleda generalized or a weak solution of (2.1.1), if it satisfies the integral form ofthe conservation law, i.e.,

d

dt

∫ x2

x1

u(x, t)dx = f(u(x1, t)) − f(u(x2, t)), (2.1.7)

holds for any fixed interval x1 ≤ x ≤ x2 and time t.It may be noticed that the integral form (2.1.7) is more fundamental than

the differential form (2.1.1) in the sense that unlike (2.1.1), it makes senseeven for piecewise continuous functions; under the additional assumption ofsmoothness, these two forms are equivalent.

An alternative integral formulation of the weak solution by integrating(2.1.1) against smooth test functions with compact support, namely

∫

R

∫(uφ,t + f(u)φ,x)dxdt = 0, (2.1.8)

where R is an arbitrary rectangle in (x, t) plane, φ is an arbitrary test functionwith continuous first derivatives in R and φ = 0 on the boundary of R, hasa similar feature; indeed, there is a direct connection between (2.1.7) and(2.1.8), and it can be shown that the two integral forms are mathematicallyequivalent (see LeVeque [106]).

We shall now consider solution of (2.1.1) in the sense of distributions thatare only piecewise smooth and, thus, admit discontinuities. Let C : x = X(t)be a smooth curve representing the line of discontinuity in the (x, t) planesuch that x1 < X(t) < x2. Suppose u and f, and their first derivatives arecontinuous in x1 ≤ x < X(t) and in X(t) < x ≤ x2, and have finite limits asx→ X(t)±. Then (2.1.7) may be written

f(u(x1, t)) − f(u(x2, t)) =d

dt(

∫ X(t)

x1

+

∫ x2

X(t)

)u(x, t)dx

=

∫ X(t)

x1

u,tdx+

∫ x2

X(t)

u,tdx+ (ul − ur)dX

dt,

where ul(t) = limx→X(t)−

u(x, t) and ur(t) = limx→X(t)+

u(x, t); now, if we let


x2 → X(t) and x1 → X(t), the integrals tend to zero in the limit as u,tis bounded in each of intervals separately, and we are left with the followingRankine-Hugoniot (R-H) jump condition, expressing the velocity of propaga-tion of the discontinuity in terms of the solution on either side

s = dX/dt = [f(u)]/[u], (2.1.9)

where [u] = ul − ur and [f(u)] = f(ul) − f(ur) denote the jumps in u andf(u) across x = X(t).

If we write the Burgers equation (2.1.5)1, as a conservation law with fluxfunction f(u) = u2/2, i.e.,

u,t + (u2/2),x = 0, x ∈ IR, t > 0 (2.1.10)

then the R-H condition (2.1.9) yields

s =ul + ur

2. (2.1.11)

Example 2.1.2 (Rarefaction Wave): Consider the Burgers equation (2.1.10)with the initial data

uo(x) =

0, x ≤ 0x/α, 0 < x ≤ α1, x > α.

(2.1.12)

As the vertical characteristics issuing from x ≤ 0 carry the value u = 0 fromthe initial data, we obtain the solution u(x, t) = 0 for x ≤ 0. Similarly, thesolution for x > α is u(x, t) = 1. It may be noticed that the characteristicsissuing from the interval 0 < x ≤ α fan out; indeed, the solution is smootheverywhere and has the parametric form

x = ξ + tu0(ξ), u(x, t) = u0(ξ), 0 ≤ ξ ≤ α. (2.1.13)

Equation (2.1.13)1 implies that ξ = αx/(α+ t), which together with (2.1.13)2,shows that u(x, t) = x/(α + t) in the region R : 0 < x ≤ α, t > 0 (see Figure2.1.1(a)). Thus, we have obtained a weak solution of the initial value problem(2.1.10) and (2.1.12) in the form

u(x, t) =

0, x ≤ 0x/(α + t), 0 < x ≤ α+ t1, x > α+ t.

(2.1.14)

It may be noticed that u is continuous along the rays t > 0, x = 0 andx = t+α, but u,x and u,t are not. In this solution, known as the expansion orrarefaction wave, the wave amplitude u increases from 0 to 1 on an x−intervalwhose length increases with an increase in t; this wave does not break and thesolution is valid for all time.


Figure 2.1.1: (a) Solution of (2.1.10) with initial data (2.1.12); (b) Solution of

(2.1.10) with initial data (2.1.15).

It may be remarked that in the limit α→ 0, the initial function in (2.1.12)has a jump discontinuity at x = 0, i.e.,

uo(x) =

0, x ≤ 01, x > 0,

(2.1.15)

and the corresponding solution is obtained from (2.1.14) by letting α tend tozero (see Figure 2.1.1(b)), i.e.,

u(x, t) =

0, x ≤ 0x/t, 0 < x ≤ t1, x > t.

(2.1.16)

Example 2.1.3 (Compression Wave): Consider the Burgers equation (2.1.10)with the initial data

u0(x) =

1, x ≤ 01 − (x/α), 0 < x ≤ α0, x > α.

(2.1.17)

Again, in view of (2.1.13), the initial value problem has the solution (see Figure2.1.2(a))

u(x, t) =

1, x ≤ t(x− α)/(t− α), t < x ≤ α for 0 < t < α.0, x > α.

(2.1.18)


In this solution, known as compression wave, the wave amplitude decreasesfrom 1 to 0 on an x−interval whose length tends to zero as t increases toα. At the point (α, α) a continuous solution is overdetermined, because thecharacteristic lines issuing from the interval 0 ≤ x ≤ α meet at (α, α), andeach carries different values of u. Hence, no solution exists in the classical sensefor t ≥ α. Indeed, the solution can be extended beyond t = α by introducing adiscontinuity at (α, α), and for this, we use (2.1.11); since ul = 1 and ur = 0,we get s = 1/2. So, for t ≥ α, there is a discontinuous solution, given by

u(x, t) =

1, x ≤ (t+ α)/20, x > (t+ α)/2.

(2.1.19)

Thus, the solution of the initial value problem has two distinct time intervals0 < t < α and t ≥ α, describing two different states, given by (2.1.18) and(2.1.19), respectively.

In the limit as α → 0, the initial profile (2.1.17) has a jump discontinuityat x = 0, i.e.,

uo(x) =

1, x ≤ 00, x > 0,

(2.1.20)

and the corresponding solution is obtained from (2.1.19) by letting α tend tozero (see Figure 2.1.2(b)).

u(x, t) =

1, x ≤ t/20, x > t/2.

(2.1.21)

Figure 2.1.2: (a) Solution of (2.1.10) with initial data (2.1.17); (b) Solution of

(2.1.10) with initial data (2.1.20).


2.1.2 Entropy condition and shocks

Introducing generalized or weak solutions makes it possible for an IVP topossess many different solutions; this can be illustrated with the help of scalarconvex equation (2.1.10) and the initial data (2.1.15). For this IVP, we haveobtained a rarefaction wave solution (2.1.16). Now we check that there aremany other weak solutions of this IVP. In fact, the condition (2.1.11) suggeststhat we obtain a weak solution to this problem by introducing a discontinuity,propagating with s = (ul + ur)/2 = 1/2, which gives (see Figure 2.1.3)

u(x, t) =

0, x < t/21, x > t/2.

(2.1.22)

Figure 2.1.3: A discontinuous solution of (2.1.10) with initial data (2.1.15).

It is easy to check that (2.1.22) satisfies (2.1.10) and (2.1.15) along with(2.1.9), and therefore, it is also a weak solution of (2.1.10), (2.1.15). In fact,it can be shown that there are infinite number of weak solutions to this IVP.For instance, if β is any number, satisfying 0 < β < 1, then the function (seeFigure 2.1.4)

u(x, t) =

0, x ≤ βt/2β, βt/2 < x ≤ (β + 1)t/21, x > (β + 1)t/2,

(2.1.23)

is also a weak solution with two discontinuities propagating with speeds β/2and (β+1)/2. Thus, there is a need to have some mechanism to decide aboutthe admissible weak solution. A criterion that selects the right solution wasproposed by Lax [101], according to which a discontinuity in the solution isadmissible only if it prevents the intersection of characteristics coming fromthe points of the initial line on the two sides of it, i.e.,

ul(t) > s(t) > ur(t). (2.1.24)

It may be noticed that the condition (2.1.24) is violated in the solutions givenby (2.1.22) and (2.1.23), which show that the characteristics originating from


the discontinuity line diverge into the domain on the two sides of it as tincreases. This is clearly a nonphysical situation, since the solution shoulddepend upon the initial conditions, not on conditions at the discontinuity;and, therefore, the solutions (2.1.22) and (2.1.23) have to be discarded. Theonly admissible solution is the continuous one, namely (2.1.16).

Figure 2.1.4: Weak solutions of (2.1.10) with initial data (2.1.15).

For the general scalar conservation law (2.1.1) with convex or concave f,a generalized or weak solution is called admissible if for each discontinuity,propagating with speed s defined by (2.1.9), the inequality

f ′(ul) > s > f ′(ur), (2.1.25)

is satisfied. This is known as the Lax’s entropy condition for the flux equationf ′′(u) 6= 0; for convex f , the condition (2.1.25) requires that u` > ur, whereasfor concave f , the requirement is u` < ur. In the context of gasdynamics,the corresponding condition excludes discontinuities across which the entropydoes not increase.

Definition 2.1.1 A discontinuity satisfying the jump condition (2.1.9) andthe entropy condition (2.1.25) is called a shock wave.

It may be remarked that shocks are not associated with only discontinuousinitial data; indeed, Example 2.1.1 illustrates that even with smooth initialdata, the continuity of the solution breaks down after a finite time tc = 8/33/2.

2.1.3 Riemann problem

The conservation law together with piecewise constant data having a singlediscontinuity is known as the Riemann problem; as an example, consider the


conservation law (2.1.1) with piecewise constant initial data

u(x, 0) =

ul, x ≤ 0ur, x > 0,

(2.1.26)

where ul and ur are constants.It may be noticed that the stretching transformation x → ax, t → at

leaves both the equation (2.1.1) and the initial data in (2.1.2) invariant. Thus,if u(x, t) is an entropy solution of (2.1.1), (2.1.2), then for every constanta > 0, the function ua(x, t) = u(ax, at) is also a solution. Since the entropysolution for a given initial condition is unique, we choose a = 1/t, and obtainu(x, t) = u(x/t, 1). We, therefore, look for self-similar solutions of the formu(x, t) = φ(x/t), and consider the Riemann problem (2.1.1), (2.1.26) withthe uniformly convex flux function, and ul < ur; then the convexity impliesf(ul) < f(ur) and the rarefaction wave is given by

u(x, t) =

ul, x ≤ tf ′(ul)φ(x/t), tf ′(ul) < x ≤ tf ′(ur)ur, x > tf ′(ur),

(2.1.27)

where φ(x/t) solves f ′(φ(x/t)) = x/t. However, if ul > ur, then the weaksolution, satisfying the entropy condition, is a shock wave, and is given by

u(x, t) =

ul, x < stur, x > st,

(2.1.28)

where s = (f(ul) − f(ur))/(ul − ur).Thus, the Riemann problem (2.1.1), (2.1.26) with f

′′

> 0 (respectively,f

′′

< 0) is solved uniquely by a rarefaction wave if ul < ur (respectively,ul > ur), and a shock wave if ul > ur (respectively, ul < ur). The solutionsof the Riemann problem for a scalar equation, namely shocks and rarefactionwaves, are called elementary waves.

Example 2.1.4 (Traffic flow): Let us consider the traffic flow problem de-scribed by (1.2.2), in which the flux function f(u) = u(1 − u) is concave, i.e.,f ′′(u) < 0 for all u, together with the initial data (2.1.26).

Case 1: Let ul = 1/2 and ur = 1, so that ul < ur. The characteristics issuingfrom the negative half-line, entering into the region t > 0, are vertical linesalong which u = 1/2, whereas the characteristics starting from x > 0 arestraight lines with negative slope, −1, along which u = 1. Consequently, thecharacteristics intersect right at the origin, and there is a need to introducea shock through the point (0,0) satisfying the R-H jump condition (2.1.9),namely s = 1 − (ul + ur) = −1/2, and the stability criterion (2.1.24). Thus,the resulting weak solution to the initial value problem is u(x, t) = 1/2 forx ≤ −t/2 and u(x, t) = 1 for x > −t/2 (see Figure 2.1.5). This solution, inwhich the density of cars jumps from ul = 1/2 to ur = 1 at the shock, cor-responds to the situation where the cars moving at speed vl = (1 − ul) = 1/2


encounter a traffic jam, and suddenly have to reduce their velocity tovr = (1 − ur) = 0.

Case 2: Let ul = 1 and ur = 1/2, so that ul > ur. This initial data maymodel the situation where a traffic light turns green at t = 0. Cars to the left,leaving the traffic light, are initially stationery, vl = (1−ul) = 0, with densityhigh (ul = 1), while on the other side of the light, there is a small constantdensity. Indeed, the weak solution to this initial value problem is u(x, t) = 1for x < −t, u(x, t) = 1

2 (1− xt ) for −t < x ≤ 0, and u(x, t) = 1/2 for x > 0; see

Figure 2.1.6. The solution shows that the cars which were stationary initially,begin to accelerate when the cars in front of them start moving; and, thus, thespreading out of characteristics leads to a spreading out of cars, acceleratingout of the high density region into the low density region.

t

u = 1u 1/2

x

u = 1u = 1/2

Figure 2.1.5: Solution of (1.2.2) with initial data (2.1.26); ul = 1/2, ur = 1.

Figure 2.1.6: Solution of (1.2.2) with initial data (2.1.26); ul = 1, ur = 1/2.

2.2 Entropy Conditions Revisited 25

2.2 Entropy Conditions Revisited

We have already seen in the preceding subsections that the weak solutionsof (2.1.1), in general, are not unique, and there is a need to appeal to someextra admissibility conditions in order to decide which of the many possibleweak solutions is relevant to the physical situation; we have already noticedthat for a convex flux function f(u), a weak solution of (2.1.1) is admissibleonly if the Lax’s entropy condition (2.1.24) is satisfied across it. For nonconvexflux functions, which appear in several applications, we need more generaladmissibility conditions (see Holden and Risebro [73] and LeFloch [105]).

2.2.1 Admissibility criterion I (Oleinik)

It was Oleinik [133], who first formulated the generalized entropy condition,which applies to nonconvex scalar functions as well. According to this, thefunction u(x, t) is an entropy solution of (2.1.1), if a discontinuity propagatingwith speed s, defined by (2.1.9), satisfies the condition

(f(u) − f(ul))/(u− ul) ≥ s ≥ (f(u) − f(ur))/(u− ur), (2.2.1)

for all u between ul and ur. This condition reduces to the compressibilitycondition (2.1.25) when the flux function is strictly convex. For a generalflux, by taking the limits u → ul and u → ur in (2.2.1), it implies the weakcompressibility condition

f′

(ul) ≥ s ≥ f ′(ur). (2.2.2)

The inequality (2.2.2) has a simple geometric interpretation by noticingthat f ′(ul) and f ′(ur) are slopes of the graph of f(u) at the points (ul, f(ul))and (ur, f(ur)), while s is the slope of the chord that connects these twopoints. In the nonconvex case, the characteristic may be tangent to a discon-tinuity; in fact, the discontinuity propagating with speed s is called a semi-characteristic shock to the left or to the right according as f ′(ul) = s ≥ f ′(ur)or f ′(ul) ≥ s = f ′(ur), respectively. However, if f ′(ul) = s = f ′(ur), i.e., thestates on both sides propagate with the same speed as the discontinuity, thediscontinuity is called a contact discontinuity or a characteristic shock accord-ing as the flux function, between ul and ur, is affine or not affine (see Serre[164] and Boillat [20]).

2.2.2 Admissibility criterion II (Vanishing viscosity)

In the vanishing viscosity approach, we replace the conservation law (2.1.1)by the higher order equation

u,t + (f(u)),x = εu,xx, (2.2.3)


in which the various dissipative effects make their appearance by the presenceof the viscous term εu,xx with ε > 0. Assuming that the viscous equation(2.2.3), satisfying the initial data (2.1.2), has smooth solution uε(x, t) andu = limε→0 uε(x, t), then u is called the physically relevant weak solution of(2.1.1). Unfortunately, it is often not so easy to work with it as the solu-tions of (2.2.3) are not often known for a general f(u); we, therefore, lookfor simpler conditions, which can be more easily verified in practice. But, be-fore we proceed to the next selection principle, we would like to remark that(2.2.3) with initial data (2.1.2) can be solved explicitly when f(u) = u2/2,using the Cole-Hopf transformation (see Whitham [210]). It can be shownthat the exact solution does indeed tend to the weak solution of (2.1.1), whichwe have discussed in preceding subsections, using characteristics and shockfitting, obeying (2.1.11); indeed, with initial data (2.1.26), the vanishing vis-cosity solution in the case ul < ur is the rarefaction wave, and in the caseul > ur, it is the shock wave solution, in agreement with the Lax’s shockcondition (2.1.24).

2.2.3 Admissibility criterion III (Viscous profile)

Another method, based on specific physical arguments, is the viscous pro-file approach, in which the scalar conservation law (2.1.1) is again replaced by(2.2.3), so that the physical problem has some diffusion. For (2.2.3), a viscousprofile is a traveling wave solution u = uε(x−st) = u(x−stε ) for some constants ∈ IR, which is to be determined, and the solution u is required to approach(i) the constant states ul and ur as x− st → −∞ and +∞, respectively, and(ii) the shock wave as ε → 0+. If a discontinuity has no such profile, it is notadmissible. It is easy to see that no smooth traveling wave can exist whenε = 0, but for ε > 0, we obtain

u′ = f(u) − su+ α, (2.2.4)

where the prime denotes differentiation with respect to the argument functionx − st, α is a constant of integration, and the flux function f is assumed tobe convex. Thus, if the requirement (i) is imposed, then limx−st→±∞ u′ = 0,and (2.2.4) implies that α = sul − f(ul) = sur − f(ur), which gives us theR-H condition (2.1.9). In particular, we say that a discontinuity has a viscousprofile, or it satisfies the traveling wave entropy condition if the followingboundary value problem holds across the discontinuity

u′ = g(u); u(+∞) = ur, u(−∞) = ul,

where g(u) = −s(u− ur) + f(u) − f(ur), and s = (f(ul) − f(ur))/(ul − ur).It may be noticed that ul and ur are zeros of g(u), and therefore the aboveODE has a solution if the function g(u) is negative (respectively, positive) forul > u > ur (respectively, ul < u < ur). Thus, if g(u) < 0 for ur < u < ul,then we have u′ < 0, and consequently the graph of f(u) must be below the


straight line joining the points (ul, f(ul)) and (ur, f(ur)), i.e.,

f(u) < f(ur) + s(u− ur), (2.2.5)

for all u ∈ (ul, ur). The solution is as shown in Figure 2.2.1 with u increasingmonotonically from ur at +∞ to ul at −∞. In fact, as ε → 0, the viscousprofile in Figure 2.2.1 tends to the discontinuous hyperbolic shock of (2.1.1)− a step function increasing from ur to ul, and traveling with speed s; this isbecause the viscous shock solution is a function of (x − st)/ε, which impliesthat the shock layer is of width ε. This discontinuous shock solution is inagreement with Lax’s shock inequality (2.1.25). Similarly, if ul < u < ur, thenu′ must be positive, and the solution increases from ul at −∞ to ur at +∞;indeed, the graph of f(u) must be above the straight line joining the points(ul, f(ul)) and (ur, f(ur)), i.e.,

f(u) > f(ul) + s(u− ul), (2.2.6)

for all u ∈ (ul, ur). As ε→ 0, the solution tends to the weak solution of (2.1.1)and (2.1.26), satisfying (2.1.25). When the above two cases are combined, itcan be concluded, that the viscous profile or traveling wave entropy conditionis equivalent to

(f(u) − f(ul))/(u− ul) > s > (f(ur) − f(u))/(ur − u), (2.2.7)

for all u between ul and ur. It may be noticed that the condition (2.2.7) is thestrict version of Oleinik’s condition. By taking the limits u→ ul and u → urin (2.2.7), one obtains the Lax’s condition (2.1.25), i.e.,

f ′(ul) > (f(ul) − f(ur))/(ul − ur) > f ′(ur).

It may be remarked that the converse implication does not hold in the caseof a general (nonconvex) flux function f.

Figure 2.2.1: Viscous profile with shock structure.


It may be noticed that the condition (2.2.1) implies that

s|k − ul| ≤ (k − ul)(f(k) − f(ul)),

for all k between ul and ur; in fact, an identical inequality holds with ulreplaced by ur. These inequalities motivate another entropy condition due toKruzkov [97], which is often more convenient to work with.

2.2.4 Admissibility criterion IV (Kruzkov)

In this approach to the entropy condition, a continuously differentiablefunction η(u) is called an entropy function for the conservation law (2.1.1)with entropy flux q(u), provided η(u) is convex (i.e., η

′′

(u) > 0), as is the casewith the physical entropy in gasdynamics, and

η′(u)f ′(u) = q′(u). (2.2.8)

The pair (η, q) is called an entropy/entropy-flux pair for the conservation law(2.1.1). It may be observed that if u is a smooth solution of (2.1.1), then(2.2.8) implies that

(η(u)),t + (q(u)),x = 0. (2.2.9)

In other words, for a smooth solution to (2.1.1), not only the quantity u isconserved, but the additional conservation law (2.2.9) also holds. However,when u is discontinuous, the operations performed above in arriving at (2.2.9)are not valid, i.e., a weak solution of (2.1.1), in general, does not provide aweak solution to (2.2.9). In order to study the behavior of η for the vanishingviscosity weak solution, we deal with the corresponding dissipative equation(2.2.3), and then let the viscosity tend to zero. In fact, it turns out that givena convex entropy η for (2.1.1), with entropy flux q, the weak solution u of(2.1.1) is an entropy solution if it satisfies the following entropy inequality inthe sense of distribution (see Evans [54])

(η(u)),t + (q(u)),x ≤ 0. (2.2.10)

Following the arguments used in deriving the R-H condition for the conserva-tion law (2.1.1), we have an equivalent form of the entropy condition (2.2.10),namely

s(η(ul) − η(ur)) ≤ q(ul) − q(ur), (2.2.11)

where ul and ur are respectively the states on the left and right sidesof the discontinuity, propagating with speed s. Let us now consider theentropy/entropy-flux pair of Kruzkov

η(u) = |u− k|, q(u) = sgn (u− k)(f(u) − f(k)),

where k is a parameter taking values in IR. One can easily check that η, qsatisfy (2.2.8) at every u 6= k, and with the pair (η, q), the inequality (2.2.11)


yields

s(|ul−k|− |ur−k|) ≤ sgn (ul−k)(f(ul)−f(k))− sgn (ur−k)(f(ur)−f(k)),(2.2.12)

for all k ∈ IR, which is known as the Kruzkov entropy condition. It can beeasily shown that for ul < ur (respectively, ul > ur) and k lying between uland ur, the Kruzkov entropy condition (2.2.12) implies the Oleinik entropycondition (2.2.1).

Here, we consider the case ul < ur. Let us choose k ≥ ur and k ≤ ul,successively; then the entropy condition (2.2.12), with these choices, yieldstwo inequalities, which together express the R-H condition (2.1.9). Finally, iful < k < ur, then from (2.2.12), we have s(ur+ul−2k) ≥ f(ur)+f(ul)−2f(k).But (ur − ul)s = f(ur)− f(ul), with the result that s(ur − k) ≥ f(ur)− f(k)and s(ul − k) ≥ f(ul) − f(k), which are equivalent to the Oleinik condition(2.2.1).

2.2.5 Admissibility criterion V (Oleinik)

If the flux function in (2.1.1) is uniformly convex, i.e., there exists a numberθ > 0 such that f

′′

(u) ≥ θ > 0 for all u, f(0) = 0 and u0(x) is bounded andintegrable, then we have an explicit form of the solution u(x, t) of (2.1.1),(2.1.2) known as Lax-Oleinik formula, which satisfies another form of entropycondition given below.

According to this criterion, the function u(x, t) is the entropy solution of(2.1.1) and (2.1.2), if there exists a constant c > 0 such that for all h > 0, t > 0and x ∈ IR,

u(x+ h, t) − u(x, t)

h≤ c

t. (2.2.13)

The above form (2.2.13) turns out to be useful in studying numerical methods;for details, see Smoller [189], LeVeque [106], and Evans [54].

Remarks 2.2.1:

(i) In order to realize (2.2.13), we notice that for smooth initial data u0(x),with u′0 > 0, the solution (2.1.3) of the initial value problem (2.1.1),yields u,x = u′0/(1+tu′0f

′′

(u0)). This shows that if f is uniformly convex,

i.e., f′′

(u) ≥ θ > 0 for all u, then u,x < c/t, where c = 1/θ; this isindeed, (2.2.13). Furthermore, if we fix t > 0, and choose x and x + hon the left and right sides of the discontinuity, where h is a measure ofthe discontinuity width, much smaller than the jump in u, then simplereasoning suggests that ur − ul ≤ ch/t ≈ 0; this implies the entropycondition ul ≥ ur, i.e., if we let x go from −∞ to +∞, then the quantityu can only jump down. For details, we refer to Smoller [189] and Evans[54].


(ii) We may recall (2.1.27), as a solution of the Riemann problem (2.1.1),(2.1.26) with a uniformly convex flux function f and ul < ur. In thefan-like region f ′(ul) <

xt < f ′(ur), the solution u = φ(x/t) is given

by solving the equation f ′(φ(xt )) = xt for φ. The solution is smooth

everywhere in t > 0, but is only Lipschitz continuous on the two linesx = tf ′(ul) and x = tf ′(ur). It may be noticed that in this region, thesolution satisfies u(x + h, t) − u(x, t) = φ(x+ht ) − φ(xt ) ≤ Lip (φ)ht , iftf ′(ul) < x < x + h < tf ′(ur); here Lip (φ) is a Lipschitz constantof φ, and is finite. This inequality implies that u satisfies the entropycondition (2.2.13).

(iii) For global existence of weak solutions to the initial value problem,(2.1.1), (2.1.2) with a smooth convex flux, and bounded initial datawith bounded total variation, the reader is referred to Chang and Hsiao[31].

2.3 Riemann Problem for Nonconvex Flux Function

We have already seen in subsection 2.1.3 that when the flux function fin (2.1.1) is convex, the solution to the Riemann problem (2.1.1), (2.1.26) isalways either a shock or a rarefaction wave. For nonconvex f, the Oleinikcondition (2.2.1) suggests that the entropy solution to the Riemann problemmight involve both a shock and a rarefaction wave. In fact, the entropy solutionto the Riemann problem (2.1.1), (2.1.26) can be determined from the graphof f in a simple manner. Assume that f ∈ C2 has only one inflection pointbetween ul and ur, and consider the case

(a) (b)

Figure 2.3.1: (a) Concave hull of f ; (b) Solution of (2.1.1) and (2.1.26) with non-

convex f .

(i) ul > ur: In order to solve this problem, we consider the smallest concavefunction f∗ such that f∗(u) ≥ f(u) for all u ∈ [ur, ul]; indeed, an elastic

2.4 Irreversibility 31

string stretched above the graph from (ur, f(ur)) to (ul, f(ul)) will follow thegraph of f∗. We, thus, observe that f∗, called concave hull of f , consists of thestraight line segment, connecting the two points (ur, f(ur)) and (um, f(um)),which is tangent to the graph of f at the point (um, f(um)), together with theportion of f over [um, ul]; see Figure 2.3.1(a). It turns out that the entropysolution consists of a semi-characteristic shock joining ur to um, where um canbe calculated from f ′(um) = f(um) − f(ur)/(um − ur), which is preciselythe shock speed, and a centered simple wave joining um and ul, i.e.,

u(x, t) =

ul, x < f ′(ul)φ(x/t), tf ′(ul) ≤ x ≤ tf ′(um)ur, x > tf ′(um),

(2.3.1)

where φ solves f ′(φ(x/t)) = x/t; it may be noticed that the solution is con-tinuous across the line x = tf ′(ul), whereas it is discontinuous across the linex = tf ′(um) with left and right states um and ur, respectively (see Figure2.3.1(b)).

(ii) ul < ur: In this case, we consider the largest convex function f∗ suchthat f∗(u) ≤ f(u) for all u ∈ [ul, ur]; observe that f∗, called convex hull off , consists of the portion of f over [ul, um] together with the straight linesegment, connecting the points (um, f(um)) and (ur, f(ur)), which is tangentto the graph of f at (um, f(um)). The entropy solution consists of a centeredsimple wave joining ul and um, and a shock that connects um and ur, whereum is given by f ′(um) = f(um) − f(ur)/(um − ur); see Figure 2.3.2.

Figure 2.3.2: Convex hull of f .


2.4 Irreversibility

We have already noticed some interesting results about the solution of non-linear equations that have no counterpart for linear equations. For instance,unlike linear equations, not only that the breakdown in the solution of nonlin-ear equations can happen with smooth initial data (see Example 2.1.1), butalso a continuous solution can have discontinuous initial data (see Example2.1.2). Furthermore, the physical processes described by smooth solutions ofhyperbolic equations are reversible in time, in the sense that if we know thesolution at t = t0, then the solution at any t > t0 or t < t0 can be deter-mined uniquely. But, if the process is described by a discontinuous solution,then there is a high degree of irreversibility in the sense that the solution maynot be traced back uniquely in time. We shall illustrate this by means of anexample.

Example 2.4.1: Consider the conservation law (2.1.10) with the initial data

u(x, 0) =

1, −∞ < x ≤ −10, −1 < x ≤ 02, 0 < x ≤ 10, x > 1.

(2.4.1)

0- 1 1

x=-1+(t/2)

x =1 +

t

u = 1

u = 0 u = 2

u = x/t u = 0

x=1+(t/2)

t

x

x = t - 2t

P (2,1)1

x = 2 t

P (4+ 2 2, 6+ 4 2 )3

( )P 0, 22

Figure 2.4.1: Solution of (2.1.10) with initial data (2.4.1).

The compressive parts of the wave at x = −1 and x = 1 break immediatelyand require the introduction of shocks, originating from (−1, 0) and (1, 0), with

2.4 Irreversibility 33

speeds s = 1/2 and 1, respectively. The equations of shock fronts originatingfrom (−1, 0) and (1, 0) are x = −1 + (t/2) and x = t + 1, respectively; theshock paths are straight as long as they continue to have u = 1 (respectively,u = 2) on the left and u = 0 (respectively, u = 0) to the right; see Figure2.4.1. Since the rarefaction wave originating from (0, 0), overtakes the shockx = t+1 at point P1(2, 1), and it is overtaken by the left shock x = −1+(t/2)at P2(0, 2), the shapes and speeds of the two shocks get altered beyond P1

and P2, which we obtain by invoking the R-H condition (2.1.11). Indeed, theshock paths beyond P1 and P2 turn out to be parabolic, namely, x = t−

√2t

and x =√

2t, respectively.At P3(4+2

√2, 6+4

√2), the two shocks eventually collide and merge into a

single shock x = (t/2) + 1, leading to an entirely different state. The stabilitycriterion (2.1.24) ensures that the solution has four distinct time intervals,describing four different states:

u(x, t) =

1, x ≤ −1 + (t/2)0, −1 + (t/2) < x ≤ 0x/t, 0 < x ≤ 2t2, 2t < x ≤ t+ 10, x > t+ 1,

for 0 < t < 1;

u(x, t) =

1, x ≤ −1 + (t/2)0, −1 + (t/2) < x ≤ 0

x/t. 0 < x ≤ 2√t

0, x > 2√t

for 1 ≤ t < 2;

u(x, t) =

1, x ≤ t− 2√t

x/t, t− 2√t < x ≤ 2

√t

0 x > 2√t

for 2 ≤ t < 6 + 4√

2;

u(x, t) =

1, x ≤ 1 + (t/2),0, x > 1 + (t/2)

for t ≥ 6 + 4√

2.

(2.4.2)

Example 2.4.2: The unique admissible weak solution of (2.1.10), satisfyingthe initial data

u(x, 0) =

1, x ≤ 10, x > 1,

(2.4.3)

is

u(x, t) =

1, x ≤ 1 + (t/2)0, x > 1 + (t/2)

for t ≥ 0. (2.4.4)

It may be noticed that the two different initial value problems, in the abovetwo examples, represent the same state, namely

u(x, t) =

1, x ≤ 1 + (t/2)0, x > 1 + (t/2)

for t ≥ 6 + 4√

2.


In fact, the solution at time t ≥ 6+4√

2 forgets the initial details, and it is notpossible to trace it back uniquely in time in the sense that it corresponds totwo different initial states (2.4.1) and (2.4.3). It is in this sense that solutionsof nonlinear conservation laws are irreversible.

2.5 Asymptotic Behavior

Let us consider the conservation law (2.1.10) with the initial datau(x, 0) = u0(x), x ∈ IR, t > 0 where u0(x) is bounded, integrable, and repre-sents a single hump disturbance with u0(x) ≡ 0 outside the range x1 < x < x2,and u0(x) > 0 in the range. Then the integral form of the conservation lawsimplies that ∫ ∞

−∞u(x, t)dx =

∫ ∞

−∞u0(x)dx, for all t > 0. (2.5.1)

Since∫ x2

x1u(x, t)dx =

∫ x2

x1u0(x)dx ≡ ∆, where ∆ is the area under the initial

hump (see Figure 2.5.1(a)), holds for both the discontinuous solution in Figure2.5.1(b) and the multivalued solution in Figure 2.5.1(c), we must insert theshock into the profile at a position so as to give equal areas ∆1 = ∆2 for thetwo lobes, as shown in Figure 2.5.1(d). This is known as the equal area rule,and the procedure, known as shock fitting, is applicable when there is onlyone shock; for its implementation and justification, see Whitham [210].

Figure 2.5.1: Distortion of the initial profile and shock fitting.

The behavior of the solution as t → ∞ can be studied without goingthrough the above construction in detail. Let us consider the following exam-ples.

Example 2.5.1: Consider the solution of the conservation law (2.1.10) withthe initial data

uo(x) =

0, x < −1−A −1 < x < 0A, 0 < x < 10, x > 1,

(2.5.2)

2.5 Asymptotic Behavior 35

where A is a positive constant.

We notice that the compressive parts of the wave at x = −1 and x = 1break immediately and require the introduction of shocks; see Figure 2.5.2(b).The shock path originating at x = −1 (respectively, x = 1) is given byx = −(At/2) − 1 (respectively, x = (At/2) + 1) with the state u = 0 (respec-tively, u = A) to the left of it and u = −A (respectively, u = 0) to the right.The centered rarefaction wave u = x/t originates from (0,0) with u = −A tothe left of it and u = A to the right; the trailing (respectively, leading) frontof the rarefaction wave overtakes the left (respectively, right) running shockat the point P1: x = −2, t = 2/A (respectively, P2: x = 2, t = 2/A). Theshock path beyond P1 (respectively, P2) turns out to be parabolic, namelyx = −

√2At (respectively, x =

√2At) which separates a state ul = 0 (re-

spectively, ul =√

2A/t) from the state ur = −√

2A/t (respectively, ur = 0).Thus, the solution has two distinct time intervals describing two differentstates, namely,

u(x, t) =

0, x < −1− (At/2)−A, −1 − (At/2) < x < −Atx/t, −At < x < AtA, At < x < 1 + (At/2)0, x > 1 + (At/2),

for 0 < t < 2/A

u(x, t) =

0, x < −√

2At

x/t, −√

2At < x <√

2At

0, x >√

2At,

for t ≥ 2/A.

It may be noticed that the area of the pulse on the right, as well as the left, ofx = 0 in Figures 2.5.2(a) – (d) remains equal to the constant A, i.e., the areaunder the initial curve is conserved; this is in agreement with the property(2.5.1). Moreover, the initial profile with positive and negative parts tends toa wave pattern, which has the shape of inverted N , called an N wave, andthe shock amplitude ul − ur decays to zero like t−1/2 as t → ∞. Indeed, it isonly the area under the initial pulse that appears in the asymptotic solution.In fact, unlike the linear case, all other initial details are completely lost.

(a) (b)


(c) (d)

Figures 2.5.2(a−d): Solution of (2.1.10) with initial data (2.5.2), showing that the

area under the initial curve is conserved.

Example 2.5.2: Consider the following initial value problem in IR × (0,∞)

u,t + (u2/2),x = 0, u(x, 0) =

2, |x| ≥ 2−x, −2 < x < −1

x+ 2, |x| ≤ 14 − x, 1 < x < 2.

(2.5.3)

We examine the pattern of characteristics and associated values of u thatemerge from different portions of the x-axis.

Starting with x ≤ −2, we have u = 2 on the family of straight linesx = 2t+ ξ,−∞ < ξ ≤ −2, and thus u = 2 for x ≤ 2t− 2.

The next segment of x-axis has a compression wave u = x/(t − 1) for2(t−1) < x ≤ t−1; in this triangle shaped region u decreases from 2 to 1 on thex-interval, whose length tends to zero as t increases to 1; see Figure 2.5.3(a).At the point (0, 1), a continuous solution is overdetermined; in fact, the graphof u steepens as t ↑ 1. Thus, there is a need to introduce a shock, originatingfrom (0, 1) into the region t > 1, by invoking the R-H condition (2.1.11). Asthe next segment of x-axis has an expansion wave u = (x + 2)/(t + 1) fort− 1 ≤ x ≤ 3t+ 1, the shock originating from (0, 1) has u = 2 to the left andu = (x+ 2)/(t+ 1) to the right; its trajectory is given by x = 2t−

√2(t+ 1),

which separates a state ul = 2 from a state ur = (x + 2)/(t + 1) = 2 −√2/(t+ 1); see Figure 2.5.3(b).The next segment of x-axis has another compression wave u=(x−4)/(t−1)

for 3t+1 < x < 2t+2, across which u decreases from 3 to 2 on the x-interval,whose length tends to zero as t ↑ 1. Again, at the point (4, 1), a continuoussolution is overdetermined, and there is a need to introduce a shock by invokingthe R-H condition. As the segment of x-axis past x = 2 has u = 2 for x ≥ 2t+2,the shock emanating from (4, 1) has u = (x+ 2)/(t+ 1) to the left and u = 2to the right; its trajectory in the region t > 1 is given by x = 2t+

√2(t+ 1),

which separates a state ul = (x+ 2)/(t+ 1) = 2 +√

2/(t+ 1) from the state

2.5 Asymptotic Behavior 37

ur = 2. Thus, the solution has two distinct time intervals describing twodifferent states, namely,

u(x, t) =

2, x ≤ 2t− 2,x/(t− 1), 2(t− 1) < x < t− 1,(x+ 2)/(t+ 1), t− 1 ≤ x ≤ 3t+ 1,(x− 4)/(t− 1), 3t+ 1 < x < 2(t+ 1),2, x ≥ 2(t+ 1),

for 0 < t < 1

u(x, t) =

2, x ≤ 2t−√

2(t+ 1),

(x+ 2)/(t+ 1), 2t−√

2(t+ 1) < x < 2t+√

2(t+ 1),

2, x ≥ 2t+√

2(t+ 1).

for t ≥ 1

The wave form for t ≥ 1 is an N wave; see Figure 2.5.3(b); the shockamplitude, ul − ur, decays to zero like t−1/2, as t→ ∞.

(a) (b)

Figure 2.5.3: (a) Characteristics curves in the x − t plane and the development of

shocks; (b) Graph of the solution of (2.5.3) for t ≥ 1

Remarks 2.5.1: For the general scalar conservation law (2.1.1) with initialcondition (2.1.2), we have the following decay estimate:

If u0(x) is a bounded, integrable function having compact support, andthe flux function f is smooth and uniformly convex with f(0) = 0, then thereexists a constant k such that |u(x, t)| ≤ kt−1/2 for all x ∈ IR and t > 0.

However, it is interesting to note that the asymptotic behavior of shocksthat originated from periodic data, the shock strength, ul − ur, decays liket−1 for large t, rather that t−1/2, i.e., the solution of a periodic wave woulddecay faster than the solution in a single hump; see Lax [100], Whitham [210],Smoller [189], and Evans [54].

Chapter 3

Hyperbolic Systems in One Space

Dimension

In this chapter, we study certain properties which ensure that the Cauchyproblem for a first order system of conservation laws is well posed. We be-gin with the study of weak solutions, satisfying the system of conservationlaws in the distributional sense, and then outline the admissibility conditionswhich ensure uniqueness of solutions. Notions of simple waves and Riemanninvariants are introduced, and some general properties of shocks and rarefac-tion waves are noted. The Riemann problem for shallow water equations ispresented with some general remarks given at the end.

3.1 Genuine Nonlinearity

We begin by considering the one-dimensional system of conservation laws

u,t + (f(u)),x = 0, x ∈ IR, t ≥ 0 (3.1.1)

where u = u(x, t) and f(u) are n vectors, representing the density andthe flux functions. Assume that f(u) is a C2 smooth function, and letA(u) = (∂fi/∂uj)1≤i,j≤n be the n × n Jacobian matrix. The system (3.1.1)is called hyperbolic, if for each u the matrix A(u) is diagonalizable with realeigenvalues λ1 ≤ λ2 ≤ . . . ≤ λn; if all the eigenvalues are distinct, the systemis called strictly hyperbolic. Let ri(u) be a right eigenvector corresponding tothe eigenvalue λi(u), referred to as the ith characteristic field. The system issaid to be genuinely nonlinear (convex) in the ith characteristic field if, andonly if

∇λi(u) · ri 6= 0 for all u, (3.1.2)

where ∇ is the gradient operator with respect to the elements of u; the systemis called genuinely nonlinear, if it is so in all the characteristic fields. This im-plies that by virtue of the continuity of first derivatives, λi(u) is monotonicallyincreasing or decreasing as u varies along the integral curve of the vector fieldri(u), which is everywhere tangent to ri(u); for a scaled ri, the nonvanishingcondition (3.1.2) takes the form

∇λi · ri = 1 for all u. (3.1.3)

39

40 3. Hyperbolic Systems in One Space Dimension

If instead∇λi · ri ≡ 0 for all u, (3.1.4)

then the ith characteristic field is said to be linearly degenerate or equivalently,the system is called linearly degenerate in the ith characteristic field; indeed,λi = λi(u) is constant along every integral curve of ri.

Example 3.1.1: For n = 1, (3.1.1) represents a single conservation law andλ1(u) = f ′(u), while r1(u) = 1 for all u. The condition (3.1.2) reduces to theconvexity requirement f

′′

(u) 6= 0, which corresponds to the notion of genuinenonlinearity. On the other hand if λ1(u) = constant, this corresponds to thelinearly degenerate case of a linear equation with constant coefficients.

Example 3.1.2: The gasdynamic equations (1.2.11) for a planar motion be-come

ρ,t+(ρu),x = 0, (ρu),t+(p+ρu2),x = 0, (ρE),t+(u(p+ρE)),x = 0, (3.1.5)

with p = p(ρ, e) as the equation of state; here E = e + (u2/2) is the totalspecific energy. We know that for smooth flows, the system (3.1.5) can beequivalently written in a nonconservative form (1.2.9), namely

ρ,t + uρ,x + ρu,x = 0, u,t + uu,x + (1/ρ)p,x = 0, S,t + uS,x = 0, (3.1.6)

with p = p(p, S) as the equation of state. The Jacobian of the system (3.1.6)has distinct eigenvalues.

λ1 = u− a < λ2 = u < λ3 = u+ a, (3.1.7)

where a =√

(∂p/∂ρ)S is the speed of sound; the associated eigenvectors canbe taken as

r1 = (ρ,−a, 0)tr , r2 = (∂p/∂s, 0,−a2)tr , r3 = (ρ, a, 0)tr . (3.1.8)

Thus, we have

∇λ1 · r1 = −(a+ ρ∂a/∂ρ), ∇λ2 · r2 = 0, ∇λ3 · r3 = a+ ρ∂a/∂ρ, (3.1.9)

implying thereby that the first and third characteristic fields are genuinelynonlinear, while the second one is linearly degenerate.

3.2 Weak Solutions and Jump Condition

Because of the dependence of λi on the dependent variable u, smoothsolutions to the system (3.1.1), in general, do not exist globally in time, andone is bound to consider weak solutions.

3.3 Entropy Conditions 41

The vector function u(x, t) is called a generalized or weak solution of (3.1.1)if it satisfies the integral form of the conservation law, i.e., if

∫ x2

x1

u(x, t2)dx−∫ x2

x1

u(x, t1)dx =

∫ t2

t1

(f(u(x1, t)) − f(u(x2, t)))dt,

holds for all x1, x2, t1 and t2. An alternative formulation of the weak solution,by integrating (3.1.1) against smooth test functions with compact support,leads to the following definition.

Definition 3.2.1. A bounded measurable function u(x, t) with a given initialcondition

u(x, 0) = u0(x), (3.2.1)

is a weak solution of (3.1.1) if, and only if

∫ ∞

0

∫ ∞

−∞(φ,tu + φ,xf(u))dxdt +

∫ ∞

−∞u0(x)φ(0, x)dx = 0, (3.2.2)

for every smooth function φ(x, t) with compact support in the set (x, t)|x ∈ IR, t ≥ 0. A discontinuity in the weak solution satisfying the Rankine-Hugoniot jump condition

s[u] = [f(u)], (3.2.3)

where s is the speed of discontinuity x = x(t), follows from (3.2.2) by choosingthe test function to concentrate at the discontinuity (see Smoller [189]).

3.3 Entropy Conditions

As in the scalar case, weak solution of the Cauchy problem (3.1.1), (3.2.1) isnot necessarily unique, and therefore we need to introduce some admissibilitycriterion, which enables us to choose the physically relevant solution amongall the weak solutions.

3.3.1 Admissibility criterion I (Entropy pair)

Given a smooth convex entropy η(u) for (3.1.1) with entropy flux q(u),i.e.,

∇q = ∇η(∇f(u)), (3.3.1)

the weak solution u(x, t) of (3.1.1), (3.2.1) is an entropy solution if it satisfiesthe inequality

(η(u)),t + (q(u)),x ≤ 0, (3.3.2)

in the distributional sense.


Indeed, it can be shown that a piecewise C1 function u, which is a weaksolution of (3.1.1) and (3.2.1) satisfies (3.3.2) if, and only if, it satisfies thejump inequality

s[η(u)] − [q(u)] ≤ 0, (3.3.3)

where s is the speed of discontinuity; see, for details, Lax [102].It may be noticed that (3.3.1) is a first order system of n equations for the

scalar variables η and q; thus, for n ≥ 3, the system is overdetermined andmay not have solutions. However, in many practical problems, it is possibleto find an entropy function, which has a physical meaning.

Example 3.3.1: Let us consider the following model for isentropic or poly-tropic gasdynamics, referred to as the p-system (in Lagrangian coordinates)

V,t − u,x = 0, u,t + (p(V )),x = 0, x ∈ IR, t > 0, (3.3.4)

where V is the specific volume i.e., V = 1/ρ with ρ as density, u the velocityand p = p(V ) the pressure, which is a known function of V . This systemis strictly hyperbolic provided that we assume p′(V ) < 0; in that case, the

functions, η(V, u) = (u2/2) −∫ V0p(s)ds, and q(V, u) = up(V ) constitute an

entropy pair (η, q) for the p-system (3.3.4). Note that the Hessian matrix ofη, namely, H ≡ (∂2η/∂ui∂uj)1≤i,j≤2 is given by

H(V, u) =

[−p′(V ) 0

0 1

],

which is positive-definite and, thus, η is strictly convex.

3.3.2 Admissibility criterion II (Lax)

Another admissibility criterion due to Lax ([101]) is particularly useful asits application is not restricted to piecewise smooth solutions. For a strictlyhyperbolic system (3.1.1), whose solution u(x, t) has a jump discontinuity, itcan be stated as follows.

Let ul and ur, respectively, be the values of u on the left and right sidesof the discontinuity, moving with speed s, satisfying

s(ul − ur) = f(ul) − f(ur). (3.3.5)

Then the discontinuity is admissible, provided that for some index k, 1 ≤ k ≤n, the following inequalities hold if the kth characteristic field is genuinelynonlinear

λk(ul) > s > λk(ur), λk−1(ul) < s < λk+1(ur). (3.3.6)

The inequalities (3.3.6), which persist under small perturbations, i.e.,

|ul − ur| << 1,

3.3 Entropy Conditions 43

can be written in the form

λk−1(ul) < s < λk(ul) and λk(ur) < s < λk+1(ur),

implying thereby that n − k + 1 (respectively, k) characteristics withspeeds λk(ul), λk+1(ul), . . . , λn(ul) (respectively, λ1(ur), λ2(ur), . . . , λk(ur))impinge on the line of discontinuity from the left (respectively, right). Thus,at a point of discontinuity there are n + 1 incoming characteristics, leadingto n+ 1 scalar data; this, together with n jump conditions (3.3.5), suffices todetermine uniquely the 2n+ 1 unknowns ul,ur and s.

Remarks 3.3.1: If a discontinuity has less (respectively, more) than n + 1characteristics impinging on it, it is called undercompressive (respectively,overcompressive). Such discontinuities occur in physical phenomena like mag-netohydrodynamics (MHD), nonlinear elasticity, multiphase flows and com-bustion, where the governing system of PDEs is not strictly hyperbolic. Forsuch systems, the concept of entropy is considerably more difficult, and is stillan area of current research. It is conjectured that the entropy solutions arenecessarily unique, but the conjecture is still wide open in the case of sys-tems, even in one-dimension, when the system is nonconvex and the jump inthe initial states is not small enough. It is believed that every entropy solutioncan be considered as a viscosity limit, but the criterion of viscous profiles isalso inadequate to resolve the nonuniqueness of solutions (see, for instance,Azevedo et al. [8], Shearer [183], and Wu [213]).

For global existence of weak solutions to the initial value problem (3.1.1),(3.2.1), whose characteristic fields are all genuinely nonlinear and total vari-ation of initial data is sufficiently small, the reader is referred to the funda-mental paper of Glimm [62].

3.3.3 k-shock wave

A discontinuity satisfying (3.3.5) and (3.3.6) is called a k-shock wave, orsimply a k-shock. In fact, it is assumed that λ0 = −∞ and λn+1 = ∞, so thatthe conditions (3.3.6) assert that no two shocks of different families fit in asingle inequality, and the characteristics in the kth family, just as in the scalarcase, disappear into the shock as time advances, resulting in its stability. Itmay be remarked that for discontinuities of small amplitude (weak) shocks,the entropy condition (3.3.3) is equivalent to the Lax’s entropy condition;for details concerning uniqueness and stability of entropy weak solutions, thereader is referred to Smoller [189], Dafermos [45], and Bressan [25].

3.3.4 Contact discontinuity

A discontinuity in a linearly degenerate field is called a contact disconti-nuity. If ul and ur are connected by a discontinuity in the kth characteristic


field, which is linearly degenerate, then it can be shown that ul and ur lie onthe same integral curve of rk(u) so that

λk(ul) = s = λk(ur). (3.3.7)

Equation (3.3.7) asserts that the characteristics in the kth family are parallelto the contact discontinuity propagating with speed s.

Thus, for systems with both genuinely nonlinear and linearly degeneratefields, the Lax’s shock inequalities (3.3.6) should read

λk(u`) ≥ s ≥ λk(ur), λk−1(u`) < s < λk+1(ur). (3.3.8)

3.4 Riemann Problem

The Riemann problem consists in finding weak solutions of (3.1.1) withinitial data

u(x, 0) =

u`, x < 0ur, x > 0,

(3.4.1)

where u` and ur are constant vectors.This problem occurs naturally in the shock-tube experiment of gasdynam-

ics, where a long thin cylindrical tube has a gas in two different states, sepa-rated by a thin membrane. The gas is at rest on both sides of the membranewith constant pressures and densities on each side. The membrane is removedat time t = 0, and the problem is to determine the subsequent gas motion,which will be a solution of a Riemann problem. Here, we are concerned withthe piecewise smooth solutions of (3.1.1), which is assumed to be strictly hy-perbolic, with Riemann data (3.4.1). But before we explore these issues, weintroduce certain basic notions, which will be used in subsequent discussion.

3.4.1 Simple waves

Simple waves are solutions of (3.1.1) of the form

u(x, t) = v(ξ(x, t)), x ∈ IR, t > 0. (3.4.2)

Inserting (3.4.2) into (3.1.1), we find that v′(ξ), which denotes the derivative ofv with respect to ξ, is an eigenvector of A for the eigenvalue dx/dt = −ξ,t/ξ,x,i.e., there exists an index k, 1 ≤ k ≤ n, such that

ξ,t + λk(v(ξ))ξ,x = 0, (3.4.3)

where v solves the ordinary differential equation

dv(α)/dα = rk(v(α)), α ∈ IR. (3.4.4)

3.4 Riemann Problem 45

If (3.4.3) and (3.4.4) hold, we call the function defined by (3.4.2) a k-simplewave; equation (3.4.3) implies that the characteristics of the kth family, de-termined by the solution of dx/dt = λk, are straight lines along which ξ isconstant. Thus, for each characteristic field λk, 1 ≤ k ≤ n, one can find ak-simple wave solution u, which remains constant along the characteristics ofthe kth field.

3.4.2 Riemann invariants

Consider the system (3.1.1) with distinct eigenvalues λk , 1 ≤ k ≤ n, of theJacobian matrix A. Corresponding to these eigenvalues λk, there are n lefteigenvectors lk and n right eigenvectors rk, such that

lkA = λklk, Ark = λkrk (k-unsummed). (3.4.5)

Premultiplication of the system (3.1.1) by lk gives

lk ·du

dt= 0, along

dx

dt= λk. (3.4.6)

Suppose that there exists a smooth scalar-valued function ω(u), which remainsconstant along a jth characteristic, then ω must satisfy

∇ω · dudt

= 0, alongdx

dt= λj , (3.4.7)

where ∇ is the gradient operator with respect to (u1, u2, . . . , un) space. Equa-tion (3.4.7), in view of (3.4.6), implies that the vector ∇ω is parallel to lj .Since lk and rk are bases of IRn, and also lj · rk = 0 for j 6= k, we obtain

∇ω · rk = 0. (3.4.8)

If equation (3.4.8) has a solution for ω(u), it is called a k-Riemann invariant.By solving the first order differential equation (3.4.8) for ω, one may con-struct in the vicinity of any point u, the (n− 1) k-Riemann invariants, whosegradients are linearly independent. Further, in view of (3.4.2) and (3.4.4), itimmediately follows from (3.4.8) that a k-Riemann invariant is constant alongthe trajectories of the vector field rk. Indeed, it may be noticed that corre-sponding to each characteristic field λk there exist n− 1 Riemann invariants,which remain constant in the kth simple wave of (3.1.1); this provides a con-nection between the simple wave solutions and the Riemann invariants. For

example, one readily verifies that the functions w1(V, u) = u−∫ V √−p′(y)dy

and w2(V, u) = u+∫ V √−p′(y)dy are, respectively, the 1- and 2-Riemann in-

variants of the system (3.3.4). Similarly, it may be verified that the three pairsof the functions (S, u +

∫(c/ρ)dρ); (u, p); (S, u −

∫(c/ρ)dρ) are respectively,

the 1-, 2- and 3-Riemann invariants of the system (3.1.6).We now construct weak solutions of the Riemann problem (3.1.1), (3.4.1)

assuming that (3.1.1) is strictly hyperbolic and ul and ur are sufficiently close.


3.4.3 Rarefaction waves

Given a fixed state uo, the kth rarefaction wave curve Rk(uo) is definedto be the trajectory in IRn of the solution of (3.4.4), satisfying v(αo) = uo;thus, Rk(uo) is an integral curve of rk(u) through a given state uo. Once thesolution v of (3.4.4) is known, which exists at least locally, i.e., for α close toαo, we look for ξ as a smooth solution of (3.4.3) or the scalar conservation law

ξ,t + (fk(ξ)),x = 0 (3.4.9)

withf ′k(α) = λk(v(α)). (3.4.10)

Since

f ′′k (α) =

dλkdα

= ∇λk(v(α)) · rk(v(α)), (3.4.11)

the flux function fk in (3.4.9) is strictly convex (or concave), if the charac-teristic field λk(u) is genuinely nonlinear. Thus, with the requirement (3.1.3)of genuine nonlinearity, (3.4.11) implies that λk(u) is strictly monotone alongthe curve Rk(uo), and we may divide Rk(uo) into parts R+

k (uo) and R−k (uo)

such that Rk(uo) = R−k (uo)∪R+

k (uo), where R+k (uo) = u ∈ Rk(uo)|λk(u) ≥

λk(uo) and R−k (uo) = u ∈ Rk(uo)|λk(u) < λk(uo). Thus, for a state

ur ∈ R+k (ul), we can construct a k-rarefaction wave connecting ul on the left

and ur on the right. Indeed, we choose ξl and ξr ∈ IR with ξl < ξr, so thatul = v(ξ`),ur = v(ξr); further, since ur ∈ R+

k (u`), we have λk(ur) > λk(ul)as λk(u) is increasing along Rk(ul) and subsequently from (3.4.11), we havef ′k(ξr) > f ′

k(ξl). The case ξ` > ξr, which corresponds to the concave flux fk,can be treated similarly. In Section 2.1.3, we have already studied the scalarRiemann problem for (3.4.9) with flux function strictly convex, whose uniqueweak solution is a continuous rarefaction wave connecting the states ξ` andξr. Thus, it follows that the function

u(x, t) =

ul, x < tλk(ul)v(x/t), tλk(ul) < x < tλk(ur)ur, x > tλk(ur),

(3.4.12)

is a continuous self-similar weak solution of (3.1.1) and (3.4.1). Solution(3.4.12) is called a k-centered simple wave or a k-rarefaction wave connectingthe states ul and ur.

3.4.4 Shock waves

Here, we consider the possibility that the states ul and ur may be joinedby a shock discontinuity along which a weak solution u of (3.1.1) satisfies theR-H condition

s(u` − ur) = f(u`) − f(ur), (3.4.13)

where s is the shock speed.


Now suppose, we fix the left state ul and look for all the states ur, whichcan be connected to ul by a shock satisfying (3.4.13), then (3.4.13) is equiva-lent to a system of n equations in n+ 1 unknowns ur and s. This observationmotivates one to study the Hugoniot curves H(uo), through a fixed point uo,defined as

H(uo) = u ∈ IRn|f(u) − f(uo) = σ(u − uo),

for some σ = σ(uo,u) ∈ IR. The jump condition (3.4.13) then implies thatur ∈ H(ul) and s = σ(ul,ur). Lax ([101], [102]) has shown that the existenceof Hugoniot curves is a consequence of the implicit function theorem, i.e., insome neighborhood of uo, H(uo) consists of the union of n smooth curvesHk(uo), 1 ≤ k ≤ n, with the properties

(i) The curve Hk(uo) passes through uo with tangent rk(uo).

(ii) The Hugoniot curve Hk(uo) and the rarefaction curve Rk(uo) have sec-ond order contact at u = uo.

(iii) The shock speed σ(uo,u) tends to the characteristic speed λk(uo) asu → uo. Moreover, the shock speed is the average of the characteristicspeeds on both sides up to second order terms in |u − uo|, i.e.,

σ(uo,u) =1

2(λk(u) + λk(uo)) +O(|u − uo|2) as u → uo with u ∈ Hk(uo).

It may be noticed from the property (iii) above that Hk(uo) can be partitionedin the same way as the rarefaction curve Rk, i.e., Hk(uo) = H−

k (uo)⋃

H+k (uo),

so that λk(u) ≥ λk(uo) for u ∈ H+k (uo) and λk(u) < λk(uo) for the state

u ∈ H−k (uo). Thus, if the system (3.1.1) is strictly hyperbolic and the kth

characteristic field is genuinely nonlinear, then in a small neighborhood of agiven state ul, the curve ur ∈ R+

k (ul) is a rarefaction wave, and ur ∈ H−k (ul) is

a k-shock satisfying the Lax entropy condition (3.3.6); however, for a linearlydegenerate field, Hk(ul) = Rk(ul) and σ(ul,ur) = λk(ur) = λk(ul), i.e., thenearby states ul and ur are connected by a contact discontinuity of speed σ.

The foregoing results allow one to form the composite curves through ul

Wk(ul) =

R+k (ul)

⋃H−k (ul), kth characteristic field genuinely nonlinear

Rk(ul) = Hk(ul), kthcharacteristic field linearly degenerate,(3.4.14)

for each k, 1 ≤ k ≤ n; the wave curves Wk(ul) are used as building blockstowards the construction of a solution of the Riemann problem for (3.1.1) inthe general case. For details, besides the Lax’s fundamental paper [101], readercan consult Liu [111], Smoller [189], and Evans [54]; indeed, the followingtheorem, which we state without proof, gives us the sufficient conditions forthe initial values such that the corresponding Riemann problem for a strictlyhyperbolic system (3.1.1) has an admissible solution as a set of elementarywaves, such as rarefaction waves, contact discontinuities and shock waves.


Theorem 3.4.1 Suppose that each characteristic field is either genuinelynonlinear or linearly degenerate. Then the Riemann problem (3.1.1), (3.4.1)has a unique solution consisting of at most (n + 1) constant states separatedby elementary waves provided that the states ul and ur are sufficiently closed.

Remarks 3.4.1 The Lax’s Theorem, referred to as above, has been general-ized by Liu [110] to systems that are strictly hyperbolic, but neither convex norlinearly degenerate; the inequality σ(u,ul) > σ(ur ,ul) for each u ∈ Hk(ul)and lying between ul and ur, is referred to as the Liu’s entropy criterion. How-ever, there are examples which illustrate that Liu’s entropy condition is nolonger sufficiently discriminating to single out a unique solution for nonstrictlyhyperbolic systems (see Dafermos [45]); the current status of the theory ofRiemann problem for such systems is far from definitive with regard to bothexistence and uniqueness of solutions. Indeed, such systems often admit alongwith undercompressive and overcompressive shocks, a new type of singularity,called delta shock, which supports point masses; for further details regardingdelta shocks; see Li et al. [107] and Zheng [215].

It is shown in [93] that there are examples of strictly hyperbolic and gen-uinely nonlinear systems for which the Riemann problem for a pair of constantstates ul and ur has no solution if |ul − ur| is large.

Example 3.4.1 (Polytropic ideal gas equations): We now return to the one-dimensional gasdynamic equations (1.2.11) for a planar (m = 0) motion, whichis studied here both as motivation and an example with basic notions explic-itly discussed. The system governing the one-dimensional planar motion of apolytropic ideal gas is

ρ,t+(ρu),x = 0, (ρu),t+(p+ρu2),x = 0, (ρE),t+(u(p+ρE)),x = 0, (3.4.15)

with E = (p/ρ)(γ − 1)−1 + (u2/2) as the total specific energy, and γ > 1.The system (3.4.15) admits three characteristic families corresponding to

the eigenvalues λ1 = u−a < λ2 = u < λ3 = u+a, where a = (γp/ρ)1/2 is thesound speed. The associated right eigenvectors can be chosen to be

r1 = (ρ,−a, ρa2)tr , r2 = (1, 0, 0)tr , r3 = (ρ, a, ρa2)tr .

Thus, ∇λ1 · r1 = −(1 + γ)a/2, ∇λ2 · r2 = 0 and ∇λ3 · r3 = (1 + γ)a/2,which show that the first and the third characteristic fields, λ1 = u − a andλ3 = u + a, are genuinely nonlinear, while the second characteristic field,λ2 = u, is linearly degenerate.

i) Simple waves and Riemann invariantsAs noticed earlier in (3.4.8), any solution ω of the partial differential equa-

tion∇ω · rk = 0, 1 ≤ k ≤ 3, is a k-Riemann invariant associated with the k-th characteristic field of the system (3.4.15). It is easy to see that the pairs(pρ−γ , u+ 2a(γ − 1)−1), (u, p) and (pρ−γ , u− 2a(γ − 1)−1) are respectively


the 1-, 2- and the 3-Riemann invariants. Again, as noticed earlier that in ak-simple wave all k-Riemann invariants are constant, it follows that the flowdescribed by

x = (u− a)t+ g(u), u+ 2a/(γ − 1) = const, (3.4.16)

where the function g(u) is arbitrary, is a 1-simple wave for the genuinelynonlinear characteristic field λ1 = u − a. In fact, the functions pρ−γ andu+ 2a/(γ − 1) are the constants of the flow; they make it possible to expressthe (nonconstant) unknowns of the system as a function of one of them. Equa-tions (3.4.16) determine the velocity, and therefore all other quantities, as animplicit function of x and t. If we assume that there is a point in the simplewave region for which u = 0, as usually happens in practice, (3.4.16)2 yieldsa = ao − ((γ − 1)/2)u, where a = ao for u = 0; since pρ−γ = poρ

−γo in the

wave region, where the subscript o refers to the point where the gas is at rest,we have

ρ = ρo

(1 − (γ − 1)u

2ao

)2/γ−1

, p = po

(1 − (γ − 1)u

2ao

)2γ/γ−1

, (3.4.17)

and x = t(−ao + (γ + 1)(u/2)) + g(u), which may be written in the form

u = φ

(x−

(−ao +

(γ + 1)u

2

)t

), (3.4.18)

with φ as another arbitrary function. Equations (3.4.17) and (3.4.18) expressdensity, pressure and velocity as functions of x and t in the simple wave re-gion. Since the velocities of different points in the wave profile are different,the profile changes its shape in the course of time. For expansion waves, thesolution (3.4.17) – (3.4.18) is complete, but in compression parts of the distur-bance, breaking occurs and shocks appear, and the wave ceases to be a simplewave; when shocks appear, we need to re-examine the arguments leading topρ−γ = const and (3.4.16).

We proceed in a similar manner for a 3-simple wave associated with thegenuinely nonlinear characteristic field λ3 = u+ a. The second characteristicfield, which is a 2-Riemann invariant, is linearly degenerate. In this case, thereare no shocks or expansion waves as the characteristics of the 2-simple waveare parallel lines x = ut+ const.; in fact, these are contact discontinuities withu and p as constants.

ii) Contact discontinuities and shocksThe R-H jump condition (3.2.3) for the system (3.4.15) can be written as

s[ρ] = [ρu], s[ρu] = [p+ ρu2],s[ρ(e+ 1

2u2)] = [(ρ(e+ 1

2u2) + p)].

(3.4.19)

which can be rewritten in a more convenient form by introducing the relative


velocity v = u− s as

[m] = 0, [p+ mv] = 0, m[a2 +γ − 1

2v2] = 0, (3.4.20)

where m = ρv, the mass flux through the discontinuity curve. There are twocases according as m = 0 and m 6= 0.

For m = 0, the R-H conditions (3.4.19) admit a solution s = ul = ur, pl =pr with an arbitrary jump discontinuity in the density. Such a discontinuity,for which

[p] = 0, [u] = 0, [ρ] 6= 0, (3.4.21)

is called a contact discontinuity in the sense of Lax (see equation (3.3.7)); thisis never crossed by the gas particles. Since s = ul = ur, the characteristiccurves of the second family, dx/dt = λ2 = u, coincide on both sides with thediscontinuity curve that appears in a linearly degenerate characteristic fieldλ2 = u.

For m 6= 0, we have a shock discontinuity that may be a 1-shock or a3-shock in the sense of Lax (see inequalities (3.3.6)). For a 1-shock, we haves < ul − al and ur − ar < s < ur; since vr > 0 and vl > al > 0, we haveul, ur > s, and therefore it is a backward facing wave, which is crossed by thegas particles moving from left to right. Similarly, for a 3-shock s > ul, ur andthe gas particles cross form right to left; it is a forward facing wave. In boththe shocks, we have v2

1 > a21 and a2

2 > v22 , where the subscripts 1 and 2 refer

to the states ahead of the shock and behind the shock, respectively; in fact for1-shock the state ahead is on the left, whereas for a 3-shock, the state aheadis on the right side. Thus, for an admissible shock, the gas velocity relative tothe shock front is supersonic at the front side and subsonic at the back side.Furthermore, since

2

γ − 1a21 + a2

1 <2

γ − 1a21 + v2

1 =2

γ − 1a22 + v2

2 <2

γ − 1a22 + a2

2, (3.4.22)

it follows that a1 < a2; here, we have used a1 < |v1|, a2 > |v2| and thejump relation (3.4.20)3. Also, since a1 < a2 equation (3.4.22) implies that|v1| > |v2|, and then ρ1v1 = ρ2v2 shows that ρ2 > ρ1. Subsequently, equation(3.4.20)2 shows that p2 > p1. Thus, a shock satisfying the entropy conditionis always compressive, i.e., p2 > p1, ρ2 > ρ1, a2 > a1, |v1| > |v2|. It may benoticed that for a 1-shock, both v1 and v2 are positive, whereas for a 3-shock,both v1 and v2 are negative.

iii) Shock HugoniotSome simple algebraic identities for a shock (m 6= 0) may be derived from

the conditions (3.4.20); indeed, the elimination of m between (3.4.20)1 and(3.4.20)2 yields v2

2 − v21 = −(p2 − p1)(V2 + V1), where V = 1/ρ is the spe-

cific volume. This relation together with (3.4.20)3 yields, on eliminating thevelocity v, that

(V2 − µ2V1)p2 − (V1 − µ2V2)p1 = 0, (3.4.23)


where µ2 = (γ−1)/(γ+1). The relation (3.4.23), which involves only p and Vis called the Hugoniot relation. If we define the Hugoniot function with center(V1, p1) as

H(V, p) = (V − µ2V1)p− (V1 − µ2V )p1. (3.4.24)

then the equation (3.4.23) may be written as H(V2, p2) = 0. The curveH(V, p) = 0, with (V1, p1) as center, is a rectangular hyperbola lying in theV − p plane, and it represents all possible states which can be connected tothe state (V1, p1) through a shock. It may be noticed that the part of thehyperbola for which V > µ−2V1 has no physical meaning, as the pressurebecomes negative; in fact, along the H-curve the values of V lie between µ2V1

and µ−2V1, whereas the pressure varies between 0 and +∞. As observed fromthe general theory in Sections 3.4.3 and 3.4.4 that the Hugoniot curve Hk(uo)and the rarefaction curve Rk(uo) have second order contact at u = uo, itfollows that 1- and 3-shock curves are projected in the V − p plane onto theHugoniot curve, while the 1- and 3-rarefaction curves are projected onto theisentropic curve S(V1, p1) = So through the point (V1, p1). For further details,see Godlewski and Raviart [63].

iv) Weak and strong shocksFor a 3-shock, the flow quantities behind the shock can be expressed in

terms of s and the state-ahead parameters as follows:

u2 − u1

a1=

2(M2 − 1)

(γ + 1)M,ρ2

ρ1=

(γ + 1)M2

(γ − 1)M2 + 2,p2 − p1

p1=

2γ(M2 − 1)

(γ + 1),

(3.4.25)where M = (s− u1)/a1 is the Mach number of a 3-shock relative to the flowahead; it may be noticed that M > 1. It is often convenient to express theshock Mach number in terms of the shock-strength parameter z = (p2−p1)/p1

as M = (1 + ((γ + 1)/2γ)z)1/2, so that the right-hand sides in the shockformulas (3.4.25) can be expressed in terms of z. This enables one to deducecertain important properties of shocks; for instance, the expression for theentropy of a polytropic gas, given by (1.2.10), shows that d(S2 − S1)/dz > 0,implying thereby that for the entropy S to increase across the shock, the shockmust be compressive with p2 > p1.

For weak shocks (M → 1, equivalently z → 0), the relations (3.4.25) maybe expressed as

M = 1 +

(γ + 1

4γ

)z +O(z2),

u2 − u1

a1=z

γ+O(z2),

ρ2 − ρ1

a1=

1

γz +O(z2),

S2 − S1

cv=γ2 − 1

12γ2z3 +O(z4),

(3.4.26)

showing thereby that for shocks of weak or moderate strength, it is a rea-sonable approximation to neglect changes in the entropy and the Riemanninvariants.


For strong shocks (z → ∞), it is required that s >> u1 so that M ∼ z/a1

and M >> 1. Thus, equations (3.4.25) may be approximated by

u2 ∼ (2/(γ + 1))s, ρ2/ρ1 ∼ (γ + 1)/(γ − 1), p2 ∼ (2/(γ + 1))ρ1s2. (3.4.27)

It may be noticed that the density ratio across an infinitely strong shockremains only finite.

Example 3.4.2 (Isentropic system): Here, we consider the Riemannproblem for (3.3.4) with initial conditions (3.4.1), where ul = (Vl, ul)

tr andur = (Vr, ur)

tr . The system (3.3.4) is of the form (3.1.1) by introducingu = (V, u)tr and f(u) = (−u, p(V ))tr . We notice that the system (3.3.4)is strictly hyperbolic as its Jacobian matrix has two real distinct eigen-values λ1,2 = ∓

√−p′(V ) with the corresponding eigenvectors r1,2 =

(±1,√−p′(V ))tr . Since ∇λk · rk > 0, k = 1, 2, it follows from (3.1.3)

that the characteristic fields λ1 and λ2 are genuinely nonlinear. For thecharacteristic field λ1, we find a rarefaction wave solution using (3.4.4) as

u = (V (x, t),∫ V (x,t)√−p′(y)dy)tr , and call it the 1-rarefaction wave solu-

tion, where V (x, t) is determined from (3.4.3) as x/t = −√−p′(V ); this

gives us V as a function of x/t, using inverse function theorem. We nowfix the state ul and find states ur which can be connected with ul by 1-rarefaction wave. Since p

′′

(V ) > 0, the characteristic speed λ1 is increasingin V , we must have Vl < Vr. Moreover, since du/dV =

√−p′(V ), we have

ur − ul =∫ Vr

Vl

√−p′(y)dy, which shows that ur > ul. In other words, the

possible initial states of the form (3.4.1), which generate a 1-rarefaction wave,lie on the curve

R1 = (V, u) : u− ul =

∫ V

Vl

√−p′(y)dy;V > Vl, u > ul.

Similarly, the 2-rarefaction wave curve through the point ul is given by

R2 = (V, u) : u− ul =

∫ V

Vl

√−p′(y)dy;V < Vl, u > ul.

Next, the shock curves S1 and S2 through ul are obtained from the R-Hconditions

s(V − Vl) = −(u− ul), s(u− ul) = p(V ) − p(Vl), (3.4.28)

where s is the shock speed. As the 1-shock wave and 2-shock wave satisfy theinequalities (3.3.6), i.e.,

s < λ1(ul), λ1(ur) < s < λ2(ur) and s > λ2(ur), λ1(ul) < s < λ2(ul),(3.4.29)

we find that s < 0 for 1-shock wave and s > 0 for 2-shock wave. Indeed, for


1-shock, we find that −√−p′(Vr) < s < −

√−p′(Vl), and for a 2-shock wave√

−p′(Vr) < s <√−p′(Vl); eliminating s from (3.4.28) and using the shock

inequalities (3.4.29), the 1-shock curve S1 and 2-shock curve S2 are defined as

S1 = (V, u) : u− ul = −√

(V − Vl)(p(Vl) − p(V );Vl > V, ul > u,

S2 = (V, u) : u− ul = −√

(V − Vl)(p(Vl) − p(V );Vl < V, ul < u.

Figure 3.4.1: Rarefaction and shock curves.

Figure 3.4.2: (a) ur lies in region Ω1; (b) Wave structure in region Ω1.

Figure 3.4.3: (a) ur lies in region Ω2; (b) Wave structure in region Ω2.


We notice that the system under consideration has four elementary waves,namely the 1-rarefaction wave R1, the 2-rarefaction wave R2, the 1-shockwave S1 and the 2-shock wave S2. We have already seen that the Riemannproblem (3.3.4), (3.4.1) admits a solution in the form of a centered rarefactionwave in the two cases ur ∈ R1, Vr > Vl, or else ur ∈ R2, Vr < Vl. Onthe other hand, we get a shock connecting these constant states providedthat either ur ∈ S1, Vr < Vl, or else ur ∈ S2, Vr > Vl. We now fix ul andsuperimpose the four curves R1, R2, S1 and S2 which divide a neighborhoodof ul into four regions Ωi, i = 1, 2, 3, 4 in the V − u plane (see Fig.3.4.1); thestructure of the general solution to the Riemann problem is now determinedby the location of the state ur with respect to the curves R1, R2 and S1, S2. Ifur lies in region Ω1 between R1 and S2, then there is a point uo ∈ R1 that lieson the curve S

′

2 of constant state that can be connected to ur by a 2-shock(see Fig. 3.4.2(a)). Since uo can be connected to ul by the 1-rarefaction waveR1 and to ur by a 2-shock wave S

′

2, the resulting weak solution consists ofa 1-rarefaction wave and a 2-shock wave (see Fig. 3.4.2(b)). If ur ∈ Ω2, thenthere is a point uo ∈ R1 which lies on the curve R

′

2 of constant states thatcan be connected to ur by a 2-rarefaction wave R

′

2; thus the weak solutionconsists of two centered rarefaction waves (see Figs. 3.4.3 (a) and (b)). Forur ∈ Ω3, the weak solution consists of 1-shock S1 and a 2-rarefaction wave,whereas for ur ∈ Ω4, the solution consists of two shocks. Thus, the Riemannproblem for (3.3.4), (3.4.1) admits a weak solution consisting of three constantstates connected by rarefaction waves or shocks.

Remarks : The existence of a linearly degenerate characteristic field, sat-isfying (3.1.4), allows for the possibility of an additional type of discontinu-ous weak solution, called a contact discontinuity, the speed of which satisfies(3.3.7), and which connects the nearby states ul and ur, having the sameRiemann invariant with respect to this linearly degenerate field; for details,see [189].

3.5 Shallow Water Equations

We consider the Riemann problem for a nonlinear hyperbolic system thatarises in shallow water theory (see Kevorkian [92]). The method of analysisused here finds its origin in references [101], [189], and [149].

The system of equations, which governs the one-dimensional modified shal-low water equations, can be written in conservation form as [89]

h,t + (hu),x = 0,(hu),t + (hu2 + gHh+ gh2/2),x = 0, t > 0, x ∈ IR

(3.5.1)

where u is x-component of fluid velocity, h the water depth, g the acceleration

3.5 Shallow Water Equations 55

due to gravity and H = (k0/g) the reduced factor characterizing advectivetransport of impulse with k0 as a positive constant.

To carry out the characteristic analysis of (3.5.1), it is convenient to use theprimitive variables U = (h, u)tr , rather than the vector of conserved variables.Then for smooth solutions, system (3.5.1) is equivalent to

U,t +AU,x = 0, (3.5.2)

where A is 2 × 2 matrix with elements A11 = A22 = u, A21 = c2/h andA12 = h; here, c =

√g(h+H) is the speed of propagation of the surface

disturbance. The eigenvalues of A are λ1 = u − c and λ2 = u + c. Thus,the system (3.5.2) is strictly hyperbolic when c > 0 and, therefore, admitsdiscontinuities and piecewise continuous solutions, which are called bores anddilatation waves, respectively. Let ~r1 = (h,−c)tr and ~r2 = (h, c)tr be the righteigenvectors corresponding to the eigenvalues λ1 and λ2, respectively. For thecharacteristic field λ1, we have ∇λ1.~r1 = (−gh − 2c2)/2c which is differentfrom zero, and therefore the first characteristic field is genuinely nonlinear.Similarly, the second characteristic field λ2 is also genuinely nonlinear. Thewaves associated with ~r1 and ~r2 characteristic fields will be either bores ordilatation waves. Because the characteristic fields are genuinely nonlinear, wecan expect to solve the Riemann problem for (3.5.1) with bores and dilatationwaves.

3.5.1 Bores

Let hl, ul = u(hl) and h, u = u(h) denote, respectively, the left and theright-hand states of either a bore or a dilatation wave. Here, we computebore curves for the hyperbolic system (3.5.1). We fix, once and for all, a state(hl, ul) and compute the state (h, u) such that there exists a speed σ satisfyingthe Rankine-Hugoniot jump conditions

σ[h] = [hu], (3.5.3)

σ[hu] = [hu2 + gHh+ gh2/2], (3.5.4)

where [.] denotes the jump across a discontinuity curve x = x(t) andσ = dx/dt is the bore speed.

Lemma 3.5.1 Let B1 and B2, respectively denote 1-bore and 2-bore associ-ated with λ1 and λ2 characteristic fields. Let the states Ul and U satisfy theRankine-Hugoniot jump conditions (3.5.3) and (3.5.4). Then the bore curvessatisfy

u = ul ∓ (h− hl)φ(h, hl), (3.5.5)

where φ(h, hl) =√

g(h+hl+2H)2hhl

; indeed, on B1, we have dudh < 0 and d2u

dh2 > 0,

whilst on B2 we have dudh > 0 and d2u

dh2 < 0.


Proof: The σ-elimination of (3.5.3) and (3.5.4) yields (3.5.5) and then differ-entiating (3.5.5) with respect to h, we obtain

du

dh= ∓

(φ(h, hl) + (h− hl)

dφ(h, hl)

dh

). (3.5.6)

It is easy to show using (3.5.6) that dudh < 0 on B1, and du

dh > 0 on B2.Differentiating (3.5.6) with respect to h, we get

d2u

dh2= ∓

(2dφ(h, hl)

dh+ (h− hl)

d2φ(h, hl)

dh2

). (3.5.7)

Since 2dφ(h,hl)dh + (h− hl)

d2φ(h,hl)dh2 = − g2(hl+2H)(4hhl+(hl+2H)(h+3hl))

16h4h2lφ(h,hl)3

, which is

negative for all values of h, we obtain, in view of (3.5.7), that d2udh2 > 0 for

1-bore, and d2udh2 < 0 for 2-bore.

We now show that the bores satisfy the Lax stability conditions.

Lemma 3.5.2 Across 1-bore (respectively, 2-bore), h > hl and u < ul (re-spectively, h < hl and u < ul) if, and only if, the Lax conditions hold, i.e.,1-bore satisfies

σ < λ1(Ul), λ1(U) < σ < λ2(U), (3.5.8)

while the 2-bore satisfies

λ1(Ul) < σ < λ2(Ul), λ2(U) < σ. (3.5.9)

Proof: First let us consider 1-bore and prove σ < λ1(Ul). On 1-bore hl < h,implying thereby that hl < (hl + h)/2, which means that

cl =√g(hl +H) <

√g(h+ hl + 2H)

2. (3.5.10)

Also, since h >√hhl, it follows from (3.5.10) that cl < hφ(h, hl), which

implies that −cl > −hφ(h, hl); in view of equation (3.5.5) for 1-bore, we get

−cl > h(u−ul)h−hl

, implying thereby that

σ =hu− hlulh− hl

< ul − cl = λ1(Ul). (3.5.11)

Next, because hl < h on 1-bore, we have (h + hl + 2H)/2 < h + H , which

implies that√

g(h+hl+2H)2 < c, or equivalently

−c < −√g(h+ hl + 2H)

2. (3.5.12)


Also, because hl <√hhl, which implies that hlφ(h, hl) <

√g(h+hl+2H)

2 , we

have

−√g(h+ hl + 2H)

2< −hlφ(h, hl). (3.5.13)

In view of (3.5.12) and (3.5.13), we get −c < −hlφ(h, hl); since for 1-bore,

u− ul = −(h− hl)φ(h, hl), it follows that −c < hl(u−ul)h−hl

, and hence

u− c = λ1(U) <hu− hlulu− ul

. (3.5.14)

Also, from (3.5.12) and (3.5.13), we obtain −hlφ(h, hl) < c, which in view of

(3.5.5) yields hl(u−ul)h−hl

< c, and hence

hu− hlulu− ul

= σ < u+ c = λ2(U). (3.5.15)

Therefore, 1-bore satisfies Lax conditions; proof for 2-bore follows on similarlines. Conversely, we assume for 1-bore that the left- and right-hand statessatisfy Lax conditions (3.5.8), and show that h > hl and u < ul. Let usdefine v = σ − u; then since σ < λ1(Ul), it follows that vl < −cl, implyingthereby that vl < 0, and hence σ < ul. Similarly, by using second conditionλ1(U) < σ < λ2(U), we get u− c < σ < u+ c, which implies that −c < v < c,showing thereby that |v| < c. From (3.5.3), we have hv = hlvl. Since h and hlare positive, both v and vl must have the same sign; further, since vl < 0, wehave v < 0. For 1-bore, the fluid velocity on both sides of the bore is greaterthan the bore velocity, and therefore the particles cross the bore from left toright.

Since for 1-bore we have v2l > c2l and v2 < c2, equation (3.5.4), namely,

hlv2l +gHhl+gh2

l /2 = hv2 +gHh+gh2/2 implies that hlc2l +gHhl+gh2

l /2 <hlv

2l + gHhl + gh2

l /2 = hv2 + gHh + gh2/2 < hc2 + gHh + gh2/2, showingthereby that 3

2h2l +2Hhl <

32h

2 +2Hh; thus, it follows that hl < h. Moreover,

from (3.5.3), we have v = hlvl

h ; since hl < h, one infers that u− ul = vl − v =

vl− hlvl

h = vl(1− hl

h ), which is negative, and hence u < ul. The correspondingresults for 2-bore can be proved in a similar way, and we shall not reproducethe details.

3.5.2 Dilatation waves

Here, we construct the dilatation wave curves and recall that an n-dilatation wave (n = 1, 2), connecting the states Ul and Ur, is a solutionto (3.5.2) of the form

U(x, t) =

Ul,xt ≤ λn(Ul)

U(xt ), λn(Ul) ≤ xt ≤ λn(Ur)

Ur,xt ≥ λn(Ur),

(3.5.16)


where U(η), η = x/t, is a solution to the system of ordinary differential equa-tions (A − ηI)(h, u)tr = 0; here, I is 2 × 2 identity matrix and an overheaddot denotes differentiation with respect to the variable η. If (h, u)tr = (0, 0),then h and u are constant; but, as we are interested in nonconstant solutions,we consider (h, u)tr 6= (0, 0) and then it follows that (h, u)tr is an eigenvec-tor of the matrix A corresponding to the eigenvalue η. Since the matrix Ahas two real and distinct eigenvalues λ1 and λ2, there are two families ofdilatation waves, D1 and D2 which denote, respectively, 1-dilatation wavesand 2-dilatation waves.

First we consider 1-dilatation waves. Since, (A − λ1I)(h, u)tr = 0 with

λ1 = u− c, we have, ch+ hu = 0, implying thereby that

Π1 ≡ u+ ψ(h) = constant, (3.5.17)

where ψ(h) = 2√g(h+H) +

√gH ln

√g(h+H)−

√gH√

g(h+H)+√gH

. Equation (3.5.17)

represents D1 curves with Π1 as the 1-Riemann invariant. Similarly,2-dilatation wave curves are given by

Π2 ≡ u− ψ(h) = constant, (3.5.18)

where Π2 is the 2-Riemann invariant; indeed, the integral curves of the vectorfields ~r1 and ~r2 are nothing but the level sets of the Riemann invariants Π1

and Π2, respectively.

Theorem 3.5.1 On D1 (respectively, D2), the Riemann invariant Π1

(respectively, Π2) is constant.

Proof: Let U be an n-dilatation wave of the form (3.5.16), and let Π be an n-Riemann invariant; here n = 1, 2. Since, U is continuous and Π is assumed tobe smooth, the function Π : (x, t) → Π(U) is continuous for t > 0. Obviously,Π(U) is constant for x/t ≤ λn(Ul) and x/t ≥ λn(Ur).

Further, since η = x/t, we have

dΠ(U)

dη= ∇Π(U).U . (3.5.19)

As U is parallel to ~rn, the right-hand side of (3.5.19) is zero, and this provesthe theorem.

Theorem 3.5.2 The D1 curve is convex and monotonic decreasing, while D2

curve is concave and monotonic increasing.

Proof: The 1-dilatation curve is given by

u = ul + ψ(hl) − ψ(h), h ≤ hl (3.5.20)


which on differentiation with respect to h, yields du/dh = −c/h < 0, andsubsequently,

d2u

dh2=c2 + gH

2ch2. (3.5.21)

Since c, h, g,H are positive, it follows from (3.5.21) that d2udh2 > 0 and, there-

fore, u is convex with respect to h for 1-dilatation waves. In a similar way, wecan prove for 2-dilatation waves.

Lemma 3.5.3 Across 1-dilatation waves (respectively, 2-dilatation waves),h ≤ hl and ul ≤ u (respectively, h ≥ hl and u ≥ ul) if, and only if, thecharacteristic speed increases from left- to right-hand state.

Proof: Since dcdh = (g/2c) > 0, c is an increasing function of h; this implies that

for 1-dilatation waves, c(h) ≤ c(hl) or equivalently −cl ≤ −c. The inequalitiesul ≤ u and −cl ≤ −c imply that λ1(Ul) ≤ λ1(U). In a similar way, we canprove λ2(Ul) ≤ λ2(U) for 2-dilatation waves.Conversely, for 1-dilatation waves, since λ1(Ul) ≤ λ1(U), we have

c− cl ≤ u− ul. (3.5.22)

Further, because in 1-dilatation wave region Π1 is constant, we have u−ul =ψ(hl)−ψ(h), which in view of (3.5.22) yields c− cl ≤ ψ(hl)−ψ(h), implyingthereby that h ≤ hl, and u−ul = ψ(hl)−ψ(h) ≥ 0. Hence, h ≤ hl and u ≥ ul.In the same way, one can prove that for 2-dilatation waves h ≥ hl and u ≥ ul.

3.5.3 The Riemann problem

In what follows, we consider the Riemann problem for the system (3.5.1)with piecewise constant initial data consisting of just two constant states,which in terms of primitive variables are Ul = (hl, ul)

tr to the left of x = x0,and Ur = (hr, ur)

tr to the right of x = x0, separated by a discontinuity atx = x0, i.e.,

U(x, t0) =

Ul, x < x0

Ur, x > x0.(3.5.23)

We solve this problem in the class of functions consisting of constant states,separated by either bores or dilatation waves. The solution of the Riemannproblem consists of at most three constant states (including Ul and Ur), whichare separated either by a bore or a dilatation wave.

Theorem 3.5.3 The bore and dilatation wave curves for 1-family, i.e., B1

and D1 (respectively 2-family, i.e., B2 and D2) have the second order contactat Ul.


Proof: In order to prove that B1 and D1 have the second order contact at Ul,we have to show that B1 and D1 curves at h = hl, up to second derivatives,are equal. The equation for 1-dilatation wave is given by (3.5.20), and from(3.5.18) we obtain

u|h=hl= ul,

du

dh|h=hl

= − clhl, (

d2u

dh2)|h=hl

=c2l + gH

2clh2l

. (3.5.24)

The equation for 1-bore is given in (3.5.5) and from (3.5.6) and (3.5.7), we get

u|h=hl= ul,

du

dh|h=hl

= − clhl, (

d2u

dh2)|h=hl

=c2l + gH

2clh2l

. (3.5.25)

Thus u, dudh and d2udh2 at h = hl have the same value for 1-bore and 1-dilatation

wave curve. Therefore, B1 and D1 have the second order contact at Ul. Prooffor 2-family follows on similar lines.

When Ur is sufficiently close to Ul, the existence and uniqueness of thesolution of Riemann problem for system (3.5.1) in the class of elementarywaves follow from the general theorem of Lax, which applies to any system ofconservation laws that is strictly hyperbolic and genuinely nonlinear in eachcharacteristic field (see Lax [100] and Godlewski and Raviart [63]). We nowshow that the solution of the Riemann problem for system (3.5.1) exists forany arbitrary initial data.

We consider the physical variables as a coordinate system; let us draw thecurves B1, B2, D1 and D2, from (3.5.5), (3.5.17) and (3.5.18), respectively,for g = 1 and H = 0.1, in the (h, u)-plane as shown in Figure 3.5.1; thesecurves divide the (h, u)-plane into four disjoint open regions I , II , III andIV for a given left state Ul. Indeed, we fix Ul and allow Ur to vary; if Ur lieson any of the above four curves, then we have seen how to solve the problem.We assume that Ur belongs to one of the four open regions I , II , III and IVas shown in Figure 3.5.1.

0.5 1 1.5 2h

-10

-5

5

10

u

B1HULB2HUL

D1HULD2HULT2HUjL

UUj

I

II

III

IV

0.5 1 1.5 2h

-2

2

4

u

B1HULB2HUL

D1HULD2HUL

UF1

G1q

Χ Ur*

Figure 3.5.1: Wave curves in Figure 3.5.2: Ur is in

the (h, u)-plane region I


Following [189], we define, for U ∈ IR2, Bn(U) = (h, u) : (h, u) ∈ Bn(U),Dn(U) = (h, u) : (h, u) ∈ Dn(U), and Tn(U) = Bn(U)

⋃Dn(U), n = 1, 2.

For fixed Ul ∈ IR2, we consider the family of curves S = T2(U) : U ∈ T1(Ul).As the (h, u) plane is covered univalently by the family of curves S, i.e.,through each point Ur, there passes exactly one curve T2(U) of S and thesolution to the Riemann problem is given as follows; we connect U to Ul onthe right by a 1-wave (either bore or dilatation wave), and then we connectUr to U on the right by a 2-wave (either B2 or D2). Indeed, depending on theposition of Ur we have different wave configurations.

Theorem 3.5.4 For any Ul = (hl, ul), Ur = (hr, ur) with hl, hr > 0, theRiemann problem is solvable.

Proof: We allow Ur ∈ IR2 to vary so that it may lie in region I , II , III orIV . If Ur ∈ I , then draw a vertical line h = hr as shown in Figure 3.5.2, whichmeets D2 and B1 uniquely at F1 = (h1, u1) and G1 = (h2, u2), respectively.We notice that the sub-family of curves in S, consisting of the set T2(U) ≡T2(h, u) : hl ≤ h ≤ hr induces a continuous mapping q → χ(q) from the arcUlG1 to line segment G1F1; indeed, the region I is covered by curves in S. So,let us suppose that (hm, um) is the point which is mapped to Ur. Then

u = ul − (hm − hl)φ(hm, hl) − ψ(hm) + ψ(hr), (3.5.26)

which on differentiation yields dudhm

=−(hm−h1)dφ(hm,hl)

dhm−φ(hm, hl)− cm

hm< 0,

implying thereby that (hm, um) is unique. Similarly, we can prove uniquenessif Ur is in region II , III or IV .

Thus if Ur ∈ I , then the solution to Riemann problem consists of 1-boreand a 2-dilatation wave connecting Ul to Ur. Suppose Ur is in region II ; thenthe solution consists of bores B1 and B2 joining Ul to Ur. Let Ur ∈ III ;then the solution of the Riemann problem is obtained by joining Ul to Urby D1 followed by B2. Let Ur lie in region IV , then the solution consists of1-dilatation wave and 2-dilatation wave. Thus the set T2(U) : U ∈ T1(Ul)covers the entire half space h > 0 in a 1-1 fashion, and the Riemann problemis solvable for any arbitrary initial data.

Since 1-dilatation wave curve and 2-bore curve defined by (3.5.17) and(3.5.5), respectively, diverge to ∞ and −∞, as h → 0, and 2-dilatation wavecurve and 1-bore curve defined by (3.5.18) and (3.5.5), respectively, diverge to∞ and −∞, as h→ ∞, we can find the solution to the Riemann problem forarbitrary Ur; this means that in contrast to the p-system, the vacuum statedoes not occur in this case.

3.5.4 Numerical solution

For a given left state Ul and a right state Ur, Karelsky and Petrosyan[89] have discussed how to find the unknown state Um analytically for all thepossible cases. Following Toro [200], we give below a numerical scheme to find


the unknown state Um, and discuss the influence of H on the unknown stateUm in the (x, t)-plane.

Case a: For hl < hm and hr ≥ hm, we eliminate um from (3.5.5) and (3.5.18)to obtain

ur − ul + ψ(hm) − ψ(hr) + (hm − hl)φ(hm, hl) = 0. (3.5.27)

Case b: For hl ≥ hm and hr ≥ hm, eliminating um from (3.5.17) and (3.5.18),we get

ur − ul − ψ(hl) − ψ(hr) + 2ψ(hm) = 0. (3.5.28)

Case c: For hl ≥ hm and hr < hm, eliminating um from (3.5.17) and (3.5.5),we get

ur − ul + ψ(hm) − ψ(hl) − (hr − hm)φ(hr, hm) = 0. (3.5.29)

Case d: For hl < hm and hr < hm, we obtain from (3.5.5) that

ur − ul + (hm − hl)φ(hm, hl) − (hr − hm)φ(hr , hm) = 0. (3.5.30)

Thus, for all the four possible wave patterns (3.5.27) – (3.5.30), we obtain asingle nonlinear equation

fr(hm, Ur) + fl(hm, Ul) + ur − ul = 0, (3.5.31)

where

fl(ρm, Ul) =

(hm − hl)φ(hm, hl), if hm > hl,ψ(hm) − ψ(hl), if hm ≤ hl,

(3.5.32)

and

fr(ρm, Ur) =

(hm − hr)φ(hm, hr), if hm > hr,ψ(hm) − ψ(hr), if hm ≤ hr.

(3.5.33)

We solve (3.5.31) for hm by using Newton-Raphson iterative procedure witha stop criterion when the relative error is less than 10−8; the initial guessfor hm is taken to be the average value of hl and hr. Once hm is known, thesolution for the x-component of fluid velocity um can be obtained from (3.5.5)or (3.5.17) (respectively, from (3.5.5) or (3.5.18)) depending on whether the 1-wave (respectively, 2-wave) is a bore or a dilatation wave. In case of dilatationwaves, we have to find the solution inside the wave region. For 1-dilatationwave, the slope of the characteristic from (0, 0) to (x, t) is

dx

dt=x

t= u− c, (3.5.34)


where the fluid velocity u and the speed of propagation c of surface disturbanceare functions of the unknown h.

Since Π1 is constant in 1-dilatation wave region, we have

u = ul + ψ(hl) − ψ(h), (3.5.35)

which in view of (3.5.34), yields

ul + ψ(hl) − ψ(h) − x

t− c = 0. (3.5.36)

Equation (3.5.36) is solved for h, using Newton-Raphson method, and thenu is found from (3.5.35). In a similar way, we find the solution inside the2-dilatation wave.

When hl < hm and hr < hm, the solution of the Riemann problem consistsof 1-bore and 2-bore; indeed, for Test 1 (see Table 1), the solution profilesare shown in Figure 3.5.3 at time t = 0.1. When hl ≥ hm and hr ≥ hm,the solution consists of a 1-dilatation wave and a 2-dilatation wave, and thesolution profiles for Test 2, at time t = 0.25, are shown in Figure 3.5.4. Whenhl < hm and hr ≥ hm, the solution consists of a 1-bore and a 2-dilatation wave,and the solution profiles for Test 3, at time t = 0.17, are shown in Figure 3.5.5.Similarly, when hl ≥ hm and hr < hm, the solution of the Riemann problemconsists of a 1-dilatation wave and a 2-bore, and the solution profiles for Test4, at time t = 0.35, are shown in Figure 3.5.6. Table 2 shows how for a fixedUl and Ur (namely, Test 4), the intermediate state Um (unknown state) isinfluenced by the presence of advective transport of impulse (H); indeed, anincrease in H makes the dilatation wave weaker and bore stronger, see Figure3.5.6.

Table 1 : Four Riemann problem tests

Test hl ul hm um hr ur Result

1 1.0 1.0 2.13116 0.0 1.0 −1.0 B1B2

2 1.0 −0.5 0.65516 0.0 1.0 0.5 D1D2

3 0.8 1.1 1.32917 0.54619 1.7 0.9 B1D2

4 3.0 0.0 2.20694 0.54306 1.5 0.0 D1B2

Table 2 : Intermediate state influenced by the parameter H

H = 0 H = 0.1 H = 0.2hm um hm um hm um

2.18076 0.51062 2.20694 0.54306 2.20732 0.55705


−0.2 −0.1 0 0.1 0.20.20.9

1.215

1.53

1.845

2.152.15

position

dept

h

−0.2 −0.1 0 0.1 0.20.2−1.2

−0.6

0

0.6

1.21.2

position

velo

city

Figure 3.5.3: Exact solution for depth and velocity at t = 0.1

−0.2 0 0.2 0.4 0.50.50.75

1

1.25

1.5

1.751.751.75

position

depth

−0.2 0 0.2 0.4 0.50.50.5

0.7

0.9

1.1

1.21.2

position

velo

city


−0.65 −0.3 0.05 0.4 0.650.650.55

0.68

0.81

0.94

1.051.05

position

dept

h

−0.8 −0.45 −0.1 0.25 0.60.6−0.555

−0.275

0.005

0.285

0.550.55

position

velo

city



−0.8 −0.4 0 0.4 0.80.81.4

1.85

2.3

2.75

3.13.1

position

dept

h

H=0H=0.1

−0.8 −0.4 0 0.4 0.80.8−0.1

0.1

0.3

0.5

0.70.7

position

velo

city

H=0H=0.1


3.5.5 Interaction of elementary waves

The interaction of elementary waves, obtained from the Riemann problem(3.5.1), (3.5.23), gives rise to new emerging elementary waves. We define theinitial function, with two jump discontinuities at x1 and x2, as follows.

U(x, t0) =

Ul, −∞ < x ≤ x1

U∗, x1 < x ≤ x2

Ur, x2 < x <∞,(3.5.37)

with an appropriate choice of U∗ and Ur in terms of Ul and arbitrary x1

and x2 ∈ IR. With the above initial data, we have two Riemann problemslocally. An elementary wave of the first Riemann problem may interact with anelementary wave of the second Riemann problem, and a new Riemann problemis formed at the time of interaction. For one dimensional Euler equations, adiscussion of the interaction of elementary waves may be found in Courantand Friedrichs [41], Chorin [35], and Chang and Hsiao [31]; in Section 3.5.7, weinclude a discussion of the interaction of shock and rarefaction waves belongingto the same family; the interaction of waves belonging to the same family isreferred to as overtaking. The notation, D2B1 → B1D2, used in the sequel,means that a 2-dilatation wave D2 of the first Riemann problem (connectingUl to U∗) interacts with the 1-bore of the second Riemann problem (connectingU∗ to Ur), and leads to a new Riemann problem (connecting Ul to Ur via Um),the solution of which consists of the bore B1 and a 2-dilatation wave D2 (i.e.,B1D2). The possible interactions of elementary waves belonging to differentfamilies, referred to as collision, consist of B2B1, B2D1, D2D1 and D2B1,while the elementary wave interactions belonging to the same family consistof B2B2, B1B1, D1B1, B1D1, B2D2 and D2B2.


3.5.6 Interaction of elementary waves from differentfamilies

i) Collision of two bores (B2B1):We consider that Ul is connected to U∗ by a 2-bore of the first Riemann

problem and U∗ is connected to Ur by a 1-bore of the second Riemann problem.In other words, for a given Ul, we choose U∗ and Ur in such a way that h∗ < hl,u∗ = ul + (h∗ − hl)φ(h∗, hl) and h∗ < hr, ur = u∗ − (hr − h∗)φ(hr, h∗). Sincethe speed of the 2-bore of the first Riemann problem is greater than the speedof the 1-bore of the second Riemann problem, B2 overtakes B1. In order toshow that for any arbitrary state Ul, the state Ur lies in the region II (seeFigure 3.5.1), it is sufficient to prove that (h−h∗)φ(h, h∗)− (h−hl)φ(h, hl)+(hl − h∗)φ(hl, h∗) > 0 for h∗ < hl and h∗ < h.

Since φ is a decreasing function with respect to the second argument, andh∗ < hl, we have φ(h, h∗) > φ(h, hl), which implies that

(h− hl)φ(h, hl) − (hl − h∗)φ(hl, h∗) < (h− hl)φ(h, hl) < (h− h∗)φ(h, h∗).

Hence (h − h∗)φ(h, h∗) − (h − hl)φ(h, hl) + (hl − h∗)φ(hl, h∗) > 0, i.e., thecurve B1(U∗) lies below the curve B1(Ul), and therefore Ur lies in the regionII. Thus, in view of the results presented in the preceding section, it followsthat the outcome of the interaction is B2B1 → B1B2; the computed resultsillustrate this case in Figure 3.5.7.

0.5 1 1.5 2h

-4

-2

2

4

6

u

UU*

B1HU*LB2HUL

D1HULD2HUL

B1HUL0.5 1 1.5 2

h

-2

2

4

u

UU*

D1HU*L

B2HUL

D1HULD2HUL

B1HUL

Figure 3.5.7: Collision B2B1 Figure 3.5.8: Collision B2D1

ii) Collision of a bore and dilatation wave (B2D1):Here U∗ ∈ B2(Ul) and Ur ∈ D1(U∗). That is, for a given Ul, we choose

U∗ and Ur such that h∗ < hl, u∗ = ul + (h∗ − hl)φ(h∗, hl) and hr ≤ h∗,ur = u∗ +ψ(h∗)−ψ(hr). Since 2-bore has a greater velocity than 1-dilatationwave, it follows that B2 overtakes D1. Moreover, since for any given Ul

ψ(hl) − ψ(h∗) − (h∗ − hl)φ(h∗, hl) > 0 for h < h∗ < hl,

it follows that the curve D1(U∗) lies below the curve D1(Ul); hence Ur lies


in the region III, and subsequently B2D1 → D1B2. The computed resultsillustrate this case in Figure 3.5.8.

0.5 1 1.5 2h

-4

-2

2

4

6

u

U U*

D1HU*L

B2HUL

D1HULD2HUL

B1HUL0.5 1 1.5 2

h

-2

2

4

u

UU* B1HU*L

B2HUL

D1HULD2HUL

B1HUL

Figure 3.5.9: Collision D2D1 Figure 3.5.10: Collision D2B1

iii) Collision of two dilatation waves (D2D1):

We consider U∗ ∈ D2(Ul) and Ur ∈ D1(U∗). In other words, for a givenUl, we choose U∗ and Ur such that hl ≤ h∗, u∗ = ul − ψ(hl) + ψ(h∗) andhr ≤ h∗, ur = u∗ + ψ(h∗) − ψ(hr). Since the trailing end of the 2-dilatationwave has velocity (bounded above) greater than that of the 1-dilatation wave,the interaction will take place in a finite time. Further, since hl < h∗ and ψ isan increasing function of h, we have ψ(hl) < ψ(h∗), and therefore the curveD1(U∗) lies above the curve D1(Ul); hence Ur lies in the region IV and theoutcome of the interaction is D2D1 → D1D2. The computed results illustratethis case in Figure 3.5.9.

iv) Collision of a dilatation wave and a bore (D2B1):

Here U∗ ∈ D2(Ul) and Ur ∈ B1(U∗), i.e., for a given Ul, we choose U∗and Ur such that hl ≤ h∗, u∗ = ul − ψ(hl) + ψ(h∗) and h∗ < hr, ur =u∗−(h∗−hr)φ(h∗, hr). Since the 1-bore speed of the second Riemann problemis less than the speed of trailing end of the 2-dilatation wave of the firstRiemann problem, the bore B1 penetrates D2 in a finite time. For any givenUl, we show that Ur ∈ I ; for this, it is enough to show that

ψ(h∗) − ψ(hl) + (h− hl)φ(h, hl) − (h− h∗)φ(h∗, h) > 0. (3.5.38)

Since ψ(h) is an increasing function of h, we have ψ(h∗) > ψ(hl) for hl < h∗;hence, the inequality (3.5.38) follows, implying thereby that the curve B1(U∗)lies above the curve B1(Ul), and Ur lies in the region I. Thus the outcome ofthe interaction is D2B1 → B1D2; the computed results illustrate this case inFigure 3.5.10.


3.5.7 Interaction of elementary waves from the same family

i) A 2-bore overtakes another 2-bore (B2B2):We consider the situation in which Ul is connected to U∗ by a 2-bore of the

first Riemann problem and U∗ is connected to Ur by a 2-bore of the secondRiemann problem. In other words, for a given left state Ul, the intermediatestate U∗ and the right state Ur are chosen such that h∗ < hl and u∗ =ul + (h∗ − hl)φ(h∗, hl) with Lax stability conditions

λ1(Ul) < σ2(Ul, U∗) < λ2(Ul), λ2(U∗) < σ2(Ul, U∗), (3.5.39)

and hr < h∗, ur = u∗ + (hr − h∗)φ(hr , h∗) with

λ1(U∗) < σ2(U∗, Ur) < λ2(U∗), λ2(Ur) < σ2(U∗, Ur), (3.5.40)

where σ2(Ul, U∗) is the speed of the bore connecting Ul to U∗, and similarlyσ2(U∗, Ur) is the speed of the bore connecting U∗ to Ur. From (3.5.39) and(3.5.40), we obtain σ2(U∗, Ur) < σ2(Ul, U∗), i.e., the 2-bore of second Riemannproblem overtakes the 2-bore of the first Riemann problem after a finite time,and gives rise to a new Riemann problem with data Ul and Ur. In order to solvethis problem, we must determine the region in which Ur lies with respect toUl. We claim that Ur lies in region III so that the solution of the new Riemannproblem consists of D1 and B2. In other words, to prove our claim, we need toshow that B2(U∗) lies entirely in the region III; to show this we are requiredto prove that for h < h∗ < hl,

(hl − h)φ(h, hl) − (hl − h∗)φ(hl, h∗) − (h∗ − h)φ(h, h∗) > 0. (3.5.41)

Let us define a new function

f1(h) = (hl − h)φ(h, hl) − (hl − h∗)φ(hl, h∗) − (h∗ − h)φ(h, h∗),

so that f1(h∗) = 0, and differentiate f1(h) with respect to h to obtain

df1dh

=(hl − h)dφ(h, hl)

dh−φ(h, hl) −

((h∗ − h)

dφ(h, h∗)

dh− φ(h, h∗)

). (3.5.42)

Now we define

f2(h, hl) = (hl − h)dφ(h, hl)

dh− φ(h, hl). (3.5.43)

Since f2(h, hl) is a decreasing function with respect to the second variable hlfor h < hl, we have f2(h, hl) < f2(h, h∗), and it follows from (3.5.42) thatf1(h) is a decreasing function of h, implying thereby that f1(h) > f1(h∗) = 0.Hence, B2B2 → D1B2; the computed results illustrate this situation in Figure3.5.11.


ii) A 1-bore overtakes another 1-bore (B1B1):The analytical proof that Ur lies in the region I, so that B1B1 → B1D2,

is similar to the previous case.

iii) A 1-dilatation wave overtakes a 1-bore (D1B1):In this case, Ul is connected to U∗ by a 1-dilatation wave of the first

Riemann problem and U∗ is connected to Ur by a 1-bore of the second Riemannproblem. That is, for a given Ul, we choose U∗ and Ur in such a way thath∗ ≤ hl, u∗ = ul + ψ(hl) − ψ(h∗) and h∗ < hr, ur = u∗ − (hr − h∗)φ(hr , h∗).

First we show that D1(Ul) lies above the curve B1(U∗) for h∗ < h ≤ hl; inother words, for h∗ < h ≤ hl

ψ(h∗) − ψ(h) + (h− h∗)φ(h, h∗) > 0. (3.5.44)

Let us define f3(h) = ψ(h∗) − ψ(h) + (h − h∗)φ(h, h∗), so that f3(h∗) = 0.We claim that f

′

3(h) > 0. Let us assume on the contrary that f′

3(h) ≤ 0,

which implies that (h − h∗)dφ(h,h∗)

dh + φ(h, h∗) ≤ ch ; squaring both sides and

simplifying, we obtain

(h− h∗)2((h∗ + 2H)2 + 4h(h+ h∗ + 2H)) ≤ 0, (3.5.45)

which is a contradiction, because the left-hand side of the inequality (3.5.45)is strictly positive. Thus f

′

3(h) > 0, implying thereby that f3(h) > f3(h∗) = 0;hence D1(Ul) lies above the curve B1(U∗) for h∗ < h ≤ hl. Next, we provethat B1(Ul) lies above the curve B1(U∗) for hl < h; for this, it is sufficient toprove that

ψ(h∗) − ψ(hl) + (h− h∗)φ(h, h∗) − (h− hl)φ(h, hl) > 0, ∀ hl < h. (3.5.46)

Let us define f4(h) = ψ(h∗) − ψ(hl) + (h− h∗)φ(h, h∗) − (h− hl)φ(h, hl), sothat f4(hl) = f3(hl) > 0. It may be noticed that

f′

4(h) = φ(h, h∗) + (h− h∗)dφ(h, h∗)

dh− φ(h, hl) − (h− hl)

dφ(h, hl)

dh> 0,

because φ(h, h∗)+ (h−h∗)dφ(h,h∗)

dh is a decreasing function with respect to h∗for h∗ < h; thus, f

′

4(h) > 0 for h∗ < hl < h, and hence ψ(h∗) − ψ(hl) + (h−h∗)φ(h, h∗) − (h− hl)φ(h, hl) > 0.

Lastly, we show that B2(Ul) and B1(U∗) intersect at some point (h1, u1),where h∗ < h1 < hl. To prove this, we define a new function

f5(h) = ψ(hl) − ψ(h∗) − (h− h∗)φ(h, h∗) − (h− hl)φ(h, hl) for h∗ ≤ h ≤ hl.

Since f5(hl) = −f3(hl) < 0 and f5(h∗) > 0, by virtue of monotonicity andthe intermediate value property, there exists a unique h1, between h∗ and hl,such that f5(h1) = 0. Thus, the intersection of B2(Ul) and B1(U∗) is uniquelydetermined; the computed results are shown in Figure 3.5.12. Thus, dependingon the value of hr we distinguish three cases:


a) When hr < h1, Ur ∈ III and the outcome of the interaction is D1B1 →D1B2; indeed, the 1-bore is weak compared to the 1-dilatation wave.

b) When hr = h1, Ur lies on B2(Ul) and the outcome of the interaction isD1B1 → B2; thus, when two waves of first family interact, they annihi-late each other, and give rise to a wave of the second family.

c) When hr > h1, Ur ∈ II and the outcome of the interaction isD1B1 → B1B2; this means that the 1-bore of the second Riemannproblem, which is strong compared to the 1-dilatation wave of the firstRiemann problem, overtakes the trailing end of the 1-dilatation wave,producing a reflected bore B2(Um, Ur) that connects a new constantstate Um on the left to the known state Ur on the right. The transmit-ted wave, after interaction, is the 1-bore that joins Ul on the left to thestate Um on the right.

iv) A 1-bore overtakes a 1-dilatation wave (B1D1):Here U∗ ∈ B1(Ul) and Ur ∈ D1(U∗). That is, for a given Ul, we choose

U∗ and Ur such that hl < h∗, u∗ = ul − (h∗ − hl)φ(h∗, hl) and hr ≤ h∗,ur = u∗ + ψ(h∗) − ψ(hr). In the (x, t) plane, the speed λ1(U∗) of the trailingend of the 1-dilatation wave is less than the velocity σ1(Ul, U∗), and thereforethe 1-dilatation wave from right overtakes the 1-bore from left after a finitetime.

First we show that B1(Ul) lies above the curve D1(U∗) for hl < h < h∗; forthis we need to show ψ(h)− ψ(h∗)− (h− hl)φ(h, hl) + (h∗ − hl)φ(h∗, hl) > 0for hl < h < h∗. To prove this, we define a new function

f6(h) = ψ(h) − ψ(h∗) − (h− hl)φ(h, hl) + (h∗ − hl)φ(h∗, hl) for hl < h < h∗.

Then, one can show that f′

6(h) = ch − φ(h, hl) − (h − hl)

dφ(h,hl)dh < 0 for

hl < h < h∗, implying thereby that f6(h) > f6(h∗) = 0.Next, we show that D1(U∗) lies below the curve D1(Ul) for h ≤ hl < h∗,

i.e., ψ(hl)−ψ(h∗)+(h∗−hl)φ(h∗, hl) > 0 for h ≤ hl < h∗. Since the left-handside of this inequality, for h ≤ hl < h∗, turns out to be f6(hl), which hasalready been shown to be positive, the conclusion follows.

Lastly, we show that B2(Ul) and D1(U∗) intersect uniquely at some point,say, (h2, u2); for this, it is enough to show that the equation

ψ(h) − ψ(h∗) + (h− hl)φ(h, hl) + (h∗ − hl)φ(h∗, hl) = 0,

has a unique root h2 such that h2 < hl. To establish this, we define a newfunction f7(h) = ψ(h) − ψ(h∗) + (h − hl)φ(h, hl) + (h∗ − hl)φ(h∗, hl); sincef7(hl) > 0, and f7(h) takes negative values as h is close to zero, in view ofmonotonicity and the intermediate value property, it follows that the curvesD1(U∗) and B2(Ul) intersect uniquely; the computed results are shown inFigure 3.5.13. Here, again, we distinguish three cases depending on the valueof hr:


a) When hr > h2, Ur ∈ II and the outcome of the interaction is B1D1 →B1B2, i.e., the 1-bore is sufficiently strong compared to the 1-dilatationwave which, after interaction, produces a new elementary wave.

b) When hr = h2, Ur ∈ B2(Ul) and the outcome of the interaction isB1D1 → B2, i.e., the interaction of elementary waves of the first familygives rise to a new elementary wave of the second family.

c) When hr < h2, Ur ∈ III and the outcome of the interaction is B1D1 →D1B2.

v) A 2-bore overtakes a 2-dilatation wave (B2D2):The B2D2 interaction takes place when U∗ ∈ B2(Ul) and Ur ∈ D2(U∗). In

other words, for a given Ul, we choose U∗ and Ur in such a way that h∗ < hl,u∗ = ul +(h∗ − hl)φ(h∗, hl) and h∗ ≤ hr, ur = u∗ − ψ(h∗) + ψ(hr).

First we show that for h∗ < h < hl, B2(Ul) lies above D2(U∗), i.e.,

ψ(h∗)−ψ(h)+(h−hl)φ(h, hl)+(hl−h∗)φ(h∗, hl) > 0, ∀ h ∈ (h∗, hl]. (3.5.47)

To prove this, we define a new function f8(h) = ψ(h∗)−ψ(h)+(h−hl)φ(h, hl)+(hl − h∗)φ(h∗, hl) so that f8(h∗) = 0. Since f

′

8(h) > 0, we have f8(h) >f8(h∗); it follows that f8(h) > 0, implying thereby that B2(Ul) lies aboveD2(U∗).

Next, we show that the curve D2(Ul) lies above the curve D2(U∗) forh∗ < hl ≤ h; for this it is enough to prove ψ(h∗)−ψ(hl)+(hl−h∗)φ(h∗, hl) > 0for h∗ < hl ≤ h. We notice that the left-hand side of this inequality is f8(hl),which has already been shown to be positive, and hence the curve B2(Ul) liesabove the curve D2(U∗) for h∗ < hl ≤ h.

Lastly, we show that D2(U∗) and B1(Ul) intersect uniquely, say, at (h3, u3)for h∗ < hl < h3. Now, we define f9(h) = ψ(h)−ψ(h∗)+(h−hl)φ(h, hl)+(h∗−hl)φ(h∗, hl) for h∗ < hl ≤ h so that f9(hl) < 0, and we can choose a constantK > 0 such that f9(h) > 0 for all h > K. Then there exists an h3 such thatf9(h3) = 0. Thus, B2(U∗) and B1(Ul) intersect uniquely at (h3, u3), as D2(U∗)and B1(Ul) are monotone; the computed results are shown in Figure 3.5.14.Here, again, the following cases arise:

a) When hr < h3, Ur ∈ II and the outcome of the interaction is B2D2 →B1B2; this means that the strength of D2 is small compared to theelementary waveB2, and B2 annihilatesD2 in a finite time. The strengthof reflected B1 wave is small compared to the incident waves B2 and D2.

b) When hr = h3, Ur ∈ B1(Ul) and the outcome of the interaction isB2D2 → B1, showing thereby that the reflected shock B1 is weak com-pared to the incident waves D2 and B2.

c) When hr > h3, Ur ∈ I and the outcome of the interaction is B2D2 →B1D2, implying thereby that D2 is stronger than B2.


0.5 1 1.5 2h

-10

-5

5

10

u

UU*

B2HU*L

B2HUL

D1HULD2HUL

B1HUL0.5 1 1.5 2

h

-4

-2

2

4

6

u

UU*

B1HU*LB2HUL

D1HULD2HUL

B1HUL

Figure 3.5.11: B2 overtakes B2 Figure 3.5.12: B1 overtakes D1

Here U∗ ∈ D2(Ul) and Ur ∈ B2(U∗). Thus, for a given Ul, we choose U∗and Ur such that hl ≤ h∗, u∗ = ul − ψ(hl) + ψ(h∗) and hr < h∗, ur =u∗ + (hr − h∗)φ(hr , h∗).

Now, we show that D2(Ul) lies above B2(U∗) for hl ≤ h < h∗, i.e.,

ψ(h) − ψ(h∗) − (h− h∗)φ(h, h∗) > 0, ∀ hl ≤ h < h∗. (3.5.48)

To prove this, we define a new function f10(h) = ψ(h)−ψ(h∗)−(h−h∗)φ(h, h∗)for hl ≤ h ≤ h∗ so that f10(h∗) = 0; this, in view of the expressions for c(h)

and φ(h, h∗), yields df10(h)dh = −[(h−h∗)2((h∗+2H)2+4h(h+h∗+2H))]

4h2h∗φ(h,h∗) < 0, implying

thereby that f10(h) > f10(h∗) = 0. Hence, the result.

0.5 1 1.5 2h

-5

-2.5

2.5

5

7.5

10

u

U U*

D1HU*L

B2HUL

D1HULD2HULB1HUL

0.5 1 1.5 2h

-4

-2

2

4

6

8

u

UU*

D2HU*L

B2HUL

D1HULD2HUL

B1HUL

Figure 3.5.13: D1 overtakes B1 Figure 3.5.14: D2 overtakes B2


0.5 1 1.5 2h

-30

-20

-10

10

u

UU*

B2HU*LB2HULD1HUL D2HUL

B1HUL

Figure 3.5.15: B2 overtakes D2

vi) A 2-dilatation wave overtakes a 2-bore (D2B2):Next, we show that B2(Ul) lies above the curve B2(U∗) for h < hl < h∗; for

this it is sufficient to prove ψ(hl)−ψ(h∗)−(h−h∗)φ(h, h∗)+(h−hl)φ(h, hl) > 0for h < hl < h∗. In order to prove this inequality, we define a new functionf11(h) = ψ(hl) − ψ(h∗) − (h− h∗)φ(h, h∗) + (h − hl)φ(h, hl) for h ≤ hl < h∗so that f11(hl) = f10(hl) > 0. Since φ(h, hl) + (h−hl)φ(h, hl) is an increasingfunction with respect to the second argument for hl > h, it follows thatdf11(h)dh < 0, implying thereby that f11(h) > f11(hl) > 0.Lastly, we show that B1(Ul) and B2(U∗) intersect uniquely at a point, say,

(h4, u4), where hl < h4 < h∗. The proof for this follows on similar lines asdiscussed earlier; the computed results are shown in Figure 3.5.15. Here, also,we encounter the following possibilities:

a) When hr > h4, Ur ∈ I and the outcome of the interaction is D2B2 →B1D2; this means that D2 is strong compared to the elementary waveB2, and the strength of reflected B1 is small compared to the incidentwaves B2 and D2.

b) When hr = h4, Ur ∈ B1(Ul) and the outcome of the interaction isD2B2 → B1.

c) When hr < h4, Ur ∈ II and the outcome of the interaction is D2B2 →B1B2, implying thereby that the elementary wave B2 is strong comparedto D2.

Chapter 4

Evolution of Weak Waves in

Hyperbolic Systems

We have already noticed in Chapter 2 that the solutions of conservation law(1.1.1) with m = 1, which are piecewise smooth, admit discontinuities acrosscertain smooth curves in the (x, t) plane. In this chapter, we study systems(1.1.1) in 1- and 3-space dimensions. We consider only piecewise smooth so-lutions so that there exists a smooth orientable surface

∑: φ(x, t) = 0, with

outward space-time normal (φ,t, φ,1, φ,2, φ,3), across which u suffers a jumpdiscontinuity satisfying the R-H condition

G[u] = [f,j ]nj ,

and outside of which u is a C1 function; here G = −φ,t/(φ,jφ,j)1/2 is thepropagation speed of

∑in the direction nj = φ,j/(φ,iφ,i)

1/2, 1 ≤ i, j ≤ 3.Explicit formulae for the discontinuities in the first and higher order deriva-tives of u across

∑, which are of the nature of compatibility conditions for

the existence of discontinuities, have been derived by Thomas [196] as well asTruesdell and Toupin [201]. A surface

∑across which the field variable u or

its derivative is discontinuous is called the singular surface or a wave-front; itis only such a wave that is being studied here. Indeed, the use of compatibilityconditions that hold on

∑enable us to obtain some interesting results in the

general theory of surfaces of discontinuity in continuum mechanics.

4.1 Waves and Compatibility Conditions

A wave may be conceived as a moving surface across which some of thefield variables or their derivatives, describing the material medium, undergocertain kinds of discontinuities that are carried along by the surface. Thediscontinuities across the surface are found to be interrelated. The relationsconnecting the field variables, or their derivatives on the two sides of the dis-continuity surface are usually referred to as compatibility conditions, and theyarise as a direct consequence of the dynamical conditions which determine thebehavior of the material medium. The first set of compatibility conditions,known as the R-H jump conditions, is essentially a consequence of the con-servation laws that hold across the discontinuity surface. The relations which

75

76 4. Evolution of Weak Waves in Hyperbolic Systems

connect the first order derivatives of the field variables on the two sides of thediscontinuity surface with its speed of propagation are known as compatibilityconditions of the first order. In a similar manner, the compatibility conditionsof second and higher order, which relate respectively to the second and higherorder derivatives, follow from the assumption of a smooth wave front.

Let φ(x, t) = 0 or xi = xi(y1, y2, t) represent the moving surface

∑; here

yα(α = 1, 2) are the curvilinear coordinates of the surface∑

(t) which ismoving with speed G. Our convention is that G > 0, which corresponds toa singular surface propagating in the direction of n. Let z be a field variablewhich suffers a jump discontinuity across

∑(t), defined as z = z2 − z1, where

z1 and z2 are the values of z immediately ahead of and behind the wave front∑(t), respectively. The compatibility conditions can now be stated as (see

Thomas [196])

[z,i] = d(1)ni + gαβ[z],αxi,β ; [z,t] = −d(1)G+δ

δt[z],

[z,ij ] = d(2)ninj + gαβ(d(1),α + gστ bασ[z],τ )(nixj , β + njxi,β)

+gαβgστ ([z],ασ − d(1)bασ)xi,βxj,τ , (4.1.1)[∂2z

∂xi∂t

]=

(−d(2)G+

δ

δtd(1) − gαβ [z],αxk,β

δnkδt

)ni + gαβ

[∂z

∂t

]

,α

xi,β ,

[∂2z

∂t2

]= d(2)G2 −G

δ

δtd(1) +Ggαβ [z],αxi,β

δniδt

+δ

δt

[∂z

∂t

],

where d(1) = [z,i]ni, d(2) = [z,ij ]ninj , and δ/δt = ∂/∂t + Gni∂/∂xi denotes

the time derivative as apparent to an observer moving with the wave frontalong the normal n; the quantities gαβ = xi,αxi,β and bαβ are respectively thecovariant components of the first and second fundamental surface tensors ofthe wave front

∑(t).

Here, summation convention on repeated indices applies; the Latin indices(i, j, k) have the range 1,2,3 and the Greek indices (α, β, σ, τ) the range 1, 2,and a comma followed by a Greek index, say α, denotes partial derivativewith respect to yα. Quantities φi,α are the components of a covariant vector,and xi,α is the tangent vector to the surface

∑(t) having the direction of the

corresponding yα curve. We also recall the following relations which we shallbe using in our subsequent discussion

nixi,α = 0; xi,αβ = bαβni; ni,β = −gασbαβxi,σ ,δni/δt = −gαβG,αxi,β ; gαβbαβ = bββ = 2Ω,

(4.1.2)

where Ω and gαβ are respectively the mean curvature and the contravariantcomponents of the metric tensor of the surface

∑; the Gaussian curvature of∑

is given by K = det(bαα).Equations (4.1.1) are called the geometrical and kinematical conditions

of compatibility of the first and second order; these conditions are iteratedto yield higher order compatibility conditions for the derivatives of z (see

4.1 Waves and Compatibility Conditions 77

Nariboli [130], Grinfeld [65], Estrada and Kanwal [53], and Shugaev andShtemenko [184]).

The singularity∑

(t) is called a wave of order N if all the derivativesof z of order less than N are continuous across

∑(t), while at least an N th

order derivative of z suffers a jump discontinuity across∑

(t). When N = 0,the singular surface is called a shock wave, while for N = 1, it is calleda weak discontinuity wave or a first order discontinuity, or an accelerationwave; a wave of order N ≥ 2 is called a mild discontinuity (see Varley andCumberbatch [205], Coleman et al. [39], and Ting [198]).

The analysis of acceleration waves and mild discontinuities for a nonlinearsystem proceeds on similar lines.

The above compatibility conditions, when applied to the system of partialdifferential equations, which describe the variation of field variables in spaceand time, yield in succession the normal speed of propagation and then thetransport equation that governs the variation of wave amplitude. When thespeed of propagation G is independent of x and n, the propagation is calledisotropic and homogeneous, and the transport equation for the variation ofwave amplitude can be easily integrated along the normal trajectories of

∑(t).

This is the simplest case, and it corresponds to the physical situation whenthe wave propagates into a medium which is in a uniform state at rest, orwhen the medium ahead is nonuniform but the problem is mathematicallyone-dimensional, such as planar, cylindrically or spherically symmetric waves(see, e.g., Chen [32] and the references therein, Singh and Sharma [186], Pai etal. [139], Schmitt [160], and Sharma and Radha [169]). The wave propagationis called anisotropic only if the speed G depends on the normal to the surface∑

(t). If the wave is propagating into a nonuniform medium then the speedG depends on the spatial coordinates also, and the propagation phenomenonis designated as anisotropic and nonhomogeneous. In this case, the transportequation for the variation of the wave amplitude along the normal trajectoriesinvolves the surface derivative terms, which pose a real difficulty in obtain-ing its solution; but if we transform the transport equation to a differentialequation along bicharacteristic curves of the governing system of partial dif-ferential equations, this difficulty disappears. In order to proceed further, weneed to observe certain relations involving the bicharacteristic curves or rayson∑

(t).

4.1.1 Bicharacteristic curves or rays

As for the wave front∑

: φ(x, t) = 0, the δ-time derivative vanishes, itsnormal speed of propagation G, which in general is a function of x and n,satisfies the first order partial differential equation:

H ≡ G(φ,iφ,i)1/2 + φ,t = 0. (4.1.3)

The characteristic curves of (4.1.3), known as bicharacteristic rays, are thecurves in space-time whose parametric representation is obtained by solving


the ODEs:

dxidt

=∂H

∂φ,i/∂H

∂φ,t,

d

dt(φ,i) = −∂H

∂xi/∂H

∂φ,t,

which can be rewritten as

dxidt

= Gni + (δij − ninj)∂G

∂nj,

dnidt

= (ninj − δij)∂G

∂xj, (4.1.4)

where ni = φ,i/(φ,jφ,j)1/2. In equation (4.1.4), the vector dxi/dt ≡ Vi is

identified as ray velocity vector; equation (4.1.4)2 describes the time variationof the normal ni as we move along the rays.

Thus the time rate of change of the field variable z, as apparent to anobserver moving with the wave front in the ray direction with velocity Vi, isgiven by dz/dt = ∂z/∂t+Viz,i; equation (4.1.4)1 shows that the component ofVi along normal to the surface

∑is just G, while its component tangential to∑

is given by Vα = Vixi,α = xi,α∂G∂ni

and, thus, the ray derivative is relatedto the δ-time derivative as

dz

dt=δz

δt+ V αz,α, (4.1.5)

where V α = gαβVβ . Equation (4.1.5), which will be used in our subsequentanalysis, shows that the rays coincide with the normal trajectories only if Gis independent of ni.

4.1.2 Transport equations for first order discontinuities

We consider the quasilinear system (1.1.2) for m unknowns ui =ui(x1, x2, x3, t), 1 ≤ i ≤ m,

u,t + AJu,J = g, J = 1, 2, 3, (4.1.6)

where AJ and g are known functions of u, and (x1, x2, x3) are the Cartesiancoordinates of a point in Euclidean space.

For the system (4.1.6), it is possible to consider a particular class of solu-tions that characterizes the first order discontinuity waves or in the languageof continuum mechanics, the acceleration waves. We consider the case that(4.1.6) admits constant solution u = uo, and we require that g(uo) = 0. Let∑

(t) : φ(x, t) = 0 be the moving surface (wave front) that separates the spaceinto two subspaces; ahead of the wave front, we have the known uniform fielduo and behind the wave front, unknown perturbed field u(x, t). Across φ = 0,the field u is continuous (i.e., [u] = 0) but its derivatives are allowed to havesimple jump discontinuities; the compatibility conditions can now be stated


as follows

[u,I ] = π(1)nI ; [u,t] = −π(1)G,

[u,IJ ] = π(2)nInJ + gαβ(nIxJ,β + nJxI,β)π(1),α − π(1)bασg

αβgστxI,βxJ,τ ,

[∂2u

∂xI∂t

]= −

(π(2)G+ δtπ

(1))nI − (π(1)G),αg

αβxI,β,

[∂2u

∂t2

]= π(2)G2 − 2Gδtπ

(1) − π(1)δtG, (4.1.7)

[u,IJK ] = π(3)nInJnK + gαβ(π(2),α + 2gγδbδαπ(1)

,γ )P IJKβ

+gαβgγδ(π(1),γα − bγαπ(2) − gστ bασbγτπ

(1))QIJKβδ

−gαβgγδgστ bασπ(1),γ M

IJKβτδ + π(1)

,γ gαβgγδ,α (nIxJ,β + nJxI,β)xK,δ

−π(1)gαβgστ (gγδ,σ bγα + gγδbγα,σ)xI,βxJ,τxK,δ ,

[∂3u

∂xI∂xJ∂t

]= gαβ(nIxJ,β + nJxI,β)

(−π(2)G+ δtπ

(1)),α

+gαβgστ (nIxJ,β + nJxI,β)bασ(−π(1)G),τ

+(−π(3)G+ δtπ

(2) + 2gαβπ(1),α G,β

)nInJ

+(

π(2)G− δtπ(1))bασ − (π(1)G),ασ

gαβgστxI,βxJ,τ ,

where I, J,K take values 1,2,3 and δt ≡ δ/δt; (n1, n2, n3) denotes unit nor-mal to the wave front

∑, π(1) = [u,I ]nI , π(2) = [u,IJ ]nInJ , π(3) =

[u,IJK ]nInJnK , and

P IJKβ = nInJxK,β + nJnKxI,β + nKnIxJ,β ,

QIJKβδ = nIxJ,βxK,δ + nJxK,δxI,β + nKxI,βxJ,δ ,

M IJKβτδ = xI,βxJ,τxK,δ + xJ,δxK,τxI,β + xK,βxI,δxJ,τ .

Taking a jump in equation (4.1.6) across the singularity surface∑

, and usingthe compatibility conditions (4.1.7)1 and (4.1.7)2 together with the continuityof u, we obtain

(AJnJ −GI)π(1) = 0, (4.1.8)

where the superscript- denotes evaluation just ahead of the wave-front atu = uo, and I is the m×m unit matrix. If π(1) 6= 0, its coefficient matrix in(4.1.8) must be singular, i.e.,

det(AJnJ −GI) = 0, (4.1.9)


which implies that the singularity surface φ = 0 must be one of the character-istic surfaces for the system (4.1.6); in other words, discontinuities in the firstderivatives of u can occur only on characteristic surfaces. The characteristicequation (4.1.9) is also called the eikonal equation. Hyperbolicity of (4.1.6)ensures that there is at least one possible speed of propagation G; strict hy-perbolicity ensures that there are m possible speeds. If the characteristic rootG of Ao

InI has multiplicity r(≤ m), then there exist r linearly independentright eigenvectors Ro

q , 1 ≤ q ≤ r, of AoInI such that

π(1) = σ(1)q Ro

q, (4.1.10)

where σ(1)q are scalars, which remain undetermined at this stage. The evolu-

tionary behavior of first order discontinuity waves can be studied merely by

determining the transport equations for σ(1)1 , σ

(1)2 , . . . , σ

(1)r .

The characteristics of the eikonal equation (4.1.9) are called bicharacteris-tic curves or rays of the system (4.1.6); if the multiplicity of the root G doesnot depend on n, these curves are given by (see Varley and Cumberbatch[205])

(Lo` ·Ro

p

) dxIdt

= Lo`AoIR

op, `, p = 1, 2, . . . r

where Lo` are r linearly independent left eigenvectors of AoInI associated with

the wave speed G so that

Lo`(AoInI −GI) = 0. (4.1.11)

Next, we differentiate the equation (4.1.6) partially with respect to xK , takejumps across

∑, multiply the resulting equations by nK , and then make use

of equations (4.1.7)1, (4.1.7)3 and (4.1.7)4 together with the identity [U ·V] =[U] · Vo + Uo · [V] + [U] · [V] to obtain

−(AoJnJ −GI)π(2) = δtπ

(1) + gαβxJ,βAoJπ

(1),α

+π(1) · (∇AJ)oπ(1)nJ − π(1) · (∇g)o,(4.1.12)

where ∇ denotes the gradient operator with respect to the field variablesu1, u2, . . . , um.

Equation (4.1.12), on pre-multiplying by the left eigenvectors Lop and using(4.1.5), (4.1.10) and (4.1.11), yields

αpqdσ

(1)q

dt+ µ(1)

pq σ(1)q + ν

(1)pq`σ

(1)q σ

(1)` = 0, (4.1.13)

where

αpq = LopRoq, ν

(1)pq` = Lop(R

oq · (∇AJ)onJ )Ro

` ,

µ(1)pq = Lop

(δtR

oq + Ao

JRoq,αg

αβxJ,β −Roq · (∇g)o

),


and d/dt is the ray derivative defined in (4.1.5).Equations (4.1.13), which constitute a system of r coupled nonlinear or-

dinary differential equations for the variation of σ(1)q at an acceleration wave-

front along bicharacteristic curves, are analogous to the transport equationsderived by Varley and Cumberbatch [205], Ting [198], Boillat and Ruggeri[21], and Estrada and Kanwal [53]. For r = 1, equation (4.1.13) reduces to asingle Bernoulli equation of the form

dσ

dt+ µ(t)σ = β(t)σ2, t > 0 (4.1.14)

where σ is the wave amplitude, and the coefficients µ and β are, in general,functions of t which are known if uo is known in the region into which thefront is spreading.

4.1.3 Transport equations for higher order discontinuities

Transport equations for the discontinuities in the second order derivativesof the field variables at an acceleration front can be obtained as an extension ofthe preceding analysis. For this, we consider the enlarged system of equationscomprising (4.1.6) together with the equations that result when (4.1.6) isdifferentiated twice with respect to the space coordinates x, i.e.,

∂

∂tu,JK + u,JK · (∇AI)u,I + 2u,J · (∇AI)u,IK + u,Ju,K(∇∇AI)u,I

+AIu,IJK = u,JK · (∇g) + u,Ju,K(∇∇g), (4.1.15)

where

u,JK · (∇AI) =∂2ui

∂xJ∂xK

∂AI

∂ui, u,Ju,K · (∇∇AI ) =

∂ui∂xJ

∂uj∂xK

∂2AI

∂ui∂uj.

Next, we take a jump in (4.1.15) across the acceleration front∑

, multiplythe resulting equation by nJnk, and use the compatibility conditions (4.1.7)1,(4.1.7)3, (4.1.7)6 and (4.1.7)7 to obtain

−(AoJnJ −GI)π(3) = δtπ

(2) + gαβxI,βAoIπ

(2),α + π(2) · (∇AI)

oπ(1)nI

+2π(1) · (∇AI)oπ(2)nI − π(2) · (∇g)o

+2π(1) · (∇AI)ogαβπ(1)

,α xI,β − 2gαβπ(1),α G,β

+2gαβgγδbδαAoIπ

(1),γ xI,β (4.1.16)

+π(1)π(1) · (∇∇AI)oπ(1)nI − π(1)π(1) · (∇∇g)o,

where use has been made of (4.1.2)1 and (4.1.2)4. We notice from (4.1.12) and(4.1.16) that

π(2) = σ(2)q Ro

q + e(1), π(3) = σ(3)q Ro

q + e(2), (4.1.17)


where e(1) (respectively, e(2)) is a particular solution depending on π(1) andits derivatives (respectively, on π(1), π(2) and their derivatives) as well as onthe geometrical and kinematical characteristics of the discontinuity surface.

Equation (4.1.16), on pre-multiplying by the left eigenvector Lop, and using(4.1.4), (4.1.10), (4.1.11) and (4.1.17)1 yields the following transport equations

for σ(2)q .

αpqdσ

(2)q

dt+ µ(2)

pq σ(2)q + ν(2)

p = 0, (4.1.18)

where µ(2)pq and ν

(2)p , which depend on σ

(1)q and its derivatives as well as on

the geometric and kinematic characteristics of the discontinuity surface, aredefined as

µ(2)pq = Lop

δtR

oq + Ao

IRoq,αg

αβxI,β + Roq · ∇AI)

onIRo`σ

(1)`

+2Ro` · (∇AI)

onIRoqσ

(1)` −Ro

q · (∇g)o,

ν(2)p = Lop

2Ro

q · (∇AI)ogαβ(Ro`σ

(1)`,α +Ro`,ασ

(1)` )σ(1)

q xI,β − 2gαβ(Roqσ

(1)q,α

+Roq,ασ

(1)q )G,β + 2gαβgγδbδαAo

I(Roqσ

(1)q,α + Ro

q,ασ(1)q )xI,β

+RoqR

o` · (∇∇AI )

onIRomσ

(1)q σ

(1)` σ(1)

m −RoqR

o` · (∇∇g)oσ(1)

q σ(1)`

+δte(1) + gαβxI,βA

oIe

(1),α + e(1) · (∇AI)

oσ(1)q Ro

qnI

+2Roq · (∇AI)

oe(1)σ(1)q nI − e(1) · (∇g)o

.

It may be noticed that unlike (4.1.13), the system of equations (4.1.18) islinear along the rays.

Transport equations for the discontinuities in higher order derivatives atan acceleration front can be obtained following the same procedure; indeed,it can be shown that for n ≥ 2

π(n) = σ(n)q Ro

q + e(n−1),

αpqdσ

(n)q

dt+ µ(n)

pq σ(n)q + ν(n)

p = 0, (4.1.19)

where e(n−1), µ(n)pq and ν

(n)p depend on σ

(1)q , σ

(2)q . . . , σ

(n−1)q and their deriva-

tives as well as on the geometric and kinematic characteristics of the surface∑. It may be remarked that the transport equations for σ

(1)q are nonlinear,

whereas the transport equations for σ(n)q , n ≥ 2 are linear.

4.1.4 Transport equations for mild discontinuities

For second order waves (N = 2),u and its first order derivatives are con-tinuous, while discontinuities in the second and higher order derivatives are


permitted, and the compatibility conditions now become

[u,IJ ] = π(2)nInJ ;

[∂

∂tu,I

]= −π(2)GnI ,

[u,IJK ] = π(3)nInJnK + gαβπ(2),α P

IJKβ − gαβgγδbγαπ(2)QIJKβδ ;

[u,IJKL] = π(4)nInJnKnL + gαβ(π(3),α + 3gγδbδαπ(2)

,γ )nInJnKxL,β

+gαβ(π(3)nL + gγδπ(2)

,γ xL,δ),α + 2gγδbδα(π(2)nL),γ

P IJKβ

+gαβgγδ(π(2)nL),γα − bγα

(π(3)nL + gστπ(2)

,σ xL,τ

)

−gστbασbγτπ(2)nL

QIJKβδ − gαβgγδgστbασ(π

(2)nL),γMIJKβτδ

+(π(2)nL),γgαβgγδ,α (nIxJ,β + nJxI,β)xK,δ

−π(2)nLgαβgστ (gγδ,σ bγα + gγδbγα,σ)xI,βxJ,τxK,δ ,[

∂

∂tu,IJK

]=

(−Gπ(4) + δtπ

(3) + 3gαβπ(2),α G,β

)nInJnk

+gαβ(−π(3)G+ δtπ

(2)),α + 2gγδbδα(−π(2)G),γ

P IJKβ

+gαβgγδ(−π(2)G),γα − bγ,α

(−π(3)G+ δtπ

(2))

−gστbασ(−π(2)G)QIJKβδ + gαβgγδgστ bασ(π

(2)G),γMIJKβτδ

−(π(2)G),γgαβgγδ,α (nIxJ,β + nJxI,β)xK,δ

+π(2)Ggαβgστ(gγδ,σ bγα + gγδbγα,σ

)xI,βxJ,τxK,δ .

For such waves, equations, (4.1.17)1, and (4.1.18) apply with e(1) = 0,

ν(2)p = 0, showing that the transport equations for σ

(2)q are linear and ho-

mogeneous; however, the variations of σ(n)q with n > 2 are governed by ODEs

which are linear and nonhomogeneous. For instance, following the procedure

used in the preceding subsection, the transport equations for σ(3)q are

αpqdσ

(3)q

dt+ µ(3)

pq σ(3)q + ν(3)

p = 0,

where µ(3)pq = Lop(δtR

oq + gαβxI,βA

oIR

oq,α −Ro

q · (∇g)o), and

ν(3)p = Lop

δte

(2) + gαβAoIe

(2),α xI,β + 3gαβ(σ(2)

q Roq),αG,β

+3Roq · (∇AI)

oRol σ

(2)q σ

(2)` nI − e(2) · (∇g)o

+3gαβgγδbβγAoI(σ

(2)q Ro

q),αxI,δ

,

with e(2) depending on π(2) and its derivatives, as well as on the geomet-rical and kinematical characteristics of the discontinuity surface. It may be


remarked that for a wave of order N ≥ 2, the transport equations for σ(N)q

are linear and homogeneous, whereas the transport equations for σ(n)q , n > N

are linear and nonhomogeneous.

4.2 Evolutionary Behavior of Acceleration Waves

In the study of acceleration waves in material media, and elsewhere, aBernoulli or Riccati type equation of the following form (see equation (4.1.14)is frequently encountered,

ds

dt= −µ(t)s+ β(t)s2, t ≥ 0; (4.2.1)

where s(t) is the wave amplitude, µ(t) depends upon the condition (state) ofthe medium ahead of the wave and the geometry of the wave surface, and β(t)depends on the elastic response of the material (see, e.g., McCarthy [120], Chen[32], Ruggeri [158], and Lou and Ruggeri [113]). The present work is concernedwith the general behavior of the wave amplitude s(t), and especially with itslimiting behavior; that is with the conditions under which it ultimately dampsout (s(t) → 0), attains a stable wave from (s(t) → const.), or develops into ashock wave in a finite time t(|s(t)| → ∞ as t→ t).

When

β(t) = 0 for all t, (4.2.2)

which corresponds to a linear mechanical response of the material, equation(4.2.1) reduces to a linear equation. Such a situation arises in a number ofphysical problems. For example, shear waves in nonlinear viscoelastic fluids inequilibrium (see Coleman and Gurtin [38]), homothermal and frozen homen-tropic transverse waves in laminated bodies (see Bowen and Wang [24]) andwaves in unstrained isotropic materials (see Scott [162]). In all these refer-ences, it has been shown that the corresponding wave amplitude s(t) satisfiesthe differential equation (4.2.1) with β(t) = 0. On the other hand, in the mostgeneral case known so far, the function β(t) is restricted to be of constantsign, so that

either β(t) < 0 for all t, or , β(t) > 0 for all t, (4.2.3)

and the results of Bailey and Chen [9], Bowen and Chen [23], and Wright [211],all of which are subject to these conditions, have been reported as theorems3.2.1 to 3.2.6 in Chen [32]; see also Menon and Sharma ([122], [175]). Noticealso that here the function is not allowed to vanish anywhere.

However, in an inhomogeneous medium interspersed with layers showing alinear response corresponding to (4.2.2) it is more reasonable to assume that

4.2 Evolutionary Behavior of Acceleration Waves 85

the function β(t) does not change sign, so that

either β(t) ≤ 0 for all t, or , β(t) ≥ 0 for all t. (4.2.4)

The condition (4.2.4), which allows the function β(t) to vanish over the partsof the range, includes both conditions (4.2.2) and (4.2.3) as special cases.

Here, we discuss the behavior of s(t) under the condition (4.2.4); thepresent account is largely based on our paper [123]. Our results lead to someimprovements over the known results even in the special case (4.2.3). In thefollowing subsections, the function β is assumed to satisfy condition (4.2.4).Our theorems contain all the known results and, in some instances, when spe-cialized to the case of an existing theorem, they provide a sharper result; it isshown that even in the simpler case, where β satisfies (4.2.3), our results arestronger and more extensive. Finally, we offer some comments on an alterna-tive but equivalent method of analyzing the behavior of wave amplitudes andthe development of shock solutions in quasi-linear hyperbolic systems due toJeffrey [80].

4.2.1 Local behavior

In this subsection, which will be needed at the end of subsection 4.2.3,we restate Theorem 3.2.1 due to Chen [32]. There are no restrictions on thefunctions µ and β at present.

Define the function λ by

λ(t) = µ(t)/β(t), (4.2.5)

which is defined only for those t for which β(t) 6= 0. Equation (4.2.1) thenbecomes

dsdt = β(t)s(t)[s(t) − λ(t)], if β(t) 6= 0;

= −µ(t)s(t), if β(t) = 0.(4.2.6)

The following result is trivial.

Proposition 4.2.1 Consider equation (1.1)

(i) At any instant, if β(t)s(t) < 0, then

ds

dt> 0 if s(t) < λ(t);

ds

dt< 0 if s(t) > λ(t).

(ii) At any instant, if β(t)s(t) 6= 0, then

ds

dt= 0, if, and only if, s(t) = λ(t).

(iii) At any instant, if β(t)s(t) > 0, then

ds

dt< 0 if s(t) < λ(t);

ds

dt> 0 if s(t) > λ(t).


Thus we may say at the instants where β(t)s(t) < 0, the function s tendsto approach the function λ.

In Theorem 3.2.1 of [32], the above statements (i) and (ii) take the form“(i) if either β(t) > 0 and s(t) < λ(t) or β(t) < 0 and s(t) > λ(t), thends(t)/dt < 0,” which is not true because it is possible that s(t) = 0; “(ii) ifs(t) 6= 0, s(t) = λ(t) if, and only if ds(t)/dt = 0,” which is again not truebecause β(t) could be zero, in that case ds/dt = −µ(t)s(t).

4.2.2 Global behavior: The main results

The integral F (t) =∫ t0 µ(τ)dτ appears in the solution s(t) below in (4.2.8).

We shall assume that either F (t) exists for all t ∈ [0,∞), or F (t) becomesinfinite at some finite t∗ > 0. When F (t) exists for all t (i.e., µ(t) is integrableover every finite sub-interval of [0,∞)), then from the well known definitionof an integral over [0,∞), it is possible that F (t) may converge, diverge to±∞, or not tend to a limit at all as t→ ∞. Further, we shall assume that

either β(t) ≥ 0 for all t (i.e., sgnβ = +1),or β(t) ≤ 0 for all t (i.e., sgnβ = −1),

(4.2.7)

in the range [0,∞) or [0, t∗) according to the condition satisfied by F (t) above.The integral of β(t) exp(−F (t)) appearing in (4.2.8) then exists if the integralof β exists, because exp(−F (t)) ≥ 0. The condition (4.2.7) may be expressedin words by asserting that “the function β does not change sign.” Notice thatif the integral of β vanishes over any interval, the solution of equation (4.2.1)reduces to that of the equation ds/dt = −µs over the interval. In [9] and [32],the function β is not allowed to vanish.

The solution of (4.2.1) is easily shown to be ([32])

s(t) =exp

(−∫ t0µ(τ)dτ

)

(1/s(0)) −∫ t0 β(τ) exp

(−∫ τ0 µ(s)ds

)dτ, t ≥ 0. (4.2.8)

Since sgnβ is either +1 or −1 from (4.2.7), it follows immediately, that

s(t) = ( sgn s(0)) exp(−F (t))/|s(0)|−1 +G(t), (4.2.9)

if sgn s(0) = − sgnβ; and

s(t) = ( sgn s(0)) exp(−F (t))/|s(0)|−1 −G(t), (4.2.10)

if sgn s(0) = sgnβ, where

F (t) =

∫ t

0

µ(τ)dτ, G(t) =

∫ t

0

|β(τ)| exp(−F (τ))dτ. (4.2.11)

We now state the main results.


Theorem 4.2.1 Consider equation (4.2.1) with s(t) 6= 0. Suppose that µ andβ are integrable on every finite subinterval of [0,∞), the function β does notchange sign (see (4.2.7)) on [0,∞), and sgn s(0) = − sgnβ. Let F,G be as in(4.2.11). Then

(a) s(t) is continuous, nonzero and of constant sign on [0,∞);

(b) |s(t)| is bounded above if F is bounded below (i.e., lim inft→∞

F (t) > −∞);

(c) s(t) is bounded away from zero if F is bounded above(i.e., lim sup

t→∞F (t) <∞) and G(∞) <∞;

(d) the limiting behavior of s is as follows:

(i) lim inft→∞

s(t) = 0 if lim supt→∞

F (t) = ∞.

(ii) Suppose that F (t) tends to a finite or infinite limit F (∞) as t→∞,then

limt→∞

s(t) =

0, if F (∞) = +∞;∞, if F (∞) = −∞ and G(∞) <∞;limt→∞

|µ/β|, if F (∞) = −∞ and G(∞) = ∞ and |µ/β|tends to an infinite or finite limit as t→∞;

|1/s(0)| +G(∞)−1exp(−F (∞)), if |F (∞)| <∞.

(iii) Suppose that F (t) does not tend to a limit as t→∞, due to theoscillation of F (t) as t→∞. If G(∞) < ∞, then s(t) oscillatesat t→∞. Moreover |s(t)| is bounded away from zero (respectively,bounded above) if, and only if, F (t) is bounded above (respectively,bounded below). If G(∞) = ∞ and F is bounded below, thenlimt→∞ s(t) = 0.

Theorem 4.2.2 Consider equation (4.2.1), with s(0) 6= 0. Suppose that µand β are integrable on every finite sub-interval of [0,∞); the function β doesnot change sign on [0,∞) and sgn s(0) = sgnβ. Let F,G be as in (4.2.11).Define the quantity α by

α = (G(∞))−1 = limt→∞

(G(t))−1 . (4.2.12)

The quantity α exists and α ≥ 0. Moreover α = 0 if, and only if, G(∞) = ∞.

Case 1. Suppose that either |s(0)| > α, or, |s(0)| = α and∫∞tβ(τ)dτ = 0 for

some t <∞, then there exists a finite time t such that

∫ t

0

β(t) · exp(−F (t)dt = |1/s(0)|, 0 < t <∞, (4.2.13)


andlimt→t

|s(t)| = ∞.

Case 2. Suppose that |s(0)| < α. If F (t) tends to a finite or infinite limitF (∞) as t→∞, then

limt→∞

s(t) =

0, if F (∞) = ∞,∞, if F (∞) = −∞,(

1|s(0)| − 1

α

)−1

F (∞), if |F (∞)| <∞.

If F (t) oscillates as t→∞, then s(t) also oscillates as t→∞, and |s(t)| isbounded away from zero (respectively, bounded above) if, and only if, F isbounded above (respectively, bounded below).

Case 3. Suppose that |s(0)| = α and∫∞t β(τ) 6= 0 for all finite t. Then

(i) limt→∞ |s(t)| = ∞ if F is bounded above;

(ii) If F (t) tends to a finite or infinite limit F (∞) as t→∞, then

limt→∞

|s(t)| =

∞, if F (∞) <∞,limt→∞ |µ/β|, if F (∞) = ∞ and |µ/β| tendsto a finite or infinite limit as t→∞;

(iii) If F (t) oscillates and is bounded above as t→∞, then limt→∞ |s(t)| = ∞.

Moreover, over the interval [0, t) in case 1, and over [0,∞) in cases 2, 3,the function s(t) is continuous and nonzero.

Theorem 4.2.3 Consider equation (4.2.1) with s(0) 6= 0. Let there exist afinite t∗ > 0 such that µ and β are integrable on every sub-interval of [0, t∗);the function β does not change sign on [0, t∗) and F (t∗) = +∞ or −∞, whereF,G are as in (4.2.11). Then s(t) is continuous and nonzero over [0, t∗).

Define the quantity α by α = G(t∗)−1 = limt→t∗G(t)−1; the quantityα exists and α ≥ 0.

Case 1. Suppose that sgn s(0) = − sgnβ. Then

limt→t∗

s(t) =

0, if F (t∗) = ∞;∞, if F (t∗) = −∞ and α > 0;limt→t∗ |µ/β|, if F (t∗) = −∞ and α = 0, and |µ/β| tendsto a finite or infinite limit as t→t∗.

Case 2. Suppose that sgn s(0) = sgnβ.

(i) If |s(0)| < α, then limt→t∗ |s(t)| =

0, if F (t∗) = ∞;∞, if F (t∗) = −∞.


(ii) If either |s(0)| > α, or, |s(0)| = α and∫ t∗t β(τ)dτ = 0 for some t < t∗,

then there exists a time t as in case 1 of Theorem 4.2.3, with 0 < t < t∗

and limt→t |s(t)| = ∞.

(iii) If |s(0)| = α and∫ t∗t β(τ)dτ 6= 0 for all t < t∗, then t = t∗ and

limt→t∗ |s(t)| =

∞, if F (t∗) = −∞,limt→t∗ |µ/β|, if F (t∗) = +∞ and |µ/β| tends to afinite or infinite limit as t→t∗.

Moreover, over the interval [0, t∗) in cases 1, 2(i), 2(iii), and over [0, t) in case2 (ii), the function s(t) is continuous and nonzero.

4.2.3 Proofs of the main results

In the following, the interval [0, T ) will stand for [0,∞) for Theorems 4.2.1and 4.2.2, and for [0, t∗) for Theorem 4.2.3. Since µ and β are both integrableover all finite subintervals of [0, T ), the functions F and G of (4.2.11) arewell-defined and continuous over [0, T ). The following Lemma is obvious sincea bounded monotone function over [0, T ) must have a limit as t→T.

Lemma 4.2.1 If µ and β are integrable on all finite sub-intervals of [0, T ),and β does not change sign on [0, T ), then

(a) the function G(t) of (4.2.11) is continuous, nonnegative and monotoni-cally nondecreasing over [0, T ), and G(0) = 0. Moreover, G(t) tends toa finite or infinite limit G(t) as t→T.

Therefore,α = G(T )

−1(4.2.14)

exists and α ≥ 0. Also, α = 0 if, and only if, G(T ) = ∞.

(b) The function exp(−F (t)) is continuous and strictly positive over [0, T ),and its limit as t→T exists if, and only if, F tends to a (finite or infi-nite) limit as t→T. Moreover, exp(−F (t)) is bounded above (respectively,bounded away from zero) if, and only if, F is bounded below (respectively,bounded above) on [0, T ).

Notice that, if T = ∞, the condition “F is bounded below” is equivalentto the condition “ lim inf

t→∞F (t) > −∞,” because F is continuous on [0,∞).

Similar remarks hold for the other conditions on F (t).

Proof: The proofs of all our theorems are based on Lemma 4.2.1.First, consider Theorem 4.2.1 (i.e., T = ∞) and case 1 of Theorem 4.2.3

(i.e. T = t∗), so that the equation (4.2.9) holds. The denominator of theright-hand side of (4.2.9) is bounded away from zero; in fact it exceeds unity,


from Lemma 4.2.1. It is also bounded above if G(∞) < ∞. The statements(a), (b), (c), d(i), d(iii) of Theorem 4.2.1, the first two statements in case 1of Theorem 4.2.3, and the last statement of Theorem 4.2.3 are all easy conse-quences of Lemma 4.2.1. Consider now the statement d(ii) of Theorem 4.2.1and the last statement in case 1 of Theorem 4.2.3. The only statement requir-ing a proof is the third one, where both the denominator and the numeratoron the right-hand side of (4.2.9) tend to ∞ as t→∞. In this situation we canuse l’Hopital’s rule, so that

limt→T

s(t) = ( sgn s(0)) limt→T

(ddt (exp(−F (t)))

ddtG(t)

)

= ( sgn s(0)) limt→T

(−µ/|β|), (4.2.15)

provided the last limit exists. Hence, if |µ/β| tends to a finite or infinite limitas t→T, then

limt→T

|s(t)| = limt→T

|µ/β|,

by taking the absolute values of both sides of (4.2.15).

Consider now the situation when sgn s(0) = sgnβ, so that the equation(4.2.10) holds. A possible complication is introduced because the denominatoron the right-hand side of (4.2.10) may vanish at some time t. If t exists, thent > 0 because G(0) = 0. Since G is monotonic nondecreasing and continuous,there are three possibilities for the denominator (|1/s(0)| −G(t)) :

(a) G(T ) < |1/s(0)|; in this case the denominator is bounded away fromzero over [0, T ), and we have the situation similar to Theorem 4.2.1;

(b) G(T ) > |1/s(0)|; in this case there must exist a t < T, where G(t) =|1/s(0)|, so that (4.2.13) is satisfied;

(c) G(T ) = |1/s(0)|; in this case, either G(t) = |1/s(0)| for some t < Twhich is case (b) above or, G(t) < |1/s(0)| for all t ∈ [0, T ).

Since the numerator exp(−F (t)) is nonzero over [0, T ), G(t) = G(T ) im-

plies that∫ Ttβ(τ)dτ = 0, and the limits of s(t) as t→t in case (b), and

as t→T in cases (a) and (c), are easily obtained, except when both the nu-merator and denominator tend to zero. In this exceptional case, thereforeF (t)→ + ∞, G(t)→|1/s(0)| as t→T, and G(t) < |1/s(0)| for t < T, and wemay apply l’Hopital’s rule again to obtain from (4.2.9),

limt→T

s(t) = ( sgn s(0)) limt→T

(ddt (exp(−F (t)))

ddtG(t)

)

= ( sgn s(0)) limt→T

(−µ/− |β|), (4.2.16)


provided the last limit exists. Therefore, as before, if |µ/β| tends to a finite orinfinite limit as t→T, we have the required limit of |s(t)|, by taking absolutevalues on both sides of (4.2.16).

This completes the proof of Theorem 4.2.2, by taking t = ∞, and case 2of Theorem 4.2.3, by taking T = t∗.

We may remark that there is no confusion with signs in (4.2.15) and(4.2.16). If (µ/|β|)→0 as t→T, there is no problem because 0 can be regardedas having either sign, while if µ/|β| tends to a finite nonzero or infinite limit,then µ is also of constant sign for all sufficiently large t as t→T. Therefore, iflimt→T F (t) = −∞ as in (4.2.15) and µ|β| tends to a finite nonzero or infinitelimit, µ is negative as t→T, i.e., −µ = |µ|. Similarly, in (4.2.16), µ is positivefor t sufficiently close to T.

4.2.4 Some special cases

We first derive a few simple consequences of Theorem 4.2.1.

Corollary 4.2.1 Consider equation (4.2.1). Let µ and β be integrable on ev-ery finite sub-interval of [0,∞), and let the function β not change sign on[0,∞). Suppose that s1(t), s2(t) are two solutions of (4.2.1) with s1(0) 6= 0,s2(0) 6= 0; and sgn s1(0) = sgn s2(0) = − sgnβ. Let P denote any one of thefollowing properties of a continuous function s(t) defined on [0,∞) :(a) |s(t)| is bounded on [0,∞); (b) |s(t)| is bounded away from zero on [0,∞);(c) limt→∞ s(t) = 0; (d) limt→∞ |s(t)| = ∞; (e) lim inf

t→∞s(t) = 0;

(f) s(t) tends to a finite nonzero limit as t→∞; (g) s(t) does not tend to alimit as t→∞.

If s1 satisfies the property P, then s2 also satisfies the Property P. More-over,

(i) if limt→∞

s1(t) = 0, then limt→∞

s2(t) = 0, so that limt→∞

|s1(t) − s2(t)| = 0;

(ii) if lim inft→∞

s1(t) = 0, then lim inft→∞

s2(t) = 0, and lim inft→∞

|s1(t) − s2(t)| = 0.

Proof: From Theorem 4.2.1, the limiting behavior of s(t) is determined byµ, β and the sign of s(0), in all cases except when both F (∞) and G(∞) arefinite, i.e., in the last line of the case d(ii). Even in this exceptional case, s(t)tends to a finite nonzero limit as t→∞, so that s(t) is bounded and boundedaway from zero over [0,∞). It follows that the property P is satisfied by bothor neither of s1(t) and s2(t). This also proves statement (i). The situation (ii)arises when lim sup

t→∞F (t) = ∞, and

s1(t) − s2(t) =

((1

|s1(0)| +G(t)

)−1

−(

1

|s2(0)| +G(t)

)−1)

exp(−F (t)),


where G(t) ≥ 0, so that (ii) is also proved. This concludes the proof of thecorollary.

Consider now the situation in Theorem 4.2.1, where sgn s(0) = − sgnβ,and suppose that lim inf

t→∞|β(t)| 6= 0. From the proposition in Subsection 4.2.1,

the function s will tend to approach the function λ = µ/β as t→∞. Since s isof constant sign and never 0 or ∞ on [0,∞), we therefore know that s→0, ifsgn s(0) = − sgnλ(t) for all sufficiently large t, whereas s(t) approaches λ(t)if sgn s(0) = sgnλ(t) for all sufficiently large t. Notice that since sgn s(0) =− sgnβ, for any instant t, if sgn s(0) = − sgnλ(t) then µ(t) > 0, and ifsgn s(0) = sgnλ(t) then µ(t) < 0. We are now in a position to investigatemore closely the approach of the function s to the function λ. The followingpossibilities arise.

Case 1. Suppose that, for all sufficiently large t, sgnλ(t) = − sgn s(0). Thenthere is a t0 such that µ(t) ≥ 0 for all t ≥ t0. Thus F (t) is monotoni-cally nondecreasing when t ≥ t0. From Lemma 4.2.1, the integral G(t) in(4.2.11) is nondecreasing and either converges, or diverges to +∞, on [0,∞). IfG(∞) = ∞, it follows that limt→∞ s(t) = 0 because the numerator exp(−F (t))in (4.2.9) is bounded above. If G(∞) <∞, it follows that exp(−F (t))→0, be-cause lim inft→∞ |β| 6= 0, and the integral of β exp(−F ) converges over [0,∞),and exp(−F ) is monotone over (t0,∞). Thus, from (4.2.9), in this case alsolimt→∞ s(t) = 0.

Case 2. Suppose that limt→∞ λ(t) = 0. Since lim inft→∞

|β(t)| 6= 0, for any suffi-

ciently small ε > 0, there exists t0 such that

|λ(t)| < ε/2 and |β(t)| > ε/2 for all t ≥ t0.

We shall prove that |s(t)| ≤ ε for all sufficiently large t. First we show thatif |s(t′)| ≤ ε for some t′ ≥ t0, then |s(t)| ≤ ε for all t ≥ t′. Since sgn s(t) =− sgnβ for all t, case (i) of Proposition 4.2.1 in Subsection 4.2.1 is applicableand the function s tends to approach the function λ, while |λ(t)| < ε/2 forall t ≥ t′. Therefore, the only possibility remaining to be examined is that|s(t)| > ε for all t ≥ t0 : in this case we find from (4.2.6) that for t ≥ t0,

∣∣∣∣d

dt

∣∣∣∣ = |β(t)| |s(t)| |s(t) − λ(t)| ≥ (ε/2)(ε)(ε/2),

which is bounded away from zero, while the function s tends to approachthe function λ where |λ| < ε/2. Hence in a finite time t′ > t0 we will have|s(t′)| ≤ ε. We have therefore proved that |s(t)|→0.

Case 3. Suppose that λ(t) tends to a finite nonzero or infinite limit L as t→∞,where sgn s(0) = sgnL. Then for all sufficiently large t, say t ≥ t1, we havesgnλ(t) = sgnL, that is, µ(t) < 0. Therefore exp(−F (t)) is monotonicallyincreasing over [t1,∞). Since µ/β→L 6= 0 as t→∞, and lim inf

t→∞|β| 6= 0, it


follows from (4.2.11) that F (t)→ − ∞ and G(t)→∞ as t→∞. Therefore, asin (4.2.15),

limt→∞

s(t) = sgn s(0) limt→∞

[− µ

|β|

]= lim

t→∞

(µ

β

),

as a consequence of applying l’Hopital’s rule.

Case 4. Suppose that s(0) > 0 and λ is bounded above, say by λ(t) ≤ Kfor all t. Then the argument used in case 2 shows that s(t) ≤ K + ε for allsufficiently large t, i.e., s(t) is bounded above. A similar result holds whens(0) < 0.

The following result has thus been established.

Corollary 4.2.2 Consider equation (4.2.1), with s(0) 6= 0. Suppose thatµ and β are integrable on [0,∞), the function β does not change sign on[0,∞), sgn s(0) = − sgnβ, and lim inf

t→∞|β(t)| 6= 0.

(a) If, for all sufficiently large t, sgnλ(t) = − sgn s(0), then limt→∞

s(t) = 0.

(b) If λ tends to a finite or infinite limit L as t→∞ and sgn s(0) = sgnL,then lim

t→∞s(t) = L.

(c) (i) If s(0) > 0 and λ is bounded above, then s is bounded above.

(ii) If s(0) < 0 and λ is bounded below, then s is bounded below.

Remarks 4.2.1: We explain briefly the relationship of our theorems to The-orems 3.2.1 through 3.2.6 of Chen [32]. The remarks (ii) – (v) refer to thespecial case where the function β satisfies the assumption (4.2.3).

(i) The proposition 4.2.1 of Subsection 4.2.1 is the corrected version ofTheorem 3.2.1 of [32].

(ii) The Corollary 4.2.2 in Subsection 4.2.4 above is the modified version ofTheorem 3.2.2 of [32]; the case (a) is ignored in [32]. The proof of case(b) is from [32].

(iii) The Corollary 4.2.1 in Subsection 4.2.4 corresponds to Theorem 3.2.3 of[32]; however, we have included more possibilities than are mentioned in[32].

(iv) Theorem 4.2.2 in Subsection 4.2.2 corresponds to Theorem 3.2.4 of [32];however, the case 3 is ignored in [32].

(v) Our Theorem 4.2.3 of Subsection 4.2.2 is a modified version of Theo-rems 3.2.5 and 3.2.6 of [32] where the possibility F (t∗) = +∞ and thecase 2(iii) were ignored. There is a slip in the last step of the proof of


Theorem 3.2.6 of [32]; it is asserted there that if F (t)→ − ∞ as t→t∗,i.e. µ(t)→−∞ as t→t∗, then |µ(t)/β(t)|→∞ as t→t∗. This is obviouslynot true, because |β(t)| could be unbounded above (e.g., β = µ2) so that|µ/β| could tend to a finite limit.

4.3 Interaction of Shock Waves with Weak Discontinu-

ities

The problem of the interaction of an acoustic wave with a shock has beenstudied by Swan and Fowles [193] as well as Van Moorhem and George [203]among several others. A detailed study directed towards gaining a better un-derstanding of the wave interaction problem within the context of hyperbolicsystems has been carried out by Jeffrey [80]. Brun [26] also developed a methodby which to study the interaction problem within a general framework, theapplication of which to elasticity and magnetofluiddynamics has been car-ried out by Morro ([126], [127]). A further contribution to the study of waveinteractions, which enables the evaluation of the reflected and transmittedamplitudes when a weak discontinuity wave encounters a shock wave, may befound in the paper by Boillat and Ruggeri [22]. The application of this work tointeraction with a contact shock and a spherical blast wave has been carriedout by Ruggeri [157] and Virgopia and Ferraioli [208], respectively.

It is a known fact that a shock undergoes an acceleration jump as a conse-quence of an interaction with a weak wave [98]; this fact has been accountedfor in the work of Brun [26] and Boillat and Ruggeri [22]. The present discus-sion, which is largely based on [148], takes into account this physical fact, andshows that the general theory of wave interaction which originated from thework of Jeffrey leads to the results obtained by Brun and Boillat.

4.3.1 Evolution law for the amplitudes of C1 discontinuities

The Bernoulli equation examined in the previous sections arises when theevolution law for weak discontinuities is derived directly from the governingsystem of equations, as in the references already cited or, for example, as in[81]. An alternative approach developed by Jeffrey [80] starts from a generalfirst order quasilinear hyperbolic system

u,t + A(u, x, t)u,x + b(u, x, t) = 0, (4.3.1)

where the vector u has n components and A is an n× n square matrix withreal eigenvalues and a full linearly independent set of eigenvectors.

The analysis proceeds by introducing a discontinuity in a derivative of

4.3 Interaction of Shock Waves with Weak Discontinuities 95

u, and then determining the evolution law governing its development as itpropagates along a wavefront. Of central importance to this approach is achange of variable which is introduced from x and t to semi-characteristiccoordinates t′ and φ, thereby leading to a linear system of equations governingthe evolution of the discontinuity in the derivative of u across the wavefront.

The method leads both to the determination of the time and place ofshock formation when this occurs on the wavefront, and also to conditionswhich ensure that no shock will ever evolve (the exceptional condition) on thewavefront. Also arising from this approach is the identification of a bifurcation-type phenomenon in the event that eigenvalues change their multiplicity.

However, the system of linear equations arising from this method appearsformally to be quite different from the Bernoulli equation which is found whenthe equation governing the amplitude of the weak discontinuity is derived di-rectly. This problem has been considered by Boillat and Ruggeri [21] whoshowed them to be equivalent. In the process they also established how thenonlinear Bernoulli evolution equation and the linear system of equations arerelated by a discontinuous mapping involving the semi-characteristic coordi-nates.

Since system (4.3.1) is assumed to be hyperbolic, in general the matrixA possesses p distinct real eigenvalues λ(i), i = 1, 2, . . . , p (assumed to beordered so that λ(p) < λ(p−1) < · · · < λ(1)) with multiplicities mi such thatp∑

i=1

mi = n, together with n linearly independent left (respectively; right)

eigenvectors L(i,k) (respectively; R(i,k)), k = 1, 2, . . . ,mi corresponding to theeigenvalues λ(i).

Let φi(x, t) = 0 be the equation of the ith characteristic curve which isdetermined as the solution of dx/dt = λ(i) passing through (x0, t0). Afterintroducing the characteristic coordinates (φi, t), if Πi and Xi are the jumpsin the derivatives of u and x with respect to φi across the ith characteristiccurve, the transport equations for a C1 discontinuity vector Πi across the ith

characteristic curve are given by the following system of ordinary differentialequations (here we recall the main results of Jeffrey [80] which will be necessaryin what follows)

L(j,k) (Πi − u0,xXi) = 0, for j = 1, 2, . . . , i− 1, i+ 1, . . . , pand k = 1, 2, . . . ,mj

(4.3.2)

dXi

dt=(∇λ(i)

)0Πi + λ

(i)0,xXi , (4.3.3)

(∇L(i,k)

)0Πi

tr du0

dt+ L

(i,k)0,x

du0

dtXi + L

(i,k)0

dΠi

dt

+(∇(L(i,k)b

))0Πi +

(L(i,k)b

)0,x

Xi = 0, for k = 1, 2, . . . ,mi , (4.3.4)

where the index i is fixed, the subscript 0 refers to the solution ahead of theith characteristic curve, ∇ denotes the gradient operator with respect to the


n elements of u, and d/dt denotes material time derivative following the ith

characteristic curve. The set of n + 1 equations (4.3.2) − (4.3.4) determinethe vector Πi and the scalar Xi, using the initial conditions Πi(t0) = Πi0 andXi(t0) = 0. It should be noted that

Πi = Λi

(Xi + (x,φi

)0)

+ u0,xXi, (4.3.5)

where Λi is the jump in u,x across the ith characteristic curve. In view of

(4.3.2) and (4.3.5), it follows that there exist functions α(i)k such that

Λi =

mi∑

k=1

α(i)k (t)R

(i,k)0 . (4.3.6)

The transport equations for C1 discontinuities across the ith characteristic,in terms of the original independent variables x and t, are obtained by using(4.3.3) and (4.3.5) in (4.3.4) in the following form

L(i,k)0

dΛi

dt+ Li,k0 (u0,x + Λi)

(∇λ(i)

)0Λi +

(∇L(i,k)

)0Λi

tr du0

dt+(L

(i,k)0 Λi

)((∇λ(i)

)0u0,x + λ

(i)0,x

)+(∇(L(i,k)b

))0Λi = 0.

(4.3.7)

When the expression for Λi given in (4.3.6) is substituted into (4.3.7), follow-ing the summation convention, we arrive at the Bernoulli type equations

a(i,k)j

dα(i)j

dt+ b

(i,k)jp α

(i)j α(i)

p + C(i,k)j α

(i)j = 0, i, k − unsummed (4.3.8)

where

a(i,k)j = L

(i,k)0 R

(i,j)0 ,

b(i,k)jp = L

(i,k)0 R

(i,j)0

(∇λ(i)

)0R

(i,p)0 ,

C(i,k)j = L

(i,k)0

dR(i,j)0

dt+(∇R(i,j)

)0

du0

dt+ L

(i,k)0 u0,x

(∇λ(i)

)0R

(i,j)0

+((

∇L(i,k))

0R

(i,j)0

)tr du0

dt+ L

(i,k)0 r

(i,j)0

((∇λ(i)

)0u0,x + λ

(i)0,x

)

+(∇(L(i,k)b

))0R

(i,j)0 , k, p = 1, 2, . . . ,mi.

which enable us to determine the amplitudes α(i)k .

4.3.2 Reflected and transmitted amplitudes

In order to study the amplitudes of the reflected and transmitted C1 dis-continuities, when an incident C1 wave comes in contact with a strong dis-continuity, we shall need to consider generalized conservative systems which


are a direct consequence of the original system (4.3.1), and have the forms

G,t (x, t,u) + F,x (x, t,u) = H(x, t,u), (4.3.9)

G∗,t(x, t,u

∗) + F∗,x(x, t,u

∗) = H∗(x, t,u∗), (4.3.10)

respectively, to the left and to the right of the discontinuity curve, dx/dt = s,which propagates with speed s. In (4.3.9) and (4.3.10) u and u∗ are the solu-tion vectors to the left and to the right of the discontinuity line.

Let P (xp, tp) be the point at which fastest C1 discontinuity Π1 of (4.3.9),moving along the characteristic φ1(x, t) = 0 and originating from the point(x0, t0), intersects the discontinuity line.

At P , the C1 discontinuities Π(R)p−q+i, i = 1, 2, . . . , q (respectively;

Π∗(T )i , i = 1, 2, . . . , q∗) are reflected (respectively; transmitted) along q (re-

spectively; q∗) characteristics belonging to the system (4.3.9) (respectively;(4.3.10)) which as time increases, enter into the region to the left (respectively;right) of the discontinuity line. Thus the speed s of the discontinuity line sep-arates q∗ transmitted wave speeds ahead from q reflected wave speeds behind.As a direct consequence of this we have λ(p) < λ(p−1) < . . . < λ(p−q+1) <s < λ∗(q) < . . . < λ∗(1). Let u(R) (respectively; u∗(T )) be the solution in thereflected (respectively; transmitted) wave region, and u0 (respectively; u∗

0) bethe known solution vector ahead of the leading incident wave (respectively;leading transmitted wave) φ1(x, t) = 0 (respectively; φ∗

1(x, t) = 0). Once the

initial values Π(R)i (P ) and Π

∗(T )i (P ) of the reflected and transmitted vectors

are known, the discontinuity vectors Π(R)i and Π

∗(T )i for t > tp can be de-

termined from the system (4.3.2) – (4.3.4) after using the initial conditionsX(R)(P ) = 0 and X∗(T )(P ) = 0.

It should be noted that at the point P

Π1(P ) +

p∑

i=p−q+1

Π(R)i (P ) = u,

(R)φp−q+1

(P ) − u0,φ1(P ),

q∗∑

i=1

Π∗(T )i (P ) = u,

∗(T )φ∗

q∗(P ) − u∗

0,φ∗

1(P ).

(4.3.11)

If we introduce the following parameterizations to φ1, φp−q+i and φ∗i ,

φ1(x, t0) = x− x0, φp−q+i(x, tp) = x− xp, φ∗i (x, tp) = x− xp, (4.3.12)

equations (4.3.11) can be written as

Π1(P ) +

p∑

i=p−q+1

Π(R)i (P ) = u,(R)

x (P ) − [x,φ1 ]0 u0,x(P ),

q∗∑

i=1

Π∗(T )i (P ) = u,∗(T )

x (P ) − u∗0,x(P ).

(4.3.13)


Across the discontinuity line the following generalized jump conditions (gen-eralized Rankine Hugoniot conditions) hold

s (G−G∗) − (F− F∗) = 0.

On applying the displacement derivative•( )= ( ),t +s( ),x to these jump con-

ditions, we obtain

s

(•G −

•G

∗)+

•s (G−G∗) −

(•F −

•F

∗)= 0. (4.3.14)

Owing to the interaction of the incident wave with a shock like discontinuityat P (xp, tp), the shock undergoes an acceleration. The relations (4.3.14) holdat t = tp+ (after the interaction ) and at t = tp− (before the interaction), andwe obtain

•s (G0 −G∗

0) + s( •G0 + (∇0G0)

•u0

)− s

( •Ga

∗

0 + (∇∗0G

∗0)

•u∗0

)

−( •F0 + (∇0F0)

•u0

)+

(•F

∗0 + (∇∗

0F∗0)

•u∗0

)

t=tp−

= 0, (4.3.15)

•s(G(R) −G∗(T )

)+ s

(•G

(R)

+(∇(R)G(R)

) •u

(R))

−s(

•G

∗(T )

+(∇∗(T )G∗(T )

) •u∗(T )

)−(

•F

(R)

+(∇(R)F(R)

) •u

(R))

+

(•F

∗(T )

+(∇∗(T )F∗(T )

) •u∗(T )

)

t=tp+

= 0, (4.3.16)

where ∇0,∇∗0,∇(R) and ∇∗(T ) are, respectively, the gradient operators with

respect to the components of u0,u∗0,u

(R) and u∗(T ). The dot derivatives ofF0, G0, F

∗0, G

∗0, F

(R), G(R), F∗(T ) and G∗(T ) are evaluated keeping the cor-responding vector argument functions u0,u

∗0,u

∗(R) and u∗(T ) constant. Onsubtracting (4.3.15) from (4.3.16) and keeping in mind that

s(tp−) = s(tp+), G0(tp−) = G(R)(tp+), G∗0(tp−) = G∗(T )(tp+),

F0(tp−) = F(R)(tp+), F∗0(tp−) = F∗(T )(tp+),

•G0 (tp−) =

•G

(R)

(tp+),

•G

∗0 (tp−) =

•G

∗(T )

(tp+),•F0 (tp−) =

•F

(R)

(tp+),•F

∗0 (tp−) =

•F

∗(T )

(tp+),

∇0F0(tp−) = ∇(R)F(R)(tp+), ∇∗0F

∗0(tp−) = ∇∗(T )F∗(T )(tp+),

∇0G0(tp−) = ∇(R)G(R)(tp+), ∇∗0G

∗0(tp−) = ∇∗(T )G∗(T )(tp+),

•u

(R)− •

u0= (sI−A0)(u(R),x−u0,x),

•u∗(T )

− •u∗0= (sI−A∗

0)(u∗(T ),x−u∗

0,x),

where I is the unit matrix, we obtain the following system of equations valid


at t = tp.

[[•s]] (G −G∗)0 + (∇G)0 (sI −A0)

2(u,

(R)x −u0,x

)

− (∇∗G∗)0 (sI−A∗0)

2(u,

∗(T )x −u∗

0,x

)= 0,

(4.3.17)

where [[•s]] =

•s(tp+)− •s(tp−)

is the jump in the shock acceleration at t = tp.

Let Λ(R)i be the jump in u,

(R)x across the ith characteristic. Then equation

(4.3.5), in view of the initial condition (4.3.12)2 which implies x,(R)φi

= 1 and

X(R)i (P ) = 0, yields the result Πi(P ) = Λ

(R)i ; following a similar argument,

it follows from (4.3.5) and (4.3.12) that Π∗(T )i (P ) = Λ

∗(T )i (P ), and Π1(P ) =

Λ1(P ) + u0,x

(1− (x,φ1 )0

). Consequently, equations (4.3.13) become

Λ1(P ) +

p∑

i=p−q+1

Λi(P ) = u,(R)x (P ) − u0,x(P ),

q∗∑

i=1

Λ∗(T )i (P ) = u,∗(T )

x (P ) − u∗0,x(P ).

(4.3.18)

System (4.3.17), in view of equations (4.3.18), becomes

[[•s]] (G −G∗)0 + (∇G)0 (sI −A0)

2p∑

i=p−q+1

Λ(R)i (P )

− (∇∗G∗)0 (sI−A∗0)

2q∗∑

i=1

Λ∗(T )i (P ) = − (∇G)0 (sI−A0)

2Λ1.

(4.3.19)

In view of the relation (4.3.6), we may write,

Λ1(P ) =

m1∑

k=1

α(1)k (tp)R

(1,k)0 , Λ

(R)i (P ) =

mi∑

k=1

α(i)k (tp)R

(i,k)0 ,

Λ∗(T )i (P ) =

m∗

i∑

k=1

β(i)k (tp)R

∗(i,k)0 ,

(4.3.20)

which are valid on the discontinuity line at P ; here α(i)k and β

(i)k are constant

coefficients, and R∗(i,k) are the right eigenvectors of A∗ corresponding to theeigenvalue λ∗(i) with multiplicity m∗

i . Because of (4.3.20), equations (4.3.19)become

[[•s]] (G−G∗)0 + (∇G)0

p∑

i=p−q+1

(mi∑

k=1

α(i)k

(s− λ(i)

)2

R(i,k)0

)

− (∇∗G∗)0

q∗∑

j=1

m∗

j∑

k=1

β(j)k

(s− λ∗(j)

)2

R∗(j,k)0

= −(∇G)0∑m1

k=1 α(1)k

(s− λ(1)

)2R

(1,k)0 ,

(4.3.21)


where all quantities are evaluated at t = tp. Matrix equation (4.3.21) rep-resents a system of n inhomogeneous algebraic equations for the unknowns

[[•s]], α

(i)k and β

(j)k . In the case in which the discontinuity is an evolutionary

shock [101], there exists an integer ` in the interval 1 ≤ ` ≤ p such that thefollowing inequalities hold

λ(p) < λ(p−1) < . . . < λ(`+1) < s < λ(`) < . . . < λ(1),

λ∗(p) < λ∗(p−1) < . . . < λ∗(`) < s < λ∗(`−1) < . . . < λ∗(1).(4.3.22)

In this situation, the system (4.3.21), with q = ` + 1 and q∗ = ` − 1, canhave at most n unknowns, and admits a unique solution provided the rankof the coefficient matrix and that of the augmented matrix are equal to thenumber of unknowns. For a strictly hyperbolic system, equation (4.3.21) withq = `+ 1, q∗ = `− 1 and p = n becomes

[[•s]] (G −G∗)0 + (∇G)0

p∑

i=n−`α(i)

(s− λ(i)

)2

R(i)0 (4.3.23)

− (∇∗G∗)0

`−1∑

j=1

β(j)(s− λ∗(j)

)2

R∗(j)0 = −(∇G)0α

(1)(s− λ(1)

)2

R(1)0 ,

which for G(u, x, t) ≡ u, reduces to the result obtained by Brun [26] and Boil-

lat and Ruggeri [22]. Once the coefficients α(i)k and β

(j)k have been determined,

equations (4.3.20) yield the initial strength of the reflected and transmittedwaves.

4.4 Weak Discontinuities in Radiative Gasdynamics

In this section, we illustrate the applicability of singular surface theory toinvestigate the evolutionary behavior of weak discontinuities headed by wavefronts of arbitrary shape in a thermally radiating inviscid gas flow, governed bythe system (1.2.30). The governing system of PDEs clearly shows the existenceof radiation induced waves, which are followed by modified gasdynamic waves.The transport equations, representing the rate of change of discontinuities inthe normal derivatives of the flow variables, show that the radiation inducedwaves are ultimately damped. Taking into account the unsteady behavior ofthe flow ahead of a modified gasdynamic wave, it is shown that it is naturalto consider the transport of discontinuities along bicharacteristic curves in thecharacteristic manifold of the governing system of PDEs. An explicit criterionfor the growth and decay of modified gasdynamic waves along bicharacteristicscurves is given, and the spatial reference is made of diverging and convergingwaves. It is found that all compressive waves, except in one special case of

4.4 Weak discontinuities in radiative gasdynamics 101

converging waves in which both the initial principal curvatures are positive andequal, grow without bound only if the magnitude of the initial discontinuityassociated with the wave exceeds a critical value. In this case, it is shown thata compressive wave, no matter how weak initially, always grows into a shockbefore the formation of the focus; emphasis is given on a description of thegasdynamic phenomenon involved rather than on numerical results. In thiscontext, we refer to the reader the work carried out on weak discontinuitywaves by Thomas [196], Helliwell [70], Elcrat [52], and Sharma and Shyam[167].

Let us consider a moving singularity surface∑

, across which the flowparameters are essentially continuous but discontinuities in their derivativesare permitted; it thus follows that the quantities p, ρ, ui, qi, ER, S and T arecontinuous across

∑, and they have their subscript o values at the wave head.

Taking jumps across∑

, in equations (1.2.30), we find, on using (4.1.7)1, and(4.1.7)2 that

(G− uno)ζ = ρoλini; ρo(G− uno

)λi = ξni + (θ/3)ni,(4/3)ERo

λini = ρoTo(G− uno)χ− uno

θ,Gθ = εini; Gεi = (c2/3)θni,

(4.4.1)

where λi = [ui,j ]nj , ξ = [p,i]ni, ζ = [ρ,i]ni, χ = [S,i]ni, θ = [ER,i]ni, εi =[qi,j ]nj and uno

= uioni are the quantities defined on∑

. Also, the equationsof the state p = p(ρ, S) and T = T (ρ, S), yield

ξ = p,ρoζ + p,So

χ; η = T,ρoζ + T,So

χ, (4.4.2)

where η = [T,i]ni, p,ρo≡ (∂p/∂ρ)o, p,So

≡ (∂p/∂S)o, T,ρo≡ (∂T/∂ρ)o and

T,So≡ (∂T/∂S)o.

Equations (4.4.1) and (4.4.2) constitute a set of ten homogeneous equa-tions in ten unknowns λi, εi, ξ, ζ, χ and θ. Hence, the necessary condition forthe existence of nontrivial solutions is that the determinant of the coefficientmatrix must vanish; this yields

G = uno, G = ±c/

√3, G = uno

± ao,

where a2o = p,ρo

+ (4ERo/3Toρ

2o)p,So

is the square of modified sound speed.The case G = uno

, in which the surface moves with the fluid, is discarded ashaving no physical interest. For an advancing wave surface, we shall take Gto be positive. We thus find that there are two types of waves present in thegas; one that propagates with the speed G = c/

√3 has attributes which are

basically due to radiation, and is referred to as a radiation induced wave. Theother, which propagates with the speed G = uno

+ ao is essentially a forwardmoving modified gasdynamic wave.

4.4.1 Radiation induced waves

It is convenient to take the flow upstream from the radiation induced wave∑to be uniform and at rest. In this case, equations (4.4.1), and (4.4.2), using


G = c/√

3, yield

θ = ε√

3/c; ξ = (εa2o

√3/c3)1 − (3a2

o/c2)−1;

λ = (ε/ρoc2)1 − (3a2

o/c2)−1;

ζ = (ε√

3/c3)1 − (3a2o/c

2)−1;

χ = (ε√

3/c3)a2o − p,ρo

(1− (3a2o/c

2))p,So−1;

η = (ε√

3/c3)(T,S/p,S)o(a2o − a∗2o )1− (3a2

o/c2)−1,

(4.4.3)

where ε = εini, λ = λini and a∗2o = p,ρo− (p,ST,ρ/T,S)o.

When (1.2.30)5 is differentiated partially with respect to t and use is madeof (1.2.30)4, we get

(3

c2

)∂2qj∂t2

+

(6kpc

)∂qj∂t

− qj,ii + 16σkpT3T,j + 3k2

pqj = 0.

If we multiply this equation by nj and take jumps across∑

, we find on usingequations (4.1.7)2, (4.1.7)3, (4.1.7)4, (4.1.2), (4.4.3)4 and (4.4.3)6 that

δζ

δt= −(Λ1 − Λ2)ζ, (4.4.4)

where Λ1 = kp − (Ω/√

3)c and Λ2 = (8σkpT3o /c

3)(T,S/p,S)o(a2o − a∗2o )1 −

(3a2o/c

2)−1. Equation (4.4.4) governs the propagation of ζ associated with aradiation induced wave; its behavior can, of course, be readily established. Themean curvature Ω at any point of the wave surface

∑has the representation

(Thomas [197])

Ω = (Ωo −KoGt)/(1 − 2ΩoGt+KoG2t2), (4.4.5)

where Ωo = (κ1 + κ2)/2 and Ko = κ1κ2 are respectively the mean and Gaus-sian curvatures of

∑at t = 0 with κ1 and κ2 being the principal curvatures,

and G = c/√

3.When k1 and k2 are nonpositive, the wave is divergent. On the other hand

if one or both the principal curvatures are positive, then it corresponds to thecase of a convergent wave. Integration of (4.4.4), after using (4.4.5) yields

ζ = ζoI exp(−kpct), (4.4.6)

where I = (1 − (κ1ct/√

3))(1 − (κ2ct/√

3))−1/2, and ζo is the value of ζ att = 0. The term Λ2 in (4.4.4) has been neglected in comparison with Λ1 inpresenting the result (4.4.6).

For diverging waves, it is apparent from (4.4.6) that ζ→0 as t→∞; thismeans that the radiation induced waves are ultimately damped out and theformation of a front, carrying discontinuities in flow quantities, is not possi-ble from a continuous flow. From the expressions (4.4.3), it follows that thequantities λ, ξ, ζ, χ and η are small compared with ε and θ as it should be inradiation induced waves.

For converging waves, it follows from (4.4.6) that there exists a finite timet∗ given by the smallest positive root of (

√3−κ1ct)(

√3−κ2ct) = 0 such that

|ζ|→∞ as t→t∗; this corresponds to the formation of a focus.


4.4.2 Modified gasdynamic waves

In general, if the flow ahead of a modified gasdynamic wave∑

, whichpropagates with the speed G = uno

+ao, is nonuniform, equations (4.4.1) and(4.4.2) yield

aoζ = ρoλ, ρoa0λ = ξ, χ = (4ERo/3T 2

o ρ2o)ζ,

η = (T,S/p,S)o(a2o − a∗2o )ζ, θ = ε = 0.

(4.4.7)

If we differentiate equations (1.2.30) partially with respect to xk, take jumpsacross

∑, use equations (4.1.1)3, (4.1.1)4, (4.1.2) and (4.4.7), and then elim-

inate from the resulting equations the quantities involving jumps in secondorder derivatives, we obtain

ρo(δλδt + uαoλ,α

)+ (p,ρo

/ao)(δζδt + uαo ζ,α

)

+(p,So/ao)

(δχδt + uαoχ,α

)+ Pζ +Qζ2 = 0,

(4.4.8)

where P and Q are known functions depending on the state-ahead variablesand their derivatives along with the orientation and the mean curvature ofthe wave front (see Sharma and Shyam [167]).

The terms involving surface derivatives in (4.4.8) cause some difficulty inits interpretation; but if we transform (4.4.8) to a differential equation alongbicharacteristics curves, this difficulty disappears. Thus, using (4.1.5) in (4.4.8)and substituting λ and χ in terms of ζ from (4.4.7), we get

dζ

dt+

1

2

d

dtlog Λ +

P

2ao

ζ +

Q

2aoζ2 = 0, (4.4.9)

where Λ = a3o/(ρop,So

)(p,S/p,ρ)(p,ρo/a2o).

Equation (4.4.9) is the transport equation governing the propagation ofthe discontinuity ζ, and it has the same form as the equation for the growthof discontinuities in the unsteady flow of a perfect gas (see Elcrat [52]). Onecan easily verify that in the limit of vanishing radiation energy and flux, itreduces to the equation derived by Elcrat [52].

Equation (4.4.9) can be integrated to yield

ζ = ζi(Λ/Λi)−1/2 exp(−f(t))/1 + Λ

1/2i ζiI(t), (4.4.10)

where f(t) =∫ t0 (P (τ)/2ao(τ))dτ, I(t) =

∫ t0 (Q(τ)/2aoΛ

1/2) exp(−f(τ))dτ, andthe subscript i indicates an initial value at t = 0. As (4.4.10) turns out to bea special case of (4.2.8), the general results presented in subsection 4.2.2 canbe applied to yield conditions under which the wave ultimately damps out, ortakes a stable wave form, or, forms a shock or a focus. Thus, if both f(t) orI(t) are continuous for 0 ≤ t < t and have finite limits f(t), I(t) as t→t andif signζi = signI(t), then the right-hand side of (4.4.10) will not only remaincontinuous throughout 0 ≤ t < t but will also approach a finite limit as t→t.


Also if signζi = −signI(t), (4.3.10) will remain finite throughout [0, t] provided

|ζi| < ζ∗c , where ζ∗c is a positive quantity defined as ζ∗c = (Λ1/2i |I(∞)|)−1. But

if signζi = −signI(t), and |ζi| > ζ∗c , then it follows from (4.4.10) that there

exists a finite time t∗ < t, given by I(t∗) = −1/ζiΛ1/2i , such that ζ→∞ as

t→ t∗. This signifies the appearance of a shock wave at an instant t∗. Further,if |ζi| = ζ∗c and sign ζi = −signI(t), then we find that ζ is continuous for t in[0, t) but approaches infinity as t→t.

4.4.3 Waves entering in a uniform region

We have already noticed in subsection 4.3.1 that for radiation inducedwaves the quantities ξ, λ, ζ, χ and η are much smaller as compared to ε andθ, and hence the main effects of the radiative transfer on the growth anddecay behavior of the gasdynamic wave

∑can be seen for the special case in

which ui = 0 and p, ρ, T are constant ahead of the wave∑

. For such a wave,propagating through a uniform flow of an ideal gas, equation (4.4.8) becomes

δζ

δt+ (ω − aoΩ)ζ + ψζ2 = 0, (4.4.11)

where ω, ao, and ψ are positive constants given by

ω = (8σkpT3o /ρ0a

2oR)(γ − 1)(a2

o − a∗2o )1 + (a2

o/(γ − 1))(c2 − 3a20)

−1,

a20 − (γpo + 4(γ − 1)pRo

)/ρo, ψ = (γ + 1)ao/(2ρo),

with R as the gas constant, γ the adiabatic index, a∗o the isothermal speedof sound defined as a∗o = (po/ρo)

1/2, and pRothe radiation pressure, which is

one third of the radiation energy ER. The term (a2o/(γ− 1))/(c2 − 3a2

o) in theexpression for ω will be neglected eventually in order to be consistent withthe nonrelativistic form of equations (1.2.30); here, it is retained temporarilysince it throws some light on the physical nature of the problem.

Equation (4.4.11), after substituting from (4.4.5), can be integrated toyield

ζ = ζo exp(−ωt)I1(t)/(1 + ψI2(t)ζo), (4.4.12)

where

I1 = (1 − κ1aot)(1 − κ2aot)−1/2, I2(t) =

∫ t

o

I1(τ) exp(−ωτ)dτ, (4.4.13)

and ζ0 is the value of ζ at t = 0.

Diverging waves: For diverging waves both κ1 and κ2 are nonpositive, andit is apparent from (4.4.13) that I1 and I2 both converge to finite limits ast→∞. Hence, if ζo > 0 (i.e., an expansion wave front), then ζ→0 as t→∞.This means that all expansion waves decay and damp out ultimately. But, ifζo < 0 (i.e., a compression wave), then there exists a positive critical value,


ζc, given by ζc = 1/ψ∫∞0I1(τ) exp(−ωτ)dτ such that waves with initial

discontinuity |ζo| < ζc damp to zero (i.e., ζ→0 as t→∞) and the waves withinitial discontinuity |ζo| > ζc grow without bound in a finite time tc (i.e.,|ζ|→∞ as t→tc) given by I2(tc) = 1/ψ|ζo|. Further, if ζo < 0 and |ζo| =ζc then it immediately follows that ζ→(ω/ψ) as t→∞. Thus, a divergingcompression wave for |ζo| = ζc can neither terminate into a shock wave norcan it ever completely damp out. Since ∂ζc/∂ω > 0, the critical value of theinitial discontinuity increases with ω, and thus the effect of thermal radiationis to delay the onset of a shock wave. Also ∂ζ/∂|κα| = 0, α = 1, 2, which implythat the initial principal curvatures have a stabilizing effect on the tendency ofa wave surface to grow into a shock in the sense that an increase in the valueof the initial curvatures causes an increase in the critical value ζc. For waveswith plane (κ1 = 0 = κ2), cylindrical (κ1 = −1/Ro, κ2 = 0) and spherical(κ1 = κ2 = −1/Ro) geometry, where Ro is the radius of the wave front att = 0, we find that the critical value of the initial discontinuity and the timetaken for the shock formation are given by the following relations:

Plane wave: ζc = ω/ψ, tc = ω−1 log1− (ω/ψ|ζo|)−1,

Cylindrical wave: ζc =1

ψ

(πRoωao

)−1/2exp(−ωRo/ao)erfc(ωRo/ao)1/2

,

erfc(ωtc + (ωRo/ao))1/2 = (1 − (ζc/|ζo|))erfc(ωRo/ao)1/2,

Spherical wave: ζc =ao exp(−ωRo/ao)ψRoEi(ωRo/ao)

,

Ei(ωtc + (ωRo/ao)) = (1 − (ζc/|ζo|))Ei(ωRo/ao),

where erfc(x) = (2/√π)∫∞x exp(−t2)dt and Ei(x) =

∫∞x t−1 exp(−t)dt are,

respectively, the complementary error function and the exponential integral.

Converging waves:(i) When only one of the initial principal curvatures is positive or both

are positive and k1 6= k2, in that case, there exists a finite time t∗ given bythe smallest positive root of (1 − κ1aot)(1 − κ2aot) = 0, such that I1(t)→∞as t→t∗, whereas I2(t) tends to a finite value as t→t∗. The fact that I2(t∗) isbounded follows from the argument that the singularity at t∗ of the integrandin I2(t) is of the form z−1/2g(z) as z→0, where g(z) is bounded. This canbe easily seen by a suitable transformation z = t− t∗. Hence, it follows from(4.4.12) that for ζo > 0, |ζ|→∞ as t→t∗, i.e., a converging expansive waveforms a focus. But, if ζo < 0, then it follows from (4.4.12) that there exists

a positive critical value ζc given by ζc = 1/ψI2(t∗), such that if |ζo| < ζc,then |ζ|→∞ as t→t∗, which corresponds to the formation of a focus but not

the shock. On the other hand, if |ζo| > ζc, then we find that |ζ|→∞ as t→tc,where tc is finite time given by I2(tc) = I/ψ|ζo|. It is evident that tc < t∗.Thus, when |ζo| > ζc, we find that the shock wave is formed before the focus


can. The case |ζo| = ζc corresponds to the simultaneous formation of a shockand a focus.

In this situation, it is interesting to note that not all converging compres-sive waves will grow into shock waves.

(ii) When both the initial principal curvatures are positive and κ1 = κ2 inthat case, the singularity at t∗ of the integrand in I2 is of the form z−1r(z) asz→0, where r(z) is bounded away from zero. Hence, I2→∞ as t→t∗. Thus, itfollows from (4.4.12) that if ζo > 0, then |ζ|→∞ as t→t∗, i.e., a focus is formedwithin a finite time t∗. But if ζo < 0, then there exists a finite time t < t∗ givenby I2(t) = I/ψ|ζo| for which the denominator of (4.4.12) vanishes whereasthe numerator remains finite. This means that in this particular situation aconverging compressive wave, no matter how weak initially, always grows intoa shock before the formation of the focus (see Sharma and Menon [175]).

4.5 One-Dimensional Weak Discontinuity Waves

A comprehensive review relating to the propagation of one-dimensionalwaves in a nonequilibrium medium has been given by Becker [12], Chu [36],and Clarke and McCheseny [37]. In what follows we shall use different methodsof approach to study this problem in one-dimension; the waves may be thoughtof as being produced by a moving piston. The basic equations governing themotion are same as in (1.2.21); for a one-dimensional motion with plane (m =0), cylindrical (m = 1) or spherical (m = 2) symmetry, these equations maybe written in the form

p,t + up,x + ρa2f (u,x + (mu/x)) = −ωa2

f (ρ,q + (α/T )ρ,S),

u,t + uu,x + (1/ρ)p,x = 0, S,t + uS,x = αω/T, q,t + uq,x = ω.(4.5.1)

where the spatial coordinate x is either axial (in flows with planar geometry)or radial (in cylindrically or spherically symmetric flows), ρ,q = ∂ρ/∂q, ρ,S =∂ρ/∂S, and the remaining symbols have the same meaning as in (1.2.21). Weassume that the medium ahead of the wave front is in a state of completethermodynamic equilibrium and at rest, i.e., the pressure po, the density ρo,entropy So and the progress variable qo ahead of the wave front are constant,and the velocity uo is zero.

4.5.1 Characteristic approach

Equations (4.5.1) constitute a hyperbolic system with four families of char-acteristics. Two of these characteristics, dx/dt = u ± af , represent wavespropagating in the ±x directions with the frozen sound speed af , and theremaining two, dx/dt = u, form a set of double characteristics representingthe particle path or trajectory.

4.5 One-Dimensional Weak Discontinuity Waves 107

In studying wave phenomena governed by hyperbolic equations, it is usu-ally more natural and convenient to use the characteristics of the governingsystem as the reference coordinate system. We thus introduce two character-istic variables φ and ψ such that

φ,t + (u+ af )φ,x = 0, ψ,t + uψ,x = 0.

The leading characteristic front can be represented by φ = 0, and if a materialparticle crosses the front at time t, its path will be represented by ψ = t.Keeping in view the properties of φ and ψ, it is obvious that the functionsx(φ, ψ) and t(φ, ψ) satisfy the following differential equations.

x,φ = ut,φ, x,ψ = (u+ af )t,ψ . (4.5.2)

The transformation from the (x, t) plane to the (φ, ψ) plane will be one-to-oneif, and only if, the Jacobian

J = x,φt,ψ − x,ψt,φ, (4.5.3)

does not vanish or become undefined anywhere. Hence, if F denotes any ofthe flow variables, p, ρ, u, S, or q, then

F,x = (F,φt,ψ − F,ψt,φ)/J, F,t + uF,x = −afF,φt,ψ/J. (4.5.4)

Since the overlapping of fluid particles is prohibited from physical considera-tions, it implies that t,ψ 6= 0. Consequently J = 0 if, and only if, tφ = 0 whichcorresponds to the situation when two adjoining characteristics merge into ashock wave.

In terms of the characteristic coordinates, the system of equations (4.5.1)can be transformed into the following equivalent system.

p,ψ + ρafu,ψ + a2fω((ρ,q + (α/T )ρ,S)t,ψ + (mρua2

f/x)t,ψ = 0,

(p,φ − ρafu,φ)t,ψ − p,ψt,φ = 0, (4.5.5)

S,φ = (αω/T )t,φ, q,φ = t,φω,

where use has been made of (4.5.3) and (4.5.4).Consider the case in which the wave front is an outgoing characteristic;

then the flow variables are continuous across it, and the boundary conditionswhich hold on it can be written as

p = po, ρ = ρo, S = So, q = qo, u = 0, t = ψ at φ = 0. (4.5.6)

The last condition is, of course, a consequence of the particular method oflabeling the particle paths; indeed, conditions (4.5.6) demand that

p,ψ = u,ψ = ρ,ψ = S,ψ = q,ψ = 0, t,ψ = 1 at φ = 0. (4.5.7)


Also, equations (4.5.2), (4.5.5)2, (4.5.5)3, and (4.5.5)4, when evaluated at thewave front φ = 0, yield

p,φ = ρoafou,φ, S,φ = 0 = q,φ, x,φ = 0, x,ψ = afo. (4.5.8)

Now to compute s = u,x at the wave front φ = 0, we invoke (4.5.4)1, (4.5.3)and (4.5.7)2 to obtain

s = −u,φ/(t,φafo). (4.5.9)

Differentiating (4.5.2)2 and (4.5.5)1 with respect to φ and (4.5.2)1 and (4.5.5)2with respect to ψ and using the foregoing results, we find that at the wavefront φ = 0

u,φψ = −µo +mafo

2xu,φ, (4.5.10)

t,φψ = − 1

2afo

(1 + Γo)u,φ, (4.5.11)

where µo = (a2fρ,qω,p)o/2 and Γo = 1 + ρo((a

2f ),p)o; in the derivation of

(4.5.10), we have used the fact that the state ahead of the wave front is incomplete thermodynamic equilibrium, i.e., αo = 0 = ωo (see [37]).

Differentiating (4.5.9) with respect to ψ and using (4.5.10) and (4.5.11), wefind that s, given by (4.5.9), satisfies the following Bernoulli-type differentialequation:

ds

dt+(µ0 +

mafo

2x

)s+

(1 + Γo

2

)s2 = 0, (4.5.12)

where x = x(0) + afot with x(0) as the value of x at t = 0. In obtaining

(4.5.12), we have used the fact that t = ψ at ψ = 0.Equation (4.5.12) is of the form (4.2.8) with µ = µo + (mafo

/2x) andβ = (1+Γo)/2. Note that in (4.5.12), Γo and µo are positive constants. Indeed,for an ideal gas, it may be readily verified that Γ = γ, the adiabatic index;similarly in an equilibrium state in which ωo = 0 = αo, it readily transpiresthat

µo =1

2a2fo

(ρ,qω,p)o =1

2τo

(a2fo

a2eo

− 1

), (4.5.13)

where τo = −(ω,q)−1o is a relaxation time of the medium and a2

eo is theequilibrium speed of sound given by a2

eo = ((∂p/∂ρ)S,q=qe)o (see [37]). Since

afo> aeo, it follows that in a complete equilibrium state, µo > 0.The behavior of the solution of (4.5.12) can now be obtained by apply-

ing our general results presented in subsection 4.2.2. Thus, when sgn s(0) =− sgnβ = −1 (i.e., an expansion wave front), then lim

t→∞s(t) = 0 (i.e., the wave

damps out). However, when sgn s(0) = sgnβ = +1 (i.e., a compression wavefront), then there exists a positive critical amplitude sc given by

sc =

[1 + Γo

2

∫ ∞

0

(1 + κoafoτ)−m/2 exp[−µoτ ]dτ

]−1

,


where κ0 = 1/x(0) is the initial wave front curvature, such that, for s(0) < sc,s→0 as t→∞, for s(0) > sc, s→∞ as t→tc (i.e., the wave culminates into ashock wave in a finite time tc), where tc is given by the solution of

∫ tc

0

(1 + κoafot)−m/2 exp[−µot]dt = 2/(s(0)(1 + Γo)),

and for s(0) = sc, s→2µo/(1 + Γo) as t→∞ (i.e., the wave ultimately takesa stable wave form).

4.5.2 Semi-characteristic approach

The method introduced by Jeffrey [80] for determining the evolution law ofweak discontinuities consists in introducing a change of variables from x andt to semi-characteristic coordinates. The system of linear equations arisingfrom this method appears formally to be quite different from the Bernoulliequation, which is found when the equation governing the amplitude of theweak discontinuity is derived directly. This problem has been considered byBoillat and Ruggeri [21], who showed them to be equivalent. Here, we illustratehis method by confining ourselves to the one-dimensional system (4.5.1), whichusing vector-matrix notation can be written as

u,t + A(u)u,x = b, (4.5.14)

where u is a column vector with components p, u, S and q, and the squarematrix A and the column vector b can be read off by inspection of (4.5.1).Let λ(j), j = 1, 2, 3, 4, be the eigenvalues of A, and L(j) the corresponding lefteigenvectors; then

λ(1) = u+ af , L(1) = [ 1 ρaf 0 0 ]tr ,

λ(2) = u− af , L(2) = [ 1 −ρaf 0 0 ]tr ,

λ(3) = u, L(3) = [ 0 0 1 0 ]tr ,λ(4) = u, L(4) = [ 0 0 0 1 ]tr .

Equations dx/dt = λ(1,2) represent waves propagating in the ±x directions,and the equations dx/dt = λ(3,4) represent particle path or trajectory. Letφ(x, t) = 0 be the wave front or the forward facing characteristic correspondingto the eigenvalue λ(1); the unperturbed field ahead of the wave front, whichwe denote by using the subscript o, is assumed to be at rest and in a stateof complete thermodynamical equilibrium. Let us introduce the new variablesφ = φ(x, t), t′ = t such that the Jacobian ∂(φ, t′)/∂(x, t) ≡ x,φ = 1/φ,x and itsinverse do not vanish. The leading characteristic x = xo+afo

t, issuing out fromthe point (xo, 0), is taken as a member of the family given by φ,t+λ

(1)φ,x = 0,φ(x, 0) = x− xo so that it is expressed by the equation φ(x, t) = 0.

Transforming from (x, t) to (φ, t′) coordinates, pre-multiplying by L(j) andusing the relation L(j)A = λ(j)L(j), equation (4.5.14) yields

L(j)x,φu,t′ + (λ(j) − λ(1))u,φ + L(j)bx,φ = 0, (4.5.15)


where u,t′ ≡ u,t + λ(1)o u,x is the derivative along the wave front, which we

shall denote subsequently by the symbol du/dt.Across the wave front φ = 0,u and u,t′ are continuous whilst u,φ and x,φ

are discontinuous and have jumps Π and X , respectively; it may be noticedthat at the rear of φ = 0, we have

s = u,x = u,φ/x,φ. (4.5.16)

Since in the region ahead of the wave front (u,φ)o = 0, (u,t′) = 0, and bo = 0,equation (4.5.15) with λ(j) 6= λ(1) yields on differencing across φ = 0 that

L(j)o Π = 0, (j 6= 1), which in view of the left eigenvectors L(2),L(3) and L(4),

yields the following relations

π1 = ρoafoπ2, π3 = 0 = π4, (4.5.17)

where π1, π2, π3 and π4 are components of the column vector Π. We now setj = 1 in equation (4.5.15), differentiate the resulting equation with respect toφ, and then form the jumps in the usual way across φ = 0 to obtain

L(1)o Π,t′ +

∇(L(1)b)

oΠ = 0, (4.5.18)

where ∇ stands for the gradient operator with respect to the components ofthe vector u. Equation (4.5.18), in view of the eigenvector L(1), the columnvector b, and the relation (4.5.17) yields.

(π2),t′ +µo +

mafo

2x

π2 = 0, (4.5.19)

where µo is same as in (4.5.13). Also, along the wave front φ = 0 we havex,t′ = λ(1), which on differentiation with respect to φ, and on forming jumpsacross φ = 0, yields

dX

dt= (∇λ(1))oΠ =

(1 + Γo)

2π2, (4.5.20)

where Γo is same as in (4.5.11); in obtaining (4.5.20), we have made use of(4.5.17).

If we differentiate (4.5.16) with respect to t′ and use (4.5.18) and (4.5.20),we find that s satisfies (4.5.12); thus, the law of propagation of weak discon-tinuities obtained using Jeffrey’s method is in agreement with the Bernoulli’slaw found using the characteristic approach.

4.5.3 Singular surface approach

Among various approaches, the use of singular surface theory quickly leadsto the results of general significance. This is an alternative approach, withoutmarked difference in analytical extent, which leads to the same basic result


concerning the evolutionary behavior of weak discontinuity waves. Here, wedemonstrate the applicability of this method to study the development ofjump discontinuities associated with the one-dimensional system (4.5.1).

Let x = x(t), or for brevity,∑

(t), denote the weak discontinuity waveacross which the flow variables are essentially continuous but discontinuities intheir derivatives are permitted. The description of the wave front

∑(t) is such

that its speed of propagation G = dx/dt is always positive. The unperturbedfield ahead of the wave is assumed to be uniform and at rest, and in a stateof complete thermodynamic equilibrium. We infer that

p = po, u = uo, S = So, q = qo, af = afo, α = αo = 0, ω = ωo = 0, (4.5.21)

where a subscript o indicates a value in the medium just ahead of the wavefront.

In a one-dimensional case, the geometric and kinematical conditions (4.1.1)of first and second order, stated in Section 4.1, reduce to

[f,x] = d, [f,t] = −Gd, [f,xx] = d, [f,xt] = −Gd+δd

δt, (4.5.22)

where the quantity f may represent any of the variables p, u, S, q, α and ω.The square brackets stand for the value of the quantity enclosed immediatelybehind minus its value just ahead of the wave front

∑(t). The quantities d

and d are defined on the wave front∑

(t), and the δ-time derivative is definedas δf/δt = f,t + Gf,x. Thus, the δ-time derivative of any quantity, which isdefined on

∑(t) is identical with the time derivative, d/dt, of that quantity

following the wave front.Taking jumps, across

∑(t), in (4.5.1) and making use of (4.5.21), we get

Gξ = ρoa2fos, ξ = ρoGs, ζ = 0 = η, (4.5.23)

where ξ = [p,x], s = [u,x], ζ = [S,x] and η = [q,x] are quantities defined on∑(t).The case G = 0, which corresponds to a material surface is discarded as

uninteresting; we therefore assume that G 6= 0. Then it follows from (4.5.23)that G = ±afo

; for an advancing wave, we shall take

G = afo. (4.5.24)

This is the speed with which the wave∑

(t) propagates into the mediumahead. Equation (4.5.23)2, in view of (4.5.24) yields

ξ = ρoafos. (4.5.25)

If we differentiate (4.5.1)1 and (4.5.1)2 with respect to x, take jumps across∑(t) and then make use of (4.5.22) – (4.5.25), we get

δs

δt=

1

ρo(ξ − ρoafo

s) − 2s(µo +mafo

2x) − (1 + Γo)s

2, (4.5.26)

δs

δt= − 1

ρo(ξ − ρoafo

s), (4.5.27)


where ξ = [p,xx], s = [u,xx], and µo and Γo are the same as in (4.5.10) and(4.5.11). Eliminating the term (ξ − ρoafo

s) from (4.5.26) and (4.5.27), andrecalling the definition of δ-time derivative, we find that s satisfies the sameBernoulli’s law (4.5.12).

Remark 4.5.1: We thus observe that the various approaches used for study-ing the propagation law of weak discontinuities are in agreement with thefamiliar Bernoulli’s law. However, the singular surface method is more gen-eral in the sense that it is applicable to higher dimensions and quickly leadsto the basic results concerning wave propagation.

4.6 Weak Nonlinear Waves in an Ideal Plasma

The propagation of one dimensional waves in a gaseous medium may bethought of as being produced by a moving piston. When the piston recedesor advances into the gas, not all parts of the gas are affected instantaneously.A wave proceeds from the piston into the gas, and the particles which havebeen reached by the wave front are disturbed from their initial state of rest. Ifthis wave represents a continuous motion caused by the receding piston, thewave front propagates into the gas with local sound speed. But, if the pistonmoves into the gas, the characteristics, in general, coalesce to form a shockwave during propagation. An impulsive motion of the piston may produce aninstantaneous unsteady shock which may grow or decay with time, dependingon the condition of the undisturbed gas and the behavior of the piston. Here,we envisage the one dimensional planar motion of a plasma, which is assumedto be an ideal gas with infinite electrical conductivity and to be permeatedby a magnetic field orthogonal to the trajectories of gas particles; the motionis conceived of as being produced by a piston moving with a small velocityas compared to the magnetoacoustic speed. We follow Chu [36] to study themain features of weak nonlinear waves, namely the expansion waves and shockwaves. The trajectories of these waves and particle paths are determined, andthe effects of magnetic field strength and the adiabatic heat exponent on thewave propagation are discussed.

The basic equations governing the motion are same as in (1.2.24) withm = 0 = n, and can be written as

ρ,t + uρ,x + ρu,x = 0,u,t + uu,x + ρ−1(p+ h),x = 0,p,t + up,x + ρa2u,x = 0,h,t + uh,x + 2hu,x = 0,

(4.6.1)

where h = µH2/2; all other symbols have the same meaning as in (1.2.24).We assume that the medium ahead of the wave front is in a uniform state atrest, and use the subscript o to denote this state.

4.6 Weak Nonlinear Waves in an Ideal Plasma 113

Equations (4.6.1) constitute a hyperbolic system with four families of char-acteristics. Two of these characteristics, dx/dt = u± c, represent waves prop-agating with the magnetoacoustic speed c = (a2 + b2)1/2 with b = (2h/ρ)1/2

as the Alfven speed, and the remaining two, dx/dt = u, form a set of doublecharacteristics representing the particle path or trajectory. We introduce thecharacteristic variables φ and ξ such that

φ,t + (u+ c)φ,x = 0, ψ,t + uψ,x = 0.

Keeping in view the properties of φ and ψ, it immediately follows that

x,φ = ut,φ, x,ψ = (u+ c)t,ψ. (4.6.2)

The transformation from x-t plane to the φ-ξ plane will be one to one if, andonly if, the Jacobian

J = x,φt,ψ − x,ψt,φ, (4.6.3)

does not vanish or become undefined anywhere. Hence, if F denotes any ofthe flow variables, p, ρ, u or h, then

F,x = (F,φt,ψ − F,ψt,φ)/J, F,t + uF,x = −cF,φt,ψ/J. (4.6.4)

Since the overlapping of fluid particles is prohibited from physical considera-tions, it implies that t,ψ 6= 0. Consequently J = 0 if, and only if, tφ = 0 whichcorresponds to the situation when two adjoining characteristics merge into ashock wave.

In terms of the characteristic variables, the system (4.6.1) can be trans-formed into the following equivalent system

(ρu,φ − cρ,φ)t,ψ − ρu,ψt,φ = 0,((p+ h),φ − ρcu,φ)t,ψ − (p+ h),ψt,φ = 0,(cp,φ − ρa2u,φ)t,ψ + ρa2u,ψt,φ = 0,(ch,φ − 2hu,φ)t,ψ + 2hu,ψt,φ = 0.

(4.6.5)

In equations (4.6.1), we nondimensionalize time t by an appropriate timet∗ that characterizes the change in piston motion. It is easy to check thatthe form of equations (4.6.1) remains unchanged if the velocities u and care nondimensionalized by co, the distance x by cot

∗, the density ρ by ρo, thepressure p and magnetic pressure h by ρoc

2o, and the sound speed a by co. Thus,

without introducing new notations, the variables x, t, ρ, u, p, h etc., appearingin (4.6.1) will henceforth be regarded as dimensionless. The characteristicvariables φ and ψ will also be dimensionless to begin with; moreover, in theundisturbed medium, we have c = co = 1 and ρ = ρo = 1.

Indeed, a wave is weak if the magnitudes of the dimensionless velocity,pressure, etc. are small compared to 1; in particular, the piston speed is smallcompared to co. Along the piston path, x = F (t), we have u = F ′(t) whereF ′ = dF/dt. We assume that F ′ is small and take it as F ′(t) = εf ′(t) where


ε(<< 1) characterizes the amplitude of the disturbance and f ′(t) ∼ O(1);here, ε may be regarded as the maximum piston velocity divided by co. Theboundary conditions at the piston are

x = εf(φ), t = φ, u = εf ′(φ) at ψ = 0. (4.6.6)

If the wave front is an outgoing characteristic, the boundary conditions onit can be written as

ρ = ρo = 1, c = co = 1, u = 0, a = ao p = po, h = ho, t = ψ at φ = 0.(4.6.7)

As ε is a small parameter, we seek the solution of the piston problem in theform

z = z(o)(φ, ψ) + εz(1)(φ, ψ) +O(ε2), (4.6.8)

where z may denote any of the dependent variables ρ, u, p, h, x, t, a, c and z(o)

represents the value of z in the undisturbed uniform region. Writing the depen-dent variables in (4.6.2), (4.6.6) and (4.6.7) in the form (4.6.8), and collectingterms of the zeroth order in ε, we get

ρ(o) = 1, u(o) = 0, p(o) = po, h(o) = ho, a(o) = ao,

c(o) = co = 1, x(o),φ = 0, x

(o),ψ = t

(o),ψ ,

(4.6.9)

which, together with the boundary conditions

t(o) = φ, x(o) = 0 at ψ = 0,t(o) = ψ at φ = 0,

(4.6.10)

yield the zeroth order solution as

x(o) = ψ, t(o) = φ+ ψ. (4.6.11)

To this order, the characteristics in the physical plane are

x = ψ, t = x+ φ. (4.6.12)

In order to find the solution valid to the first order, we use (4.6.8) in (4.6.2)and (4.6.5), and collect terms of the order of ε; we thus obtain

x(1),φ = u(1), x

(1),ψ = t

(1),ψ + u(1) + c(1),

ρ(1),φ − u

(1),φ + u

(1),ψ = 0, u

(1),φ − (p(1) + h(1)),φ + (p(1) + h(1)),ψ = 0,

p(1),φ − a2

ou(1),φ + u

(1),ψ = 0, h

(1),φ − 2hou

(1),φ + 2hou

(1),ψ = 0.

(4.6.13)

It may be noticed that

a2o + b2o = 1, c(1) = aoa

(1) + bob(1), b2o = 2ho,

bob(1) = h(1) − hoρ

(1), a(1) = (ao/2)((p(1)/po) − ρ(1)).(4.6.14)


The first order boundary conditions from (4.6.6) are

x(1) = f(φ), t(1) = 0, u(1) = f ′(φ) at ψ = 0. (4.6.15)

Similarly, in view of (4.6.2) and (4.6.7), the first order boundary conditionsat the characteristic front are

t(1) = 0, u(1) = 0 p(1) = 0, ρ(1) = 0, h(1) = 0 at φ = 0. (4.6.16)

Solution of the system (4.6.13), satisfying (4.6.15) and (4.6.16) may be con-structed by the method of Laplace transforms. If we define the Laplace trans-form of z(φ, ψ) with respect to φ by

z(ξ, ψ) =

∫ ∞

0

z(φ, ψ)e−ξφdφ,

then we find that

ξx(1) = u(1),dx(1)

dψ=

d

dψt(1) + u(1) + c(1), (4.6.17)

ξ(ρ(1) − u(1)) = − d

dψu(1), ξ(u(1) − p(1) − h(1)) = − d

dψ(p(1) + h(1)), (4.6.18)

ξ(p(1) − a2ou

(1)) = − d

dψu(1), ξ(h(1) − 2hou

(1)) = −2hodu(1)

dψ. (4.6.19)

The transformed boundary conditions (4.6.15), in view of (4.6.17)1 yield

x(1) = f(ξ), t(1) = 0, u(1) = ξf(ξ) at ψ = 0. (4.6.20)

From equations (4.6.14), (4.6.18)1 and (4.6.19), we obtain

h(1) = b2oρ(1), b(1) = (bo/2)ρ(1), p(1) = ρ(1) − (1 − a2

o)u(1), (4.6.21)

which together with (4.6.18) yield, on eliminating ρ(1) and dρ(1)/dψ, the fol-lowing second order linear homogeneous differential equation

d2u(1)

dψ2− λ

du(1)

dψ= 0, (4.6.22)

where λ = ξ(1 + (1 + b2o)−1). The solution of (4.6.22), which satisfies (4.6.20)

and remains bounded as ψ→∞, is

u(1) = ξf(ξ),

and hence (4.6.17)1, (4.6.18)1 and (4.6.21) imply that

x(1) = f(ξ), ρ(1) = ξf(ξ), p(1) = a2oξf(ξ),

h(1) = b2oξf(ξ), b(1) = (bo/2)ξf(ξ).

(4.6.23)


Finally, equation (4.6.17)2, in view of (4.6.14)2 and (4.6.20)2, implies that

t(1) = −Λoξf(ξ)ψ, where Λo = (1 + γ) + (2 − γ)b2o/2. Thus, we can easilyobtain the complete solution for p, ρ, u, h, x and t; the solutions for u, x and tare

u(φ, ψ) =ε

2πi

∫ α+i∞

α−i∞ξf(ξ)eξφdξ +O(ε2),

x = ψ +ε

2πi

∫ α+i∞

α−i∞f(ξ)eξφdξ +O(ε2),

t = φ+ ψ − εΛoψ

2πi

∫ α+i∞

α−i∞ξf(ξ)eξφdξ +O(ε2),

(4.6.24)

where α is a positive number, satisfying the conditions that all the singular-ities of these integrals are to the left of the line ξ = α in the complex plane.The trajectories of the outgoing waves and particle paths in the x-t plane aredescribed by equations (4.6.24)2 and (4.6.24)3, which clearly exhibit the bend-ing of the characteristics. If we retain only the zeroth order terms, equation(4.6.24)1 becomes

u(x, t) =ε

2πi

∫ α+i∞

α−i∞ξf(ξ)eξ(t−x)dξ +O(ε2),

which is obviously the solution of the linearized equations (4.6.1).

4.6.1 Centered rarefaction waves

Here, we shall study the rarefaction waves caused by a receding pistonmotion; indeed, a piston receding from a gas at rest with speed which neverdecreases causes a rarefaction wave of particles moving toward the piston. Ofparticular interest is the case in which the acceleration of the piston fromrest to a constant velocity takes place in an infinitely small time interval, i.e.,instantaneously; then the family of characteristics forming the simple wavedegenerates into a centered rarefaction wave. The problem can be consideredas the special case of a receding piston x = εf(t) with f(t) = − 1

2kt2 for

t < 1/k, and f(t) = g(t) for t > 1/k, where g(1/k) = −1/(2k), and k is thedimensionless piston acceleration; in the limit k → ∞, the piston velocity willchange instantaneously from 0 to −ε. We introduce a new characteristic levelβ, defined as β = kφ for φ < 1/k, so that as φ varies from 0 to 1/k, β changesfrom 0 to 1, and seek a solution to the problem in which an instantaneouschange in the piston velocity generates a concentrated expansion wave thatspreads out into an expansion fan in the course of propagation.

In terms of β, the governing equations for the zeroth order variables become

x(o),β = 0, x

(o),ψ = t

(o),ψ , (4.6.25)


along with the boundary conditions

t(o) = β/k, x(o) = 0 at ψ = 0 for β < 1,t(o) = ψ at β = 0.

(4.6.26)

Equations (4.6.25) and (4.6.26) admit the following solution

t(o) = ψ + (β/k), x(o) = ψ. (4.6.27)

The governing equations (4.6.13) for the first order variables, and the bound-ary conditions (4.6.15) and (4.6.16), when written in terms of β, become

x(1),β = u(1)/k, x

(1),ψ = t

(1),ψ + u(1) + c(1), ρ

(1),β − u

(1),β + k−1u

(1),ψ = 0,

u(1),β − (p(1) + h(1)),β + k−1(p(1) + h(1)),ψ = 0,

p(1),β − a2

ou(1),β + k−1u

(1),ψ = 0, h

(1),β − 2hou

(1),β + (2ho/k)u

(1),ψ = 0, (4.6.28)

x(1) = −β2/2k, t(1) = 0, u(1) = −β at ψ = 0 for β < 1, (4.6.29)

t(1) = 0, u(1) = 0, p(1) = 0, ρ(1) = 0, h(1) = 0 at β = 0. (4.6.30)

The solution of the system (4.6.28) – (4.6.30) in the limit k → ∞ is deter-mined as

x(1) = 0, u(1) = −β, ρ(1) = −β, p(1) = −a20β, h(1) = −b2oβ, t(1) = Λβψ,

(4.6.31)where

Λ = (γ + 1) + (2 − γ)b2o/2. (4.6.32)

Thus, the complete solution valid up to the first order of ε is

ρ = 1 − εβ +O(ε2), u = −εβ +O(ε2), p = po − εa2oβ +O(ε2),

a = ao(1 − γ−12 εβ) +O(ε2), h = ho − εb2oβ +O(ε2),

x = ψ +O(ε2), t = ψ(1 + εβΛ) +O(ε2).

(4.6.33)

Equation of the characteristics in the expansion fan, and the flow field in thephysical plane can be obtained from (4.6.33) as

x = t(1 − εβΛ) +O(ε2), u = 1Λ

(xt − 1

)+O(ε2),

ρ = 1 + 1Λ

(xt − 1

)+O(ε2), a = ao

(1 + γ−1

2Λ

(xt − 1

))+O(ε2),

p = po +a2

o

Λ

(xt − 1

)+O(ε2), h = ho +

b2oΛ

(xt − 1

)+O(ε2).

(4.6.34)


In the absence of magnetic field, bo→0 and Λ→(γ + 1)/2, equations(4.6.34)2 and (4.6.34)4 become

u =2

γ + 1

(xt− 1)

+O(ε2), a = ao

(γ − 1

γ + 1

x

t+

2

γ − 1

)+O(ε2),

which, in terms of the dimensionless variables defined earlier, is precisely thesolution for expansion waves obtained in [210].

4.6.2 Compression waves and shock front

Let us now consider the case when a piston, initially at rest, for t ≤ 0, ispushed forward with a uniform velocity εco(> 0) for t > 0. The fan is reversedand forms a multivalued region, which can be envisaged as a fold in the x− tplane; indeed, it corresponds to an immediate breaking, and must be replacedby a shock wave propagating into the undisturbed medium. The differentialequations governing the flow remain the same as in (4.6.1); in particular thefirst order variables ρ(1), u(1), p(1), h(1), x(1) and t(1) are governed by (4.6.13).The boundary conditions at the piston are given by

x(1) = f(φ), t(1) = 0, u(1) = 1 at ψ = 0, (4.6.35)

instead of (4.6.14), and the boundary conditions at the shock front, φ = Φ(ψ)are the R-H conditions given by (see [96] and [27])

[ρ(u− s)] = 0, [ρu(u− s) + p+ h] = 0,[(u− s)( 1

2ρu2 + 1

γ−1p+ h) + u(p+ h)]

= 0, [h(u− s)2] = 0,(4.6.36)

where s is the shock speed. Since the shock path in the physical plane isrepresented by x = x(Φ, ψ), t = t(Φ, ψ), the shock velocity s and the functionΦ(ψ) are related to each other by

s = (x,φΦ′

(ψ) + x,ψ)/(t,φΦ′

(ψ) + t,ψ), (4.6.37)

where Φ′

(ψ) ≡ dΦ/dψ. It may be recalled that the variables ρ, u, p, h, x, t,etc., are all dimensionless. It is easy to see that the equations (4.6.36) re-main unchanged if the shock velocity s is nondimensionalized by co. Thus, ifδ ≡ [ρ]/ρo, is the density strength of the shock, then it follows from (4.6.36)1,(4.6.36)2 and (4.6.36)4 that

ρ = 1 + δ = s/(s− u), h = ho(1 + δ)2, p = po − hoδ(2 + δ) + s2δ(1 + δ)−1.(4.6.38)

Using (4.6.38) in the energy equation (4.6.36)3, we find that

s2 =2(1 + δ)

2 − (γ − 1)δ

1 +

(2 − γ

2

)b20δ

. (4.6.39)


In the limit of vanishing magnetic field, i.e., bo→0, (4.6.39) reduces to thecorresponding gasdynamic result (see Whitham [210]). For a weak shock waveδ = 0(ε), and consequently, to the first order of approximations in ε, we have

s = 1 + (Λδ/2) +O(ε2), (4.6.40)

where Λ is same as in (4.6.32). Thus, the shock conditions (4.6.38) valid upto the first order of approximation in ε are

ρ = 1 + δ +O(ε2), u = δ +O(ε2),h = (b2o/2)(1 + 2δ) +O(ε2), p = p0 + δa2

o +O(ε2).(4.6.41)

Along with (4.6.41), the requirement that t = ψ, and (4.6.37) must be satisfiedat the shock φ = Φ(ψ), which is yet to be determined.

Now if ε→0, the shock front tends to the characteristic φ = 0; indeed, theequation of the shock path in the characteristic plane may be written as

φ = εΦ(1)(ψ) +O(ε2), (4.6.42)

and a quantity z(φ, ψ) at the shock front, assuming that it has a Taylorexpansion in φ, can be evaluated at φ = 0, i.e.,

z|φ=Φ(ψ) = z(o)(0, ψ) + εz(1)(0, ψ) + z(0),φ (0, ψ)Φ(1)(ψ) +O(ε2).

In view of (4.6.11), (4.6.37), (4.6.40), (4.6.41) and (4.6.42), it follows that theboundary conditions for the first order variables at the shock are

ρ(1) = u(1) = 2(s− 1)/Λ, p(1) = 2a2o(s− 1)/Λ, h(1) = 2b2o(s− 1)/Λ,

t(1) = 0, x(1),ψ = t

(1),ψ + (Λ/2)ρ(1) + dΦ(1)/dψ at φ = 0. (4.6.43)

Equations (4.6.13) together with the boundary conditions (4.6.35) and (4.6.43)can be solved by using Laplace transforms; indeed, the solution of (4.6.22),which satisfies the transformed boundary condition (4.6.35), i.e., u(1) = 1/ξ,and remains bounded as ψ→∞, is

u(1) = 1/ξ, (4.6.44)

and the complete solution, similar to (4.6.24), can easily be obtained. Theshock locus can be obtained from (4.6.43)5, which together with (4.6.13)2,yields

dΦ(1)/dψ = (2 − Λ)(u(1)/2) + c(1) at φ = 0. (4.6.45)

In view of the equations (4.6.14), equations (4.6.43) yield c(1) = (Λ − 1)u(1)

and, thus, (4.6.45) becomes

dΦ(1)/dψ = (Λ/2)u(1) at φ = 0. (4.6.46)

The value of u(1) at φ = 0 can be obtained from (4.6.44) by making use of


the Tauberian relation, i.e., limφ→0

u(1) = limξ→∞

ξu(1)(ξ, ψ) = 1, and thus (4.6.46)

yields Φ(1) = (Λ/2)ψ; the shock path in the characteristic plane is, therefore,

φ = ε(Λ/2)ψ. (4.6.47)

The shock path in the physical plane can also be obtained by noting thatx = ψ + εx(1) +O(ε2) and t = φ+ ψ + εt(1) +O(ε2), which imply that

φ = t− x− ε(t(1) − x(1)) +O(ε2), (4.6.48)

and similarly (4.6.13)2, in view of the foregoing results, implies that

t(1) − x(1) = −Λψ. (4.6.49)

Thus, using (4.6.47) and (4.6.49) in (4.6.48), we obtain the shock path, to thefirst order in ε, in the x− t plane as

x = t(1 + εΛ

2) +O(ε2), (4.6.50)

which in the absence of magnetic field, i.e., bo→0 reduces exactly to the oneobtained in [210]. Since the quantity Λ depends on the dimensionless Alfvenspeed bo and the specific heat ratio γ, it may be remarked that for γ < 2, anincrease in bo (i.e., the magnetic field strength) causes an increase in the shockspeed to the first order, relative to what it would have been in the absence ofthe magnetic field; however for γ > 2, the behavior is just the opposite. It isinteresting to note that for γ = 2, the shock speed to the first order is not atall influenced by the magnetic field strength. It may be pointed out that γ = 2corresponds to an ideal plasma with transverse magnetic field, whereas γ > 2is suitable for barotropic fluids in relativistic astrophysics and cosmology.

4.7 Relatively Undistorted Waves

Varley and Cumberbatch [206] introduced the method of relatively undis-torted waves to take into account nonlinear phenomena which are governed bynonlinear equations. Based on this method, which places no restriction on thewave amplitude, Dunwoody [51] has discussed high frequency plane waves inideal gases with internal dissipation. This method, which has been discussedin detail by Seymour and Mortell [165], proposes an expansion scheme thataccounts for amplitude dispersion and shock formation.

The advantage of this method, which makes no assumption on the mag-nitude of a disturbance, lies in the fact that the solution can be obtained bysolving ordinary differential equations; a study of this problem has been madeby Radha and Sharma [146]. In order to illustrate this method, we consider

4.7 Relatively Undistorted Waves 121

disturbances in a one-dimensional unsteady flow of a relaxing gas in a ductof cross sectional area A(x), where x is the distance along the duct. The gasmolecules have only one lagging internal mode (i.e., vibrational relaxation)and the various transport effects are negligible. The governing equations, us-ing summation convention, can be written in the form (see Clarke [37])

ui,t + Aijuj,x + Bi = 0, i, j = 1, 2, 3, 4 (4.7.1)

where ui and Bi are the components of column vectors u and B defined asu = (ρ, u, σ, p)tr ,B = (ρuΩ, 0,−Q, ρa2uΩ + (γ − 1)ρQ)tr with Ω = A

′

/A.Aij are the components of 4 × 4 matrix A with nonzero components A11 =A22 = A33 = A44 = u, A12 = ρ, A24 = 1/ρ and A42 = ρa2. Here u is theparticle velocity along the x-axis, t the time, ρ the density, p the pressure,σ the vibrational energy, γ the frozen specific heat ratio, a = (γp/ρ)1/2 thefrozen speed of sound in the gas, and A′ ≡ dA/dx. The quantity Q, whichis a known function of p, ρ and σ, denotes the rate of change of vibrationalenergy. The situation Q = 0, corresponds to a physical process involving norelaxation; indeed, it includes both the cases in which the vibrational mode iseither inactive or follows the translational mode according as the flow is eitherfrozen (σ = constant) or in equilibrium (σ = σ∗), where σ∗ is the equilibriumvalue of σ evaluated at local p and ρ. Following Scott and Johannesen [163],the entity Q is given by

Q = (σ∗ − σ)/τ, σ∗ = σ0 + c(ρρ0)−1(pρ0 − ρp0), (4.7.2)

where the suffix 0 refers to the initial rest condition, and the quantities τ and c,which are respectively the relaxation time and the ratio of vibrational specificheat to the specific gas constant, are assumed to be constant. The solutionvector u is said to define a relatively undistorted wave, if there exists a familyof propagating wavelets φ(x, t) = constant, such that the magnitude of the rateof change of u moving with the wavelet is small compared with the magnitudeof rate of change of u at fixed t. Let us consider the transformation, x = x, t =T (x, φ), from (x, t) to (x, φ) coordinate system, and let u(x, t) = U(x, φ). Thena relatively undistorted wave is defined by the relation ‖U,x‖ ‖u,x‖ , where‖·‖ denotes the Euclidean norm of a vector. Further, since U,x = u,x+T,xu,t,in a relatively undistorted wave u,x ' −T,xu,t holds, and consequently

‖U,x‖ ‖u,t‖ . (4.7.3)

A special example of a relatively undistorted wave is an acceleration front,where [U,x] = 0, while in general [u,t] 6= 0, and hence (4.7.3) is triviallysatisfied.

In terms of independent variables (x, φ), equation (4.7.1) becomes

(Aij (T,x) − δij) uj,t = AijU

j,x + Bi, (4.7.4)

where δij is the Kronecker function. Equations (4.7.3) and (4.7.4) are


compatible if ‖B‖ = O (‖AU,x‖), while (T,x)−1

is an eigenvalue of A.Otherwise, (4.7.4) would completely determine the ui,t as linear forms in

the U i,x, and (4.7.3) could not hold. Accordingly, in such waves, where

det(Aij − δij (T,x)

−1)

= 0, U i must satisfy the compatibility condition

Li(AijU

j,x + Bi

)= 0, (4.7.5)

for every left eigenvector L of A corresponding to the eigenvalue (T,x)−1.

Thus, the essential idea underlying this method is based on a scheme of suc-cessive approximations to the system (4.7.4) which, to a first approximation,is replaced by (

Aij − δij (T,x)−1)U j,φ = 0. (4.7.6)

Then, if R = (Ri) is the right eigenvector of A corresponding to the eigenvalue(T )−1

,x , equation (4.7.6) implies that to a first approximation

U i,φ = k(φ, x)Ri, (4.7.7)

for some scalar k(φ, x). It is important to appreciate that equations (4.7.7) areapproximate and, in general, cannot be integrated to obtain relations in U i

which are uniformly valid for all time. The terms which have been neglected inarriving at (4.7.7) will, in general, ultimately produce first order contributionsto U i.

The matrix A has eigenvalues u±a and u (with multiplicity two); here weare concerned with the solution in the region x > x0, where a motion consistingof only one component wave, associated with the eigenvalue (T,x)

−1 = u+ a,is perturbed at the boundary x = x0 by an applied pressure p(x0, t) = Π(t). Itmay be noticed that for nonlinear systems, there is, in general, no superposi-tion principle so that when more than one wave mode is excited, the propaga-tion of the individual component waves cannot be calculated independently.Consequently, the problem involving nonlinear interaction of component wavesneeds a different approach. The left and right eigenvectors of A correspondingto the eigenvalue

(T,x)−1 = u+ a (4.7.8)

areL = (0, ρa, 0, 1), R = (ρ, a, 0, ρa2)tr . (4.7.9)

4.7.1 Finite amplitude disturbances

Let us consider the situation when the disturbance, which is headed bythe front φ(x, t) = 0, is moving into a region, where prior to its arrival thegas is in a uniform state at rest with u = 0, p = p0, ρ = ρ0 and σ = σ0. Itis possible to choose the label, φ, of each wavelet so that φ = t on x = x0;consequently the boundary conditions for p and T become

p = Π(φ), T = φ at x = x0. (4.7.10)


To a first approximation, conditions in the wave region, associated with(T,x)

−1 = u + a, are determined by the differential relations (4.7.7), whichcan be formally integrated subject to the uniform reference values u = 0,p = p0, ρ = ρ0 and σ = σ0 on the leading front φ = 0. Thus we have

ρ = ρ0(p/p0)1/γ , u = 2a0(γ − 1)−1

(p/p0)

(γ−1)/2γ − 1,

a = a0(p/p0)(γ−1)/2γ , σ = σ0,

(4.7.11)

which hold at any x on any wavelet φ = constant. Equation (4.7.5), in viewof (4.7.9)1 and (4.7.11), provides the following transport equations for thevariation of p and T at each wavelet φ = constant.

p,x +(γ − 1)ρ0

4a0F (p)

(p

p0

) 1γ

2a30Ω

γ − 1

(p

p0

) γ−1γ

(p

p0

) (γ−1)2γ

− 1

+ (γ − 1)Q

= 0,

(4.7.12)

T,x = (γ − 1)/2a0F (p), (4.7.13)

where F (p) = ((γ + 1)/2)(p/p0)(γ−1)/2γ − 1. Equation (4.7.12), on using

(4.7.2) and (4.7.11), may be integrated using the boundary condition (4.7.10)1to give p = P (x,Π(φ)); once p is known, equation (4.7.13) may be integrated,subject to (4.7.10)2, to determine t = T (x, φ). Subsequently p(x, t) may beobtained, and hence ρ(x, t), u(x, t) and a(x, t). As a matter of fact, the inte-gration of (4.7.13) leads to the determination of the location of φ wavelets in

the form T = φ +

∫ x

x0

Λ(s,Π(φ))ds, where Λ = (γ − 1)/(2a0F (p(s,Π(φ)))).

From this result, it follows immediately that a shock forms on the wavelet φs

at the point xs, where 1 + Π′∫ xs

x0

Λ,Π(X,Π(φs))dX = 0.

The above results indicate that both the amplitude dispersion and shockformation along any wavelet depend on the amplitude, Π(φ), carried by thatwavelet.

4.7.2 Small amplitude waves

In the small amplitude limit, equations (4.7.12) and (4.7.13) can be lin-earized about the uniform reference state p = p0, ρ = ρ0, σ = σ0, u = 0, toyield

p1,x + (α+ Ω/2)p1 = 0, T,x =1 − (γ + 1)(2ρ0a

20)

−1p1

a−10 , (4.7.14)

where p1 is the small perturbation of the equilibrium value p0 and

α = (γ − 1)2c/2γa0τ,

which serves as the amplitude attenuation rate on account of relaxation; it maybe noted that α is identical with the absorption rate defined by Johannesenand Scott [87] and Scott and Johannesen [163].


The ideal gas case corresponds to α = 0. The boundary conditions for p1

and T which follow from (4.7.10) can be rewritten as

p1 = Π(φ), T = φ at x = x0, (4.7.15)

where∣∣∣Π(φ)

∣∣∣ = |Π(φ) − p0| 1. Equations (4.7.14), together with (4.7.15),

yield on integration

p1 = Π(φ)ψ(x), (4.7.16)

a0(T − φ) = x− x0 − (γ + 1)(Π(φ)/2ρ0a20)J(x0, x, α), (4.7.17)

where J(x0, x, α) =

∫ x

x0

(A(s)/A0)−1/2 exp(−α(s− x0)) ds, and

ψ(x) = (A/A0)−1/2 exp(−α(x − x0)).

Equation (4.7.16) implies that for each wavelet φ = constant, which decaysexponentially, the attenuation factor is independent of the amplitude Π(φ)carried by the wavelet. However, conditions at any x on a wavelet φ1 = con-stant are determined by the signal carried by φ1, and are independent of theprecursor wavelets 0 ≤ φ < φ1.

It may be recalled that for plane (m = 0) and radially symmetric (m = 1, 2)flow configurations, (A/A0) = (x/x0)

m; and therefore the integral J convergesto a finite limit J0 as x → ∞, i.e.,

limx→∞

J(x0, x, α) ≡ J0 =

1/α, (plane)

(πx0/α)1/2 exp(αx0)erfc(√αx0), (cylindrical)

x0E(αx0) exp(αx0), (spherical)(4.7.18)

where erfc(x) = 2π−1/2∫∞x

exp(−t2) dt and E(x) =∫∞xt−1 exp(−t) dt are,

respectively, the complementary error function and the exponential integral.

Thus J can be expressed as

J = J0(1 −K(x)), (4.7.19)

where

K(x) =

exp(−α(x − x0)), (plane)erfc(

√αx)/erfc(

√αx0), (cylindrical)

E(αx)/E(αx0), (spherical).

Equation (4.7.17) indicates that a shock first forms at (xs, φ), where the min-imum value of x (i.e., xs) is given by the solution of

H(x) ≡ 1 − (Π′

(φ)/b)(1 −K(x)) = 0, (4.7.20)

where b = 2ρ0a30/(γ + 1)J0 > 0 and Π

′

(φ) ≡ dΠ/dφ; since the expression(1 −K(x)) monotonically increases from 0 to 1 as x increases from x0 to ∞,it follows that a shock can only occur if Π

′

> b > 0, and subsequently H(x)


first vanishes at that wavelet φ where Π′

(φ) is greatest. Expression for theshock formation distance on the leading wavefront φ = 0, where (4.7.20) isexact, was pointed out by Johannesen and Scott [87]. It may be noted thatthe relatively undistorted approximation is valid only if

∣∣∣Π′

/Π∣∣∣

∣∣∣(α+ Ω/2)1 + (γ + 1)(2ρ0a

20)

−1Πψ(x)a0H

∣∣∣ , (4.7.21)

which, in fact, corresponds to the slow modulation approximation (see Varleyand Cumberbatch [204]). As discussed above, a shock wave may be initiatedin the flow region, and once it is formed, it will propagate by separating theportions of the continuous region. When the shock is weak, its location canbe found from the equal area rule (see Whitham [210])

2

∫ φ2

φ1

Π(t) dt = (φ2 − φ1)

Π(φ1) + Π(φ2), (4.7.22)

where φ1 and φ2 are the wavelets ahead of and behind the shock. For a weakshock propagating into an undisturbed region, where Π(φ1) = 0 for φ1 ≤ 0,equation (4.7.22), on using (4.7.17) and (4.7.19), becomes

∫ φ2

0

Π(t) dt = (γ + 1)Π2(φ2)(1 −K(x))J0/4ρ0a30. (4.7.23)

Figure 4.7.1: The variation of the dimensionless pressure against the dimensionlessvariable ξ (defined as (x − aot)/xo, using the initial profile Ia (defined in (4.7.25)).The distortion of the profile is delineated at various distances before and after shock

formation on the leading wavelet, φ = 0 in cylindrically (m = 1) and spherically

(m = 2) symmetric flow configurations of a nonrelaxing gas (α = 0) for δ = 0.35 andγ = 1.4. Shock forms at (i) xs = 4.79 (cylindrical) and (ii) xs = 10.81 (spherical).For m = 1, x = 1 (Ia), x = 2 (IIa), x = 3 (IIIa), x = 4 (IVa), x = 4.79 (Va), x = 7(VIa); for m = 2, x = 1 (Ib), x = 2 (IIb), x = 4 (IIIb), x = 6 (IVb), x = 8 (Vb),x = 10.81 (VIb), xs = 12 (VIIb).

http://www.crcnetbase.com/action/showImage?doi=10.1201/9781439836910-c4&iName=master.img-604.jpg&w=215&h=172


This equation, in the limit α →0, yields a result which agrees fully withthe result obtained by Whitham [210] for nonrelaxing gases. Assuming thatthe integral on the left-hand side of (4.7.23) is bounded, it follows that forsufficiently large x, Π(φ2) stays in proportion to J−1/2. Consequently, in viewof (4.7.13) and (4.7.16), the pressure jump, [p], across the shock, which isdefined as the measure of shock amplitude, decays like

[ p ] ∝

exp(−αx), (plane)

x−1/2 exp(−αx), (cylindrical)x−1 exp(−αx), (spherical)

where α = αx0 and x = x/x0. For a nonrelaxing gas (α =0), we find that asx→ ∞, the shock decays like

[ p ] ∼ x0

x−1/2, (plane);

x−3/4, (cylindrical);x−1(ln x)−1/2, (spherical).

(4.7.24)

These asymptotic results for nonrelaxing gases are in full accord with theearlier results (see Whitham [210]).

In order to trace the early history of shock decay after its formation on theleading wave front φ = 0, we consider a special case in which the disturbanceat the boundary x = x0 is a pulse defined as

Π(φ) =

0, φ < 0δ sin(a0φ/x0), 0 < φ < πx0/a0 ;0, φ > πx0/a0

δ > 0. (4.7.25)

It may be recalled that H(x), in equation (4.7.20), first vanishes on thatwavelet for which Π

′

(φ) has a maximum value greater than b, i.e., δa0/x0 > b;consequently the shock first forms on the wavelet φ = 0 at a distance x = xs,nearest to x0, given by the solution of

E(x) ≡ ((γ + 1)/2)(1 −K(x))δJ0 = 1, (4.7.26)

where δ = δ/ρ0a20 and J0 = J0/x0 are the dimensionless constants. The dis-

tortion of the pulse, defined in (4.7.25), is shown in Figs 1 and 2. The usualsteepening of the compressive phase and flattening of the expansive phase ofthe wavelet are quite evident from the distorted profiles. The depression andflattening of the peaks with increasing x, which become even more pronouncedon account of relaxation or the wavefront geometry, indicate that the distur-bance is undergoing a general attenuation. Equation (4.7.20) shows that forα = 0, the pressure profile develops a vertical slope, thereby indicating theappearance of a shock at x = 4.79 and 10.81 in cylindrically and sphericallysymmetric flows respectively, while for α = 0.05, shocks in respective flowconfigurations develop at x = 5.31 and 20.92. Thus, the presence of relaxationor an increase in the wavefront curvature both serve to delay the onset of a


shock. Equation (4.7.23), in view of (4.7.25), then simplifies to give φ2 on theshock by the following relation,

sin φ2 = 2(E(x) − 1)1/2/E(x), (4.7.27)

where φ2 = φ2a0/x0 is the dimensionless variable. Equation (4.7.27), togetherwith (4.7.16) and (4.7.25), implies

[ p ] = 2δx−m/2 exp(−α(x− 1))(E(x) − 1)1/2/E(x), (4.7.28)

where p = p/ρ0a20 is the dimensionless pressure. Equation (4.7.28) shows that

the shock after its formation on φ = 0, at x = xs > x0, grows to a maximumstrength at x = x1 > xs, where x1 is given by the solution of E(x) = 2, andthen decays ultimately in proportion to x−m/2 exp(−αx) as concluded above.

Figure 4.7.2: Development of the dimensionless pressure against ξ, using the initialprofile Ia (defined in (4.7.25)). The distortion of the profile is delineated at various

distances before and after shock formation on the leading wavelet for δ = 0.35, α =0.05 and γ = 1.4. The figure in the inset shows the development on the waveheadφ = 0, for m = 2 when the profile Vb develops a vertical slope signifying theappearance of a shock at xs = 20.92 in a relaxing gas. Shock forms at (i) xs = 5.31(cylindrical). For m = 1, x = 1 and (ii) xs(Ia), x = 2 (IIa), x = 3 (IIIa), x = 4(IVa),x = 5.31 (Va), x = 7 (VIa); for m = 2, x = 1 (Ib), x = 2 (IIb), x = 3 (IIIb), x = 4(IVb), x = 5 (Vb), x = 10 (VIb), xs = 20.92 (VIIb).

Let us now consider a special case in which the small disturbance at theboundary, x = x0, has a periodic wave form given by

Π(φ) = δ sin(φ), (4.7.29)



where δ < 0 and φ = a0φ/x0, and consider the development over one cycle

0 ≤ φ ≤ 2π. In this case, the shock first forms on the wavelet φ = π at adistance x = xs, nearest to x0, given by the solution of (4.7.20); this canoccur, of course, if | δ |a0/x0 > b. Equations (4.7.17) and (4.7.22) are satisfied

on the shock, if φ1 + φ2 = 2π and φ1 − φ2 = 2θ, where θ is given by thesolution of

θ/ sin θ = (γ + 1)(1 −K(x))J0| δ |/2, (4.7.30)

with δ = δ/ρ0a20. The discontinuity in p at the shock is, therefore, given by

[ p ] = 2| δ |(x/x0)−m/2 exp(−α(x− x0)) sin θ, (4.7.31)

which shows that the shock starts with zero strength corresponding to θ → 0 atx = xs, given by the solution of (4.7.30). The shock amplitude decays to zero asθ → θm, where θm is given by the solution of θm/ sin(θm) = (γ+1)|δ|J0/2. Theshock strength grows for θ lying in the interval 0 < θ < θ∗, whereas it decaysover the interval θ∗ < θ < θm, thus exhibiting a maximum corresponding toθ = θ∗ at a distance x = x∗, where both θ∗ and x∗ can be determined using(4.7.30) and the relation

4(α+m/2x∗)(sin θ∗ − θ∗ cos θ∗) = (γ + 1)| δ |x−m/2∗ sin(2θ∗) exp(−α(x∗ − 1)).

Figure 4.7.3: (a) The variation of the dimensionless pressure against ξ using theinitial profile defined in (4.7.29). The distortion of the profile is exhibited at various

distances before and after shock formation on the wavelet, φ = π, in cylindrically

and spherically symmetric flow in a relaxing gas for δ = −0.35, α = 0.05 and γ =1.4. Shock forms at (i) xs = 5.31 (cylindrical) and (ii) xs = 20.92 (spherical). (b)development of the pressure profile against ξ using the same initial profile (as in

(a)), at the wavehead φ = π, for δ = −0.35, α = 0 and γ = 1.4. Shock forms at (i)xs = 4.79 (cylindrical) and (ii) xs = 10.81 (spherical).

The shock decays ultimately with θ = θm, x→ ∞, according to the law

[ p ] ∼ λx−m/2 exp(−αx), (4.7.32)



where λ = 4θm/(γ + 1)J0. However, in the absence of relaxation, it followsfrom (4.7.30) and (4.7.31) that as θ → π , x → ∞, the shock decays like

[ p ] ∼ 4π(γ + 1)−1

x−1, (plane);(2x)−1 (cylindrical);(x ln x)−1, (spherical).

(4.7.33)

The development of the pressure profile with initial disturbance given by(4.7.29), and the subsequent shock formation at the wavehead φ = π are ex-hibited in Figs. 3a (with relaxation) and 3b (without relaxation), showing ineffect that the influence of relaxation is to delay the onset of a shock wave.The evolutionary behavior of the pressure profile before and after the shockformation, exhibited in Fig. 4.7.3a, follows a slightly different pattern fromthat depicted in Fig. 4.7.3b, in the sense that the profile, which eventuallyfolds into itself, develops concavity with the peak slightly advanced. The evo-lutionary behavior of shocks evolving from profile (4.7.29) are depicted in Fig.4.7.4; indeed, the shock after its formation at x = xs grows to a maximumstrength at x = x∗, and then decays according to the law (4.7.32) or (4.7.33)depending on whether the gas is relaxing or nonrelaxing respectively. How-ever, in the absence of relaxation, a shock resulting from (4.7.29) decays fasterthan that evolving from the pulse (4.7.25).

Figure 4.7.4: Growth and decay of a shock wave which appears first at x = xs

on the wavelet φ = π; comparison is made with the behavior from the same initialand boundary data for a nonrelaxing (α = 0) gas. The effects of relaxation and thewavefront curvature on the shock formation distance (xs), and the distance x∗ at

which the shock strength attains maximum strength are exhibited; δ = −0.35.



4.7.3 Waves with amplitude not-so-small

We now seek to explore the predictions of subsection 4.6.1 by consideringthe amplitude limit to be not so small; in fact, here we extend the analy-sis of the preceding subsection to the next order by including the nonlinearquadratic terms in the perturbed flow quantities p1, ρ1 etc, which were other-wise ignored in the subsection 4.6.2 while writing equations (4.7.14). In thisinstance, equations (4.7.12) and (4.7.13) reduce to

p1,x + (α+ Ω/2)p1 − q(x)p21 = 0, (4.7.34)

T,x = 1− (γ + 1)(2ρ0a20)

−1p1 + 3(γ + 1)2(8ρ20a

40)

−1p21/a0, (4.7.35)

where q(x) = αγ + (3 − γ)Ω/4/2ρ0a20. Equations (4.7.34) and (4.7.35), to-

gether with the boundary conditions for p1 and T at x = x0, yield on integra-tion

p1 = Π(φ)ψ(x)

1− Π(φ)M(x)−1

, (4.7.36)

a0(T − φ) = (x− x0) +γ + 1

2ρ0a20

∫ x

x0

(3(γ + 1)

4ρ0a20

p21 − p1

)dx, (4.7.37)

where M(x) =∫ xx0ψ(x

′

)q(x′

) dx′

and φ is the parameter distinguishing the

wavelets from each other. Equations (4.7.36) and (4.7.37) indicate that in con-trast to the results of subsection 4.6.2, both the rate at which the amplitudevaries on any wavelet and the time taken to form a shock are influenced by theamplitude of the signal carried by the wavelet. Indeed, for ΠM < 0 (respec-tively, > 0) the amplitude decays more rapidly (respectively, slowly) than thatpredicted in subsection 4.6.2; the computed results are shown in Figs 5(a,b).The results computed for small amplitude disturbances in subsection 4.6.2 arealso incorporated into Figures 5(a) and 5(b) for the sake of comparison andcompleteness. The numerical results indicate that the shock arrival time on aparticular wavelet increases as compared to the case discussed in subsection4.6.2. Behind the shock the wavelets are determined by (4.7.37). Accordingto Pfriem’s rule, the shock velocity dT/dx |s of a weak shock is given by theaverage of the characteristic speeds ahead of and behind the shock. Thus, tothe present approximation

dT/dx |s =1 − (γ + 1)(4ρ0a

20)

−1(p1 − 3(γ + 1)p21/4ρ0a

20)/a0, (4.7.38)

where p1 = p1(x, φs) denotes the value at position x on the shock. Evaluating(4.7.37) along the shock, we obtain the shock trajectory T = T (x) and, thus,an alternative expression for the shock speed

dT

dx

∣∣∣∣s

=dφsdx

1 − γ + 1

2ρ0a30

∫ x

x0

p1,φ dx+3(γ + 1)2

4ρ20a

50

∫ x

x0

p1p1,φ dx

+1

a0

1 − γ + 1

2ρ0a20

p1 +3(γ + 1)2

8ρ20a

40

p21

. (4.7.39)


Eliminating dT/dx |s between (4.7.38) and (4.7.39), we obtain a differentialequation for the unknown φs, which can be integrated numerically subject tothe condition, x = xs at φ=0. Having determined φs at different locations x,the shock strength at these respective locations, and hence the evolutionarybehavior of shock decay, can be determined from (4.7.36).

Figure 4.7.5: (a) solution for pressure in the pulse region of a cylindrically symmet-ric flow of a relaxing gas at various distances when the wave amplitude is not-so-small(see the dashed lines). Comparison is shown with the corresponding situation whenthe wave amplitude is small (see the solid lines); the initial profile is defined in(4.7.25). (b) Solution for pressure in the pulse region of a spherically symmetricflow of a relaxing gas at various distances when the disturbance amplitude is not-so-small (see the dashed lines). Comparison is shown with the corresponding situationswhen the wave amplitude is small (see the solid lines); the initial profile is defined

in (4.7.25). For both (a) and (b), α = 0.05, δ = 0.35, γ = 1.4.

Chapter 5

Asymptotic Waves for Quasilinear

Systems

In this chapter, we shall assume the existence of appropriate asymptotic ex-pansions, and derive asymptotic equations for hyperbolic PDEs. These equa-tions display the qualitative effects of dissipation or dispersion balancing non-linearity, and are easier to study analytically. In this way, the study of com-plicated models reduces to that of models of asymptotic approximations ex-pressed by a hierarchy of equations, which facilitate numerical calculations;this is often the only way that progress can be made to analyze complicatedsystems. Essential ideas underlying these methods may be found in earlierpublications; see for example Boillat [19], Taniuti, Asano and their coworkers([7], [194]), Seymour and Varley [166], Germain [61], Roseau [155], Jeffrey andKawahara [82], Fusco, Engelbrecht and their coworkers ([58], [59]), Cramerand Sen [43], Kluwick and Cox [94], and Cox and Kluwick [42]. An account ofsome of the rigorous results which deal with convergence of such expansionsmay be found in [47].

5.1 Weakly Nonlinear Geometrical Optics

Here we shall discuss the behavior of certain oscillatory solutions of quasi-linear hyperbolic systems based on the theory of weakly nonlinear geometri-cal optics (WNGO), which is an asymptotic method and whose objective isto understand the laws governing the propagation and interaction of smallamplitude high frequency waves in hyperbolic PDEs; this method describesasymptotic expansions for solutions satisfying initial data oscillating with highfrequency and small amplitude. The propagation of the small amplitude waves,considered over long time-intervals, is referred to as weakly nonlinear. Mostwaves are of rather small amplitude; indeed, small amplitude high frequencyshort waves are frequently encountered. Linearized theory satisfactorily de-scribes such waves only for a finite time; after a sufficiently long time-interval,the cumulative nonlinear effects lead to a significant change in the wave field.the basic principle involved in the methodology used for treating weakly non-linear waves lies in a systematic use of the method of multiple scales. Each

133

134 5. Asymptotic Waves for Quasilinear Systems

dissipative mechanism present in the flow, such as rate dependence of themedium, or geometrical dissipation due to nonplanar wave fronts, or inhomo-geneity of the medium, defines a local characteristic length (or time) scale,that arises in a natural way. Indeed, WNGO is based on the assumption thatthe wave length of the wave is much smaller than any other characteristiclength scale in the problem. When this assumption is satisfied, i.e., the time(or length) scale defined by the dissipative mechanism is large compared withthe time (or length) scale associated with the boundary data, the wave isreferred to as a short wave, or a high frequency wave. For instance, considerthe motion excited by the boundary data u(0, t) = af(t/τ) on x = 0, where arepresents the size of the boundary data, while τ is the applied period or pulselength. When we normalize the boundary data, by a suitable nondimensionalquantity and t by the time scale τr, which is defined by the dissipative mech-anism present in the flow, the geometrical acoustics limit then correspondsto the high frequency condition ε = τ/τr 1; in fact, there is a region inthe neighborhood of the front where the nonlinear convection associated withthe high frequency characteristics is important. For both significant nonlin-ear distortion and dissipation, in this high frequency limit, we must have thenondimensional amplitude |a| = O(ε). In terms of these normalized variables,the boundary condition becomes u(0, t) = εg(t/ε), which describes oscillationsof small amplitude ε and of frequency ε−1. For data of size ε, the method ofWNGO yields approximations, which are valid on time intervals typically ofsize ε−1. Following the pioneering work of Landau [99], Lighthill [108], Keller[91] and Whitham [210], a vast amount of literature has emerged on the de-velopment, both formal and rigorous, in the theory of WNGO. However, thedevelopment of these methods through systematic self-consistent perturba-tion schemes in one and several space dimensions is due to Chouqet-Bruhat[34], Varley and Cumberbatch ([204], [206]), Parker [140], Mortell and Varley[128], Seymour and Mortell [165], Hunter and Keller [75], Fusco [60], Mazdaand Rosales [115], and Joly, Metivier and Rauch [88]. For results based onnumerical computations of model equations of weakly nonlinear ray theory,the reader is referred to Prasad [145]. Using the systematic procedure, alludedto and applied by the above mentioned authors, we study here certain aspectsof WNGO, which are largely based on our papers ([146], [171], [173], [190],and [192]).

5.1.1 High frequency processes

Consider a quasilinear system of hyperbolic PDEs in a single space variable

u,t + A(u)u,x + b(u) = 0,−∞ < x <∞, t > 0 (5.1.1)

where u and b are n-component vectors and A is a n× n matrix. Let uo bea known constant solution of (5.1.1), such that b(uo) = 0. We consider smallamplitude variations in u from the equilibrium state u = uo, which are of thesize ε, described earlier; and look for a small amplitude high frequency wave

5.1 Weakly Nonlinear Geometrical Optics 135

solution of (5.1.1), representing a single wave front, and valid for times of theorder of O(1/ε). Then the WNGO ansatz for the situation described above isthe formal expansion for the solution of (5.1.1)

u = uo + εu1(x, t, θ) + ε2u2(x, t, θ) + . . . (5.1.2)

where θ is a fast variable defined as θ = φ(x, t)/ε with φ as the phase functionto be determined. The wave number k and wave frequency ω are defined byk = φ,x and ω = −φ,t. By considering the Taylor expansion of A and b in theneighborhood of uo, and taking into account (5.1.2), equation (5.1.1) impliesthat

O(εo) : (Ao − λI)u1,θ = 0O(ε1) : (Ao − λI)u2,θ = −(∇A)ou1u1,θ − u1,t + Aou1,x + (∇b)ou1φ−1

,x ,(5.1.3)

where the subscript o refers to the evaluation at u = uo, I is the n× n unitmatrix, λ = −φ,t/φ,x, and ∇ is the gradient operator with respect to thecomponents of u. The equation (5.1.3)1 implies that for a particular choice ofthe eigenvalue λo (assuming that it is simple),

u1 = π(x, t, θ)Ro, (5.1.4)

where π is a scalar oscillatory function to be determined, and Ro is the righteigenvector of Ao corresponding to the eigenvalue λo. The phase φ(x, t) isdetermined by

φ,t + λoφ,x = 0. (5.1.5)

It may be noticed that if φ(x, 0) = x, then (5.1.5) implies that φ(x, t) = x−λot.Let Lo be the left eigenvector of Ao corresponding to its eigenvalue λo,

satisfying the normalization condition Lo·Ro = 1. Then if (5.1.3)2 is multipliedby Lo on the left, we obtain along the characteristic curves associated with(5.1.5) the following transport equation for π ;

π,τ + ν(π2/2),θ = −µπ, (5.1.6)

where ∂/∂τ = ∂/∂t+ λo∂/∂x, and

µ = Lo · ∇b|o ·Ro, ν = Lo · ∇A|o ·RoRo. (5.1.7)

If the initial condition for π is specified, i.e., π|τ=0 = πo(xo, θo) with xo =x|τ=o and θo = ε−1φ|τ=0, then the minimum time taken for the smooth solu-tion to breakdown can be computed explicitly. In fact, the initial conditionslead to a shock only when ν∂πo/∂θo < 0 and −ν∂πo/∂θo > µ; in passing,we remark that the presence of source term b in (5.1.1) makes the solutionexist for a longer time relative to what it would have been in the absence ofa source term. Further, if (5.1.1) has an associated conservative form, then(5.1.6), which is the proper conservation form, can be used to study the prop-agation of weak shocks (see [75]).


5.1.2 Nonlinear geometrical acoustics solution in a relaxinggas

The foregoing asymptotic analysis can be used to study the small ampli-tude high frequency wave solution to the one dimensional unsteady flow of arelaxing gas described by the system (4.6.1), i.e., ui,t+Aiju

j,x+Bi = 0, i, j =

1, 2, 3, 4, where the symbols have the same meaning, defined earlier. Here weare concerned with the motion consisting of only one component wave associ-ated with the eigenvalue λ = u+ a. The medium ahead of the wave is takento be uniform and at rest. The left and right eigenvectors of A correspondingto this eigenvalue are

L = (0, 1/(2a), 0, 1/(2ρa2)), R = (ρ, a, 0, ρa2), (5.1.8)

and the phase function φ(x, t) is given by φ(x, t) = x−aot, where the subscripto refers to the uniform state uo = (ρo, 0, σo, po) ahead of the wave. The trans-port equation for the wave amplitude π is given by (5.1.6) with coefficients µand ν determined as

µ = (α+ (aoΩ)/2) and ν = (γ + 1)ao/2, (5.1.9)

where α is the same as in (4.6.14)1. The characteristic field associated with πin (5.1.6) is defined by the equations

dx/dt = ao, dθ/dt = ((γ + 1)aoπ)/2. (5.1.10)

In view of (5.1.10), equation (5.1.6) can be written as dπ/dt = −ao(α+(Ω/2))πalong any characteristic curve belonging to this field, and yields on integration

π = πo(xo, θo)(A/Ao)−1/2 exp−αaot, (5.1.11)

along the rays x− aot = xo (constant). We look for an asymptotic solution ofthe hyperbolic system (4.6.1)

u = uo + εu1(x, t, θ) +O(ε2), (5.1.12)

satisfying the small amplitude oscillatory initial condition

u(x, 0) = εg(x, x/ε) +O(ε2), (5.1.13)

where g is smooth with a compact support; indeed, expansion (5.1.12) with u1

given by (5.1.4), where the wave amplitude π is given by (5.1.11), is uniformlyvalid to the leading order until shock waves have formed in the solution (seeMajda [116]). Using (5.1.11) in (5.1.10)2 we obtain the family of characteristiccurves parametrized by the fast variable θo as

θ− (γ + 1)

2aoπ

o(xo, θo)

∫ t

0

(A(xo + aot)

A(xo)

)−1/2

exp (−αaot)dt = θo. (5.1.14)

5.2 Far Field Behavior 137

It may be noticed that equations (5.1.11) and (5.1.14) are similar to the equa-tions (4.6.16) and (4.6.17), and therefore the discussion with regard to theshock formation and its subsequent propagation follows on parallel lines. Itmay be recalled that for plane (m = 0) and radially symmetric (m = 1, 2)flow configurations, (A/Ao) = (x/xo)

m, and therefore the integral in (5.1.14)converges to a finite limit as t → ∞. Thus, an approximate solution (5.1.12),satisfying (5.1.13), is given by

u = u0 + εRoπo(xo, θo)(A(xo + aot)/A(xo))

−1/2 exp (−aoαt),

where R is given by (5.1.8)2; the fast variable θo, given by (5.1.14), is chosensuch that θo = x/ε at t = 0, and the initial value of π is determined from(5.1.13) as πo(x, ξ) = g(x, x/ε)/ao. This completes the solution of (4.6.1)and (5.1.13); any multivalued overlap in this solution has to be resolved byintroducing shocks into the solution.

5.2 Far Field Behavior

When the characteristic time τ associated with the boundary data is largecompared with the time scale τr defined by the dissipative mechanism presentin the medium (i.e., δ = τr/τ << 1), the situation corresponds to the lowfrequency propagation condition. This means that the time and distances con-sidered are large in comparison to the relaxation time or relaxation length.Since at large distances, away from the source, any nonlinear convection isassociated with the low frequency characteristics, and as the principal signalin this region is centered on the equilibrium or low frequency characteristics,it is possible to introduce a reduced system of field equations which providesan approximate description of the wave process. Based on Whitham’s ideas,it was shown by Fusco [60] that for a quasilinear first order system of PDEswith several space variables involving a source term, it is possible to introducea reduced system which, in an asymptotic way, brings out the dissipative ef-fects produced by the source term against the typical nonlinear steepening ofthe waves. In fact, the wave motion is asymptotically described by a trans-port equation of Burger’s type that holds along the characteristic rays of thereduced system. We illustrate this procedure for the one dimensional nonequi-librium gas flow described by the system (4.6.1).

As the principal signal in the far field region is centered on the equilibriumcharacteristic, the system (4.6.1) is approximated by the following reducedsystem.

ρ,t + uρ,x + ρu,x + Ωρu = 0, u,t + uu,x + ρ−1p,x = 0,p,t + up,x + ρa2

∗(u,x + Ωu) = 0, Q(p, p, σ) = 0,(5.2.1)


where a∗ = (∂p/∂ρ)|1/2S,σ=σ∗

is the equilibrium speed of sound with S as specificentropy of the gas.

In order to study the influence of nonequilibrium relaxation in (4.6.1) onthe wave motion, associated with (5.2.1), we consider the following stretchingof the independent variables x = δ2x, t = δ2t. When expressed in termsof x and t, and then suppressing the tilde sign, the system (4.6.1) yields thefollowing set of equations

ρ,t + uρ,x + ρu,x + Ωρu = 0, u,t + uu,x + ρ−1p,x = 0,p,t + up,x + γp(u,x + Ωu) = −(γ − 1)ρQδ−2, (σ,t + uσ,x) = Qδ−2.

(5.2.2)In the limit δ →0, the above system yields the reduced system (5.2.1), and,therefore, the transformed system (5.2.2) may be regarded as a perturbedproblem of an equilibrium state characterized by (5.2.1); see Fusco [60].

We now look for an asymptotic solution of (5.2.2) exhibiting the characterof a progressive wave, i.e.,

f(x, t) = f0 + δf1(x, t, θ) + δ2f2(x, t, θ) + · · · , (5.2.3)

where f may denote any of the dependent variables ρ, u, σ and p; f0 refers tothe known constant state, θ = φ(x, t)/δ is a fast variable, and φ(x, t) is thephase function to be determined.

By introducing (5.2.3) and the Taylor’s expansion of Q about the uniformstate (ρ0, 0, σ0, p0) into the transformed equations (5.2.2)1 and (5.2.2)2 andcanceling the coefficients of δ0, δ1, we obtain the following system of first orderpartial differential equations for the first and second order variables

O(δ0) : ρ1,θφ,t + ρou1,θ

φ,x = 0, ρou1,θφ,t + p1,θ

φ,x = 0,O(δ1) : ρ2,θφ,t + ρou2,θφ,x = −ρ1,t − ρou1,x − mρou1

x + (u1ρ1,θ + ρ1u1,θ)φ,x,O(δ1) : ρou2,θ

φ,t + p2,θφ,x = −ρ1u1,θ

φ,t − ρou1u1,θφ,x − p1,x − ρou1,t

.(5.2.4)

Similarly, equations (5.2.2)3 and (5.2.2)4, on equating the coefficients of δ0

and δ−1, yield the following equations

O(δ0) : p1,θφ,t + ρoa

2ou1,θφ,x = −(γ − 1)ρoQ2, σ1,θ

φ,t = Q2,O(δ−1) : σ1 = c(p1 − (a2

o/γ)ρ1)/ρo,(5.2.5)

where Q2 =

c

τρ2o

(poρ

21

ρo− p1ρ1

)+cp2

τρo− cpoρ2

τρ2o

− σ2

τ

.

Eliminating Q2 between (5.2.5)1 and (5.2.5)2, and using (5.2.5)3 in theresulting equation, we get

γ1 + (γ − 1)cp1,θφ,t + ρoa

2oγu1,θ

φ,x − (γ − 1)ca2oρ1,θ

φ,t = 0. (5.2.6)

Equation (5.2.6), together with (5.2.4)1 and (5.2.4)2, constitutes a systemof three equations for the unknowns ρ1,θ

, u1,θand p1,θ

. The necessary andsufficient condition for this system to have a nontrivial solution is that the

5.2 Far Field Behavior 139

determinant of the coefficient matrix must vanish, i.e., −φt/φx = 0,±Γ ,

where Γ = a0

(cγ + γ − c)/(cγ2 − cγ + γ)

1/2. For −φt/φx = +Γ, the phase

function φ is determined as φ(x, t) = t − ((x − xo)/Γ) when φ(xo, t) = t; asthe vector of unknowns is collinear to the right null vector of the coefficientmatrix, we get

ρ(1) = Γ−2p(1), u(1) = (ρ0Γ)−1p(1), σ(1) = (γ−1)c ρ0(γ + (γ − 1)c)−1p(1).

(5.2.7)It may be noticed that Γ is a characteristic speed related to the reducedsystem (5.2.1). The system of equations for the second order variables, onmultiplying by the left null vector of the coefficient matrix, corresponding tothe value −φ,t/φ,x = Γ, and taking into account the relations (5.2.7), yieldsthe following transport equation for p(1) in the moving set of coordinates Xand ξ,

∂Xp(1) − λ p(1)∂ξp

(1) +mp(1)/2X = ω∂2ξξp

(1), (5.2.8)

where ∂X = ∂x + Γ−1∂t and the nonlinear and dissipation coefficients λ andω are given by

λ = (γ/ρ0ΓΛ1) γ + 1 + 2c(γ − 1) , ω = 2cτγΓa20 (γ − 1)/Λ12

,

with Λ1 = 2a20(cγ + γ − c).

Equation (5.2.8) is known as the generalized Burgers equation which allowsus to study in detail various effects that appear in the propagation of plane(m = 0), cylindrical (m = 1) and spherical (m = 2) waves in a dissipativemedium with a quadratic nonlinearity.

In contrast to the high frequency nonlinear solution, discussed earlier, asignificant feature of the low frequency solution in the far field is its continuousstructure, i.e., any convective steepening is always diffused by the dissipativenature of the relaxation. The reader is referred to Crighton and Scott [44] andManickam et al. [117] for a detailed discussion of the analytical and numericalsolutions of such an equation.

Remarks 5.2.1: In a special case where the coefficient ω vanishes in (5.2.8),the source term B involved in (4.7.1) does not cause dissipation; in this sit-uation, the asymptotic development (5.2.3) needs to be modified in order tosee how the higher order terms in (4.7.1) may influence the wave motion as-sociated with the characteristic speed of the reduced system. The transportequation for the wave amplitude in this case is the generalized Korteveg-deVries (K dV) equations. While considering the diffraction of a weakly nonlin-ear high frequency wave in a direction transverse to its rays for a first orderquasilinear system involving a source term, it can be shown that the wave am-plitude is governed by the Zabolotskaya-Khokhlov equation or by a modifiedKadomtsev-Petviashvili (KP) equation (see [59] and [60]).


5.3 Energy Dissipated across Shocks

We have noticed that the classical solutions of nonlinear hyperbolic con-servation laws, in one or several unknowns, break down in a finite time be-yond which the solutions have to be interpreted in the sense of distributions,known as weak solutions. These weak solutions often contain discontinuitiessupported along lower dimensional manifolds. The strength of the discontinu-ity is constrained through the Rankine-Hugoniot formula, which relates thejump in the solution to the unit normal to the discontinuity locus. Recently,motivated by their study on gravity currents, Montgomery and Moodie [125]have analyzed the effect of singular forcing terms on the Rankine-Hugoniotconditions in systems involving one space dimension. The singular forcingfunction considered, is a sum of a surface density and a volume density andthe latter does not contribute to the jump relation. Their results to severalspace dimensions were generalized in [191] using pull back of distributions; in-deed, the analysis in [191] sheds some light on the mechanism through whichthe terms involving the volume density cancel out, thereby not contributing tothe jump conditions. The analysis in this section is related to another problemin the system of conservation laws namely, the manipulation of conservationlaws through multiplication by polynomial nonlinearities. The resulting sys-tems contain distributional terms supported along singularity loci; these maybe regarded as singular forcing terms considered in [125]. Here we derive anexpression for these singular terms which have been interpreted as the energydissipated at the shocks. As an application, the expression for these singularterms is obtained from the transport equation governing the propagation ofsmall amplitude high frequency waves in hyperbolic conservation laws.

5.3.1 Formula for energy dissipated at shocks

We consider an autonomous scalar conservation law without source termfor a scalar valued function u on a domain Ω in IRn+1 namely,

n∑

j=0

∂jFj(u) = 0. (5.3.1)

Let the system (5.3.1), when multiplied by P (u), a polynomial or a smoothfunction of one variable, transform into

n∑

j=0

∂j(Fj(u)) = 0, (5.3.2)

5.3 Energy Dissipated across Shocks 141

for certain densities Fj(u) defined on Ω. The systems (5.3.1) and (5.3.2) areequivalent for smooth solutions u of (5.3.1), namely the identity

P (u)

n∑

j=0

∂jFj(u) =

n∑

j=0

∂j Fj(u),

holds for all smooth functions u. However, the class of weak solutions ofthis systems involving discontinuities, are different since the correspondingRankine-Hugoniot conditions differ. To restore the equivalence of the two sys-tems for a class of distributional solutions which are piecewise smooth with ajump discontinuity along a locus Φ(x) = 0, we modify the equation (5.3.2) byadding a distributional term to (5.3.2) namely,

P (u)

n∑

j=0

∂jFj(u) =

n∑

j=0

∂j(Fj(u)) +E, (5.3.3)

where the distribution E is supported on Φ(x) = 0. In the context of nonlineargeometrical optics, this singular term is interpreted as the energy dissipatedacross the shock locus Φ(x) = 0. We proceed to determine explicitly thedistribution E. Let φ be any arbitrary test function, then (5.3.3) gives

n∑

j=0

∫

Ω

P (u)∂jFj(u)φ =

n∑

j=0

〈P (u)∂jFj(u), φ〉

= 〈E, φ〉 +

n∑

j=0

〈∂j Fj(u), φ〉, (5.3.4)

in the sense of distributions. We proceed to simplify the term Σnj=0〈∂j Fj(u), φ〉on the right-hand side of (5.3.4):

n∑

j=0

〈∂j Fj(u), φ〉 = −∫

Ω

n∑

j=0

Fj(u)∂jφ

= −∫

Φ=0

n∑

j=0

[Fj ]njφdS +

∫

Ω

n∑

j=0

∂jFj(u)φ, (5.3.5)

where [f(u)] denotes the jump of f(u) across the locus of discontinuity Φ = 0,n = (n1, n2, . . . , nn) is the unit normal vector to Φ = 0 and dS is the areameasure on the surface. Since the function u is of class C1 up to the boundaryon either side of the discontinuity locus, the left-hand side of (5.3.4) transformsas

n∑

j=0

∫

Ω

P (u)∂jFj(u)φ =

n∑

j=0

∫

Ω

∂j Fj(u)φ, (5.3.6)


and so (5.3.4) and (5.3.5) prove that the distribution E is a smooth densitysupported on the discontinuity locus given by

E : φ 7→∫

φ=0

φ

n∑

j=0

[Fj(u)]njdS. (5.3.7)

We now prove a general result concerning the distribution E arising out ofmultiplying (5.3.1) by P (u).

Theorem 5.3.1 Assume that the conservation law

∂tF0(u) + ∂xF1(u) +G(x)u = 0,

on multiplication by P (u) becomes

∂tF0(u) + ∂xF1(u) +G(x)uP (u) +E = 0.

The distribution E, supported on the discontinuous locus t = Ψ(x), is givenby the density

−(

[F0][F1]

[F0]− [F1]

)Φ∗(δ0), (5.3.8)

where Φ∗(δ0) denoted the pull back of δ0 by Φ with Φ(x, t) = t− Ψ(x).

Proof: In the special case of one space variable, the discontinuity locus istaken to be the graph of the function t = Ψ(x). So, ndS = (−Ψ′, 1)dx which onsubstitution in (5.3.7) and using the Rankine-Hugoniot condition (see (5.3.12)below), gives (5.3.8).

Remark 5.3.1: The distribution E is of order zero and so involves only thepull back of δ0. Presumably for higher order equations, the distribution Emay be of higher order involving the pull back of derivatives of δ0.

5.3.2 Effect of distributional source terms

Recall the formula for pull back of a distribution by a submersion [74]

∂jΦ∗H = (∂Φj)Φ

∗δ0 =(∂jΦ)dS

|∇xΦ| , (5.3.9)

where ∇x is the spatial gradient operator, andH denotes the Haviside functionso that Φ∗(H) is the characteristic function of the set x : Φ(x) > 0 = Ω+,implying thereby

∂jχΩ+ =(∂jΦ)dS

|∇xΦ| . (5.3.10)

Similarly, we define Ω−x : Φ(x) < 0. Let us now consider a weak solution uof a scalar conservation law (5.3.1) in n independent variables in a region Ω


enclosing a single discontinuity of the solution u. We write u = fχΩ+ + gχΩ−

for some smooth maps f, g. Then

n∑

j=0

∂jFj(u) =

n∑

j=0

∂jFj(f)χΩ+ +

n∑

j=0

∂jFj(g)χΩ−

+

n∑

j=0

Fj(f)∂jχΩ+ +

n∑

j=0

Fj(g)∂jχΩ−. (5.3.11)

The left-hand side as well as the first two terms on the right-hand sidevanish as functions in L1

loc, so that on evaluating the distribution equation(5.3.11) against a test function φ gives in view of (5.3.10), the formula

⟨n∑

j=0

(Fj(f) − Fj(g))njdS, φ

⟩:=

∫

φ=0

(Fj(f) − Fj(g))njdS

=

∫

φ=0

φ

n∑

j=0

[Fj(u)]njdS = 0. (5.3.12)

Equation (5.3.12) is the classical Rankine-Hugoniot condition that we havedemonstrated in the spirit of distribution theory. Note that if linear sourceterms G(x)u are present in the conservation law, they cancel out from ei-ther side of equation (5.3.11) since (G(x)f + Σnj=1∂jFj(f))χΩ+ and (G(x)g +Σnj=1∂jFj(g))χΩ−

both vanish, and similarly the formula is unaffected byintegro-differential terms [154]. However, the analysis shows how nonhomoge-neous terms, which are distributions singular along Ψ = 0, are to be handled.

Theorem 5.3.2 Let β(x) be a smooth function defined on the manifold Φ = 0and consider a weak solution u for the system

n∑

j=0

∂jFj(u) = β(x)Φ∗(δ0), (5.3.13)

with a jump discontinuity along Φ = 0. Then the jump in u satisfies thefollowing modified Rankine-Hugoniot condition

∫

φ=0

φ

n∑

j=0

[Fj(u)]njdS = 〈β(x)Φ∗(δ0), φ〉, (5.3.14)

where φ is an arbitrary test function.

A detailed proof of theorem (5.3.2) is available in [191]. As an applicationof theorem (5.3.2), recall that in classical potential theory, the electric fieldvector E and potential V , satisfy the Poisson’s equation

divE = ∇2xV = 4πρ. (5.3.15)


The density of the electrostatic medium ρ is often singular along manifoldsand, in the case of layer potentials is given by ρ = cΦ∗(δ0).

Applying formula (5.3.14) to the Poisson’s equation, we recover as a specialcase of the Rankine-Hugoniot condition, the following classical theorem [85]giving the jump in the normal derivative of the layer potential

∂V

∂n= [E].n = 4πρ. (5.3.16)

5.3.3 Application to nonlinear geometrical optics

We now proceed to apply the formula (5.3.8) to the transport equations ofnonlinear geometrical optics which, at leading order, govern the propagationof high frequency monochromatic waves propagating into a given backgroundstate. Let us consider the small amplitude high frequency wave solutions of aquasilinear hyperbolic system of partial differential equations in N unknownsu and n space variables:

∂u

∂t+

n∑

j=1

∂

∂xj(Fj(x, t,u)) = 0, (5.3.17)

where the n flux functions Fj are assumed to depend smoothly on their ar-guments. Let u = u0(x, t) be a smooth solution of (5.3.17) which we call thebackground state and denote by A0

j the derivative DuFj(x, t,u) evaluated

at u0 and λ0q(k,x, t,u0) the qth eigenvalue of Σnj=1kjA

0j ; the left and right

eigenvectors being respectively L0q(k,x, t,u0) and R0

q(k,x, t,u0). Without anyloss of generality (with densities and fluxes redefined, if necessary), we mayassume u0 ≡ 0. We assume that only one eigen-mode, the qth mode, in theinitial data is excited. Motivated by the linear theory we seek an asymptoticseries solution in which only the qth eigen mode is present at the leading ordernamely,

uε(φ/ε,x, t) = u0 + επ(φ/ε,x, t)R0q + ε2u2(φ/ε,x, t) + · · ·. (5.3.18)

The phase function φ satisfies the qth branch of the eikonal equation

∂φ

∂t= −λ0

q(∇xφ,x, t), (5.3.19)

with the wave number vector k = ∇xφ. The validity of (5.3.18) over timescales of order O(ε−1) requires the use of multiple time scales and we denoteby ξ the fast variable ξ = φ/ε. The equation governing the evolution of theprincipal amplitude π(ξ, t,x) is given by (see [153]):

dπ

dt+ πχ+

∂

∂ξ

(aπ +

bπ2

2

)= 0, (5.3.20)


where ddt denotes the ray derivative, namely, the directional derivative along

the characteristics of (5.3.19), χ is the smooth function 12

∑nj=1

∂(L0qA

0jR

0q)

∂xjthat

is related to the geometry of the wave front, and a and b are known functionsof x and t. On multiplying (5.3.20) by π, we get

d

dt

(π2

2

)+π2

2

n∑

j=1

∂(L0qA

0jR

0q)

∂xj+∂(aπ2/2 + bπ3/3)

∂ξ+E = 0. (5.3.21)

Formula (5.3.8) with F0 = π, F1 = aπ+ 12bπ

2, F0 = 12π

2 and F1 = 12aπ

2+ 13bπ

3

gives the following distribution density

E = −(b

4[π2]2 − b

3[π3][π]

)Φ∗(δ0)

[π]=

b

12[π]3Φ∗(δ0). (5.3.22)

Remarks 5.3.2:

(i) The motivation for multiplying (5.3.20) by π to get (5.3.21) lies in theobservation that, away from the loci of discontinuities, the expression

d

dt

(π2

2

)+π2

2

n∑

j=1

∂

∂xj(L0

qA0jR

0q) +

∂

∂ξ

(aπ2

2+bπ3

6

),

is an exact divergence, namely,

d

dt

(π2

2

)+

n∑

j=1

∂

∂xj

(π2

2L0qA

0jR

0q

)+

∂

∂ξ

(aπ2

2+bπ3

6

),

implying the blow-up of intensity π2 as the ray tube collapses.

(ii) In [153], the distribution E given by equation (5.3.22) is interpretedas the energy dissipated across the shock. We note here that (5.3.22)may be recast in zero divergence form and implies the blow-up of theintensity as the ray tube collapses.

(iii) The distributional term E in (5.3.21) may be regarded as a singularforcing term which is yet another motivation for the study carried outin [191].

In certain systems exhibiting anomalous thermodynamic behavior, namely,when the so called fundamental derivative is small [94], the nonlinear distor-tions in the solution of the Cauchy problem for (5.3.17) are noticeable overmuch longer time scale of O(ε−2). This calls for a different scaling in the fastvariable, namely ξ = φ/ε2. In place of (5.3.20) we get the following equationgoverning the evolution of π (see [94]):

dπ

dt+(γ

2π2 + Σ1π

) ∂π∂ξ

+ χπ = 0, (5.3.23)


where, γ and Σ1 are known constants. Equation (5.3.23) can be written as

d

dt

(π2

2

)+π2

2

n∑

j=1

∂(L0qA

0jR

0q)

∂xj+

∂

∂ξ

(γπ4

8+

Σ1π3

3

)+E = 0.

Applying Formula (5.3.8), we get expression for E:

E =

(γ[π2][π]2

24+

Σ1[π]3

12

)Φ∗(δ0). (5.3.24)

If there are several shocks located at discrete locations Φi = 0i, the formula(5.3.8) must be modified to

−∑

i

([F0]i

[F1]i[F0]i

− [F1]i

)Φ∗i (δ0). (5.3.25)

Equations (5.3.22) and (5.3.24) get modified to

E =∑

i

b

12[π]3iΦ

∗i (δ0), (5.3.26)

and

E =∑

i

(γ[π2]i[π]2i

24+

Σ1[π]3i12

)Φ∗i (δ0), (5.3.27)

respectively. Note that equation (5.3.27) differs from (5.3.26) by a correctionterm with coefficient γ, thereby generalizing the result of [153]. Here the no-tation [π]i denotes the jump across the ith shock with locus Φi(x, t) = 0.

5.4 Evolution Equation Describing Mixed Nonlinearity

It has been found that in certain systems with singular thermodynamicbehavior, where the so-called fundamental derivative is small, the effect ofnonlinearity is perceptible over time scales longer by an order of magnitude,necessitating the use of fast variables of a higher order of magnitude, namely oforderO(ε−2). This has been studied in an abstract setting by Kluwick and Cox[94], and on applying it to the usual equations of gas-dynamics, the authorshave found that the transport equations governing the asymptotic behaviorcontain, in addition to the usual quadratic nonlinearity, cubic correction terms.

Srinivasan and Sharma [192] have generalized the work of [94] to the caseof multiphase expansions. The analysis in [192], which parallels the one in [77],is complicated due to the fact that the amplitudes at the leading order andthe next order appear together in the transport equations, and to separate

5.4 Evolution Equation Describing Mixed Nonlinearity 147

them a two stage averaging process has to be employed rather than the singlestage process detailed in [77]. Although this complexity is absent in [77], therequisite analytic apparatus is developed in [77].

In as much as the results obtained in [94] are important and interesting,we have rederived the transport equation (equation (5.4.23) below) for theleading amplitude in a spirit closer to ([115], [77]), expressing the result interms of the Glimm interaction coefficients Γkij .

Besides, we have employed a method similar to the one used by Cramerand Sen [43] in contrast to the approach in [94], where the perturbative pa-rameter is introduced in the differential equations. In particular, we rederivethe coefficients of the nonlinear terms of the (cubic) Burgers equation (5.4.23)in terms of the coefficients Γkij . In the case of a symmetric and isotropic sys-tems, the mean curvature of the wave front, which appears as the coefficientof the linear term in the transport equation, is related to the areal deriva-tive along the bicharacteristics. It is known from physical considerations andproved rigorously in the context of linear geometrical optics that the ampli-tude near a caustic becomes unbounded inversely as the area of cross sectionof the ray tube; we provide a short proof in Section 5.4.1 imitating the clas-sical proof of Liouville type theorems on integral invariants. It turns out thatthe square of the amplitude is a multiplier in the sense of Jacobi−Poincare[144], implying thereby the blow-up of intensities in a neighborhood of caus-tics [153]. However, the analysis involves the manipulation of conservationlaws through multiplication by polynomials in the unknown which is knownto change the Rankine-Hugoniot conditions across discontinuities. It is wellknown that these manipulations are invalid across discontinuities; in order torestore the validity, one has to modify the equations so obtained by adding sin-gular terms which are distributions supported along discontinuity loci. Thesesingular terms have been interpreted as the energy dissipated across shocks in[153]. To simplify the exposition somewhat, we have assumed throughout thatthe hyperbolic system of equations are in conservation form, which has theeffect of making the coefficients Γkij symmetric in i and j. In the final sectionwe present an application to the system governing the propagation of acousticwaves through a nonuniform medium stratified by gravitational source terms.

5.4.1 Derivation of the transport equations

We derive the transport equation (5.4.23) governing the propagation of thehigh frequency monochromatic waves, that includes both quadratic and cubicnonlinearities inherent in the system of conservation laws. In the first stagewe introduce the fast variable corresponding to the frequency of the wave andset up the stage for the perturbative analysis. In the second stage we obtainan ε-approximate equation (equation (5.4.16) below), at the second level ofperturbation, balancing the error at the third level resulting in the transportequation (equation (5.4.23) below), satisfied by the primary amplitude π of thewave solution.We have in the last paragraph of this section, briefly comparedour derivation of (5.4.23) with that in [94].


We consider the hyperbolic system of conservation laws with a source term,

∂u

∂t+

n∑

i=1

∂Fi (x, t,u)

∂xi+ G (x, t,u) = 0. (5.4.1)

The small amplitude high frequency solution u propagates in a backgroundstate u0, a spatially dependent known solution of (5.4.1), i.e.,

n∑

j=1

∂Fi(x, t,u0)

∂xj+ G (x, t,u0) = 0.

The independent variable x varies over an open set in IRn, and densities Fias well as the unknown u are IRm valued. The hyperbolicity of system (5.4.1)means that for each nonzero n-vector (ϑ1, ϑ2, ..., ϑn), the matrix

∑nj=1 ϑjA

0j ,

where A0j = DuFj(u0), is diagonalizable with real eigenvalues. We assume a

highly oscillatory initial Cauchy data with one excited eigen-mode, say, theqth,

u|t=0 = u0 + επ0

(x,

φ

ε2

)R0q , (5.4.2)

where the initial amplitude πo is a smooth function which is bounded withbounded first derivative; R0

j and L0j are, respectively, the right and left eigen-

vectors corresponding to the eigenvalue λ0j (ϑ1, . . . , ϑn), 1 ≤ j ≤ n, and the

zero superscript denotes their values at u = u0. Denoting by Bk[v1,v2] andCk[v1,v2,v3], respectively, the second and third derivatives D2Fk[v1,v2] andD3Fk[v1,v2,v3], with a zero superscript indicating evaluation at u = u0, weexpand the fluxes in a Taylor series

Fj = F0j + A0

j (x, t)(u − u0) +1

2B0j (x, t)[u − u0,u− u0]

+1

6C0j (x, t)[u − u0,u − u0,u− u0] +O(|u − u0|4). (5.4.3)

Likewise the source term G (x, t,u) may be developed as

G (x, t,u) = G (x, t,u0) +DuG (x, t,u0) (u − u0) +O(|u− u0|2). (5.4.4)

Introducing the fast variable of order O(ε−2), ξ = θ(x, t)/ε2, where the phasefunction θ(x, t) is the solution, corresponding to λ0

q , of the eikonal equation

Det

n∑

j=1

∂θ

∂xjA0j +

∂θ

∂tI

= 0, (5.4.5)

the PDE (5.4.1) may be recast in the form

n∑

j=0

[A0j

∂v

∂xj+∂A0

j

∂xjv+

1

ε2∂θ

∂xj

∂

∂ξ

[A0jv+

1

2B0j [v,v]+

1

6C0j [v,v,v]

]]+E0v = 0.

(5.4.6)


In Eqs. (5.4.1), (5.4.2), and (5.4.6), the expressions such as ∂Fj/∂xj and∂A0

j/∂xj refer to the xj partial of the composite function Fj(x, t,u(x, t))

(respectively, A0j (x, t,u0(x))). In (5.4.6) we have retained the terms of order

at most O(ε3), using the notations v = u − u0, F0(v) ≡ v, x0 = t, and E0 =DuG (x, t,u0). We assume that the multiplicities of the eigenvalues remainconstant, which implies that the eigenvalues depend smoothly on (ϑ1, ..., ϑn),thereby ensuring that the various branches are algebraic functions of ϑ1, ..., ϑn.Since the excited eigen-mode is the qth, we take the qth branch namely,

∂θ

∂t= −λ0

q(x, t,∇xθ). (5.4.7)

The characteristics of (5.4.7), are the linear bicharacteristics of (5.4.1), whichwill be referred to as rays. We seek a solution to (5.4.6) as a perturbationseries

v = u−u0 = εu1+ε2u2+ε3u3+... = επ(x, t, ξ)R0q+ε

2u2+ε3u3+· · · . (5.4.8)

Substituting (5.4.8) into (5.4.6), multiplying through by ε, we get at levelsO(εk) for k = 1, 2, 3 the following equations:

n∑

j=0

A0j

∂θ

∂xj

∂π

∂ξR0q = 0, (5.4.9)

n∑

j=0

A0j

∂θ

∂xj

∂u2

∂ξ= −

n∑

j=0

∂θ

∂xjB0j [R

0q ,R

0q ]ππ,ξ , (5.4.10)

n∑

j=0

∂θ

∂xjA0j

∂u3

∂ξ= −

n∑

k=0

∂θ

∂xk

∂

∂ξ

(B0k[u1,u2] +

1

6C0k[u1,u1,u1]

)

−n∑

k=0

A0k

∂u1

∂xk−E0u1, (5.4.11)

where (5.4.9) holds by the choice of θ(x, t).

5.4.2 The ε-approximate equation and transport equation

The solvability condition for (5.4.10) obtained by pre-multiplying with L0q

is the vanishing of

Σ0 :=

n∑

j=0

∂θ

∂xjL0qB

0j [R

0q ,R

0q] = 0. (5.4.12)

In certain applications to media with mixed nonlinearity, condition (5.4.12)generally holds only approximately with an error of O(ε). This is seen fromphysical considerations of the parameters involved, and seems to have been


the main motivation in [94] for implicitly assuming an ε-dependence for thematrices A0

j allowing a development of Bk and Ck as a series in ε. Cramerand Sen [43], in a different context, cope with this problem by manipulatingthe perturbation expansion by introducing ε-corrections to the coefficients ofε2 and ε3 as follows. When (5.4.8) is substituted in (5.4.6) the result is anasymptotic expansion

εZ1 + ε2Z2 + ε3Z3 + · · · = 0, (5.4.13)

where Z1 = 0 holds since the O(ε) term in (5.4.8) is proportional to R0q ;

however Z2 = 0, which is equation (5.4.10), only holds with an error of orderO(ε). Rewriting (5.4.13) as

εZ1 + ε2(Z2 − εZ ′3) + ε3(Z3 + Z ′

3) + ... = 0, (5.4.14)

where Z ′3 = Z2/ε and Z3 = Z ′

3. In other words (5.4.10) is replaced by anapproximate equation (5.4.16) given below which we solve exactly, therebycompensating the error at the O(ε3) level. The smallness of the physical pa-rameters involved makes this procedure licit. Let us write

−n∑

k=0

∂θ

∂xkB0k[R

0q ,R

0q ] =

n∑

j=1

µjR0j , (5.4.15)

so that µq = O(ε) and

µjL0jR

0j = −

n∑

k=0

∂θ

∂xkL0jBk[R

0q ,R

0q ].

We now solve exactly the ε-approximate equation

n∑

j=0

∂θ

∂xjA0j

∂u2

∂ξ= −ππ,ξ

∑

j 6=qµjR

0j , (5.4.16)

namely,

∂u2

∂ξ= −

∑

j 6=q

n∑

k=0

∂θ

∂xkL0jB

0k[R

0q ,R

0q ]R

0j

ππ,ξ(λ0j − λ0

q)L0jR

0j

, (5.4.17)

which implies

u2 = −∑

j 6=q

n∑

k=0

∂θ

∂xkL0jB

0k[R

0q ,R

0q ]R

0j

π2

2(λ0j − λ0

q)L0jR

0j

. (5.4.18)

These expressions are unique up to an additive multiple of R0q. With the

notations

∆0j = L0

jR0j , and Γkij =

n∑

l=0

∂θ

∂xlL0kB

0l [R

0i ,R

0j ],


the formula for u2 can be stated as

u2 = −∑

j 6=q

ΓjqqR0jπ

2

2∆0j (λ

0j − λ0

q),

∂u2

∂ξ= −

∑

j 6=q

Γjqqππ,ξR0j

∆0j (λ

0j − λ0

q). (5.4.19)

In terms of Γkij , we get Σ0 = Γqqq . We note here that the general systemconsidered in [94] is not in conservation form and hence the multilinear mapsB0k,C

0k would be unsymmetric in general. Multiplying (5.4.11) by L0

q andusing (5.4.8) we get the compatibility condition

n∑

k=1

∂θ

∂xk

∂

∂ξ(L0

qB0k[u1,u2]) +

1

2

n∑

k=0

L0q

∂θ

∂xkC0k[R

0q ,R

0q ,R

0q ]π

2π,ξ

+

n∑

k=0

L0qA

0k

∂(πR0q)

∂xk+ L0

qE0R0qπ = 0. (5.4.20)

On Substituting in (5.4.20) the values of ∂u2/∂ξ and u2 from equation (5.4.19)we get

−∑

j 6=q

3ΓjqqΓqqjπ

2π,ξ

2∆0j (λ

0j − λ0

q)+

1

2

n∑

k=0

L0q

∂θ

∂xkC0k[R

0q ,R

0q ,R

0q ]π

2π,ξ

+

(dπ

dt+ (χ+ h)π

)= 0, (5.4.21)

where h = L0qE

0R0q and d/dt denotes the ray derivative along the character-

istics of (5.4.7), and

χ =

n∑

k=0

L0qA

0k

∂(R0q)

∂xk. (5.4.22)

Note that we have obtained the transport equation (32) of [94] except forthe quadratic terms of the Burgers equation. We must now incorporate the εcorrections indicated in equation (5.4.14).

Since εZ ′3 is the R0

q component of Σnj=1(∂θ/∂xj)B0j [R

0q ,R

0q], namely

Γqqqππ,ξ/∆0q , we must add Z ′

3 = (1/ε)Γqqqππ,ξ/∆0q = O(1) to the right-hand

side of (5.4.21); thus we get

(dπ

dt+ (χ+ h)π

)+γ

2π2π,ξ + Σ1ππ,ξ = 0, (5.4.23)

which is precisely the equation obtained in [94] taking into account the contri-bution due to the source term in (5.4.1), where Σ1 = Σ0/(ε∆

0q) = Γqqq/(ε∆

0q)

and

γ = −∑

j 6=q

3ΓjqqΓqqj

∆0j (λ

0j − λ0

q)+

n∑

k=0

L0q

∂θ

∂xkC0k[R

0q,R

0q ,R

0q ]. (5.4.24)


5.4.3 Comparison with an alternative approach

We compare our derivation of (5.4.23) with the derivation in [94], wherefor simplicity we have assumed that the system in [94] arises out of a systemof conservation laws. Let us denote by A(λ) the matrix Σnj=1(∂θ/∂xj)A

0j −λI

and aτ (λ) its rows. Let b be the row vector that represents the functionalΣnj=0(∂θ/∂xj)L

0qB

0j [R

0q , ·]/∆0

q . Condition (5.4.12) can be stated as vanishing

of b · R0q . The authors in [94] proceed to observe that this holds if b is in

the row space of the matrix A(λq), i.e., b = Στβτaτ (λ). The coefficients ofthe cubic Burgers equation satisfied by π, as derived in [94], involve βτ , andhence there is a need to compute them explicitly. To do this, notice thataτR

0p = (λ0

p − λ)R0p,τ , where R0

p,τ denotes the τ entry in the column vectorR0p, and similarly L0

p,τ denotes the τ component of the left eigenvector L0p.

Multiplying by βτ and summing over τ we get

b · R0p/(λ

0p − λ) = Γqqp/∆

0q(λ

0p − λ) = β · R0

p, p = 1, 2, ..., n, (5.4.25)

where β denotes a row matrix, from which it follows that

β =∑

l

ΓqqlL0l

∆0l∆

0q(λ

0l − λ)

,

and hence

βτ =∑

l

ΓqqlL0l,τ

∆0l∆

0q(λ

0l − λ)

.

On using this in equation (32) of [94], we recover (5.4.23). It may be noticedthat the summand with l = q in the above formula for βτ vanishes, since(5.4.10) is assumed in [94] at the level A0

i |ε=0.

5.4.4 Energy dissipated across shocks

Throughout this section we assume that u0 is constant, the matrices A0j

are symmetric, and ∂A0j/∂xj = 0. With the normalization L0

qR0q = 1 we

get on using the relation ∂λq/∂ϑj = L0qA

0jR

0q obtained by differentiating

Lq(−λqI + Σnj=1ϑjA0j )R

0q = 0 with respect to ϑj ,

n∑

j=0

L0q

∂

∂xj(A0

jR0qπ) =

1

2

n∑

j=0

π∂

∂xj(L0

qA0jR

0q) +

n∑

j=0

(L0qA

0jR

0q)∂π

∂xj,

=1

2

n∑

j=1

∂

∂xj

∂λq∂ϑj

∣∣∣∣∣∣ϑ=∇xθ

π +

n∑

j=0

(L0qA

0jR

0q)∂π

∂xj.

Numerous physical systems are governed by isotropic conservation laws, wherethe eigenvalues λ0

q have the form λ0q(x, t, ϑ) = |ϑ|c0q(x, t). In particular, this is


so with the example considered in ([94, 153]). For isotropic systems we have

1

2

n∑

j=1

∂

∂xj

∂λq∂ϑj

∣∣∣∣∣∣ϑ=∇xθ

π =π

2(n · ∇xc

0q + c0qχ). (5.4.26)

Here n is the unit normal vector to the wave front θ(x, t) = c and χ givenby (5.4.22) is the mean curvature of the wave front. Note that due to theabsence of explicit (x, t) dependence in system (5.4.1), the eigenvalues λqare independent of (x, t) except for their appearance through ∇xθ, namelyλ = c0q |∇xθ| with c0q a constant; here, ∇x is the spatial gradient operator. ThePDE (5.4.23) therefore reads

dπ

ds+(γπ

2+ Σ1

)π∂π

∂ξ+ (χ+ h)π = 0. (5.4.27)

Here Σ1 is the coefficient of genuine nonlinearity introduced by Lax, γcharacterizes the degree of material nonlinearity and h is due to the sourceterm in (5.4.1). The coefficient χ also has the following interpretation as theareal derivative of the wave front along rays.

Theorem 5.4.1 Consider system (5.4.1), where (x, t) do not explicitly ap-pear in the equations. Let s be the ray coordinate and A(s) the area of thecross section of the ray tube generated by the characteristics of (5.4.7). Then

χ = lim|A|→0

1

2A

dA

ds. (5.4.28)

Proof: The characteristic flow of (5.4.7) is given by the system of ODEs

dxjds

= L0qA

0jR

0j =

∂λq∂xj

,dϑjds

= 0,dθ

ds= 0, 0 ≤ j ≤ n. (5.4.29)

Let Ω0 be an element of surface area along θ = c. Assume that Ω0 evolves toΩs along the bicharacteristic flow. The area A(s) of the surface element Ωs isgiven by

A(s) =

∫

Ωs

θ,t|∇xθ|

dx1dx2...dxn,

where we have assumed that ∂θ/∂t 6= 0, and x1, x2, ..., xn are local coordinateson the surface. Denoting by (c1, c2, ..., cn) the coordinates of a general pointon the element Ω0, we get

A(0) =

∫

Ω0

θ,t|∇xθ|

dc1dc2...dcn.


If x = φs(c) denotes the bicharacteristic flow, then

A(s) =

∫

Ω0

θ,t|∇xθ|

∣∣∣∣∂x(s)

∂c

∣∣∣∣ dc1dc2...dcn,

dA

ds=

∫

Ω0

θ,t|∇xθ|

d

ds

(∣∣∣∣∂x(s)

∂c

∣∣∣∣)dc = 2

∫

Ω0

θ,tχ

|∇xθ|

∣∣∣∣∂x(s)

∂c

∣∣∣∣ dc = 2

∫

Ωs

θ,tχ

|∇xθ|dx,

where we have used (5.4.29) and the variational equation for derivatives,namely,

d

ds

(∣∣∣∣∂x(s)

∂c

∣∣∣∣)

= Trace

(∂

∂xi

∂λq∂ϑk

) ∣∣∣∣∂x(s)

∂c

∣∣∣∣ = 2χ

∣∣∣∣∂x(s)

∂c

∣∣∣∣ .

Now, χ is not constant but we may write χ = χ(p) + o(1) if p is a genericpoint of a sufficiently small element Ωs so that

dA(s)

ds= (2χ(p) + o(1))A(s), (5.4.30)

from which the result follows.

We now introduce the augmented field

X(t, x1, ..., xn, ξ) =

(1, L0

qA01R

0q , . . . , L0

qA0nR

0q ,

2

3Σ1π +

γ

4π2

), (5.4.31)

and show that π2/2 is a multiplier for this vector field, giving an integral in-variant along the bicharacteristic flow. It may be noted that integral invariantswere introduced by Poincare in connection with celestial mechanics [144].

Theorem 5.4.2 (i) π2/2 is a multiplier for the augmented vector field(5.4.31), i.e.,

Div

(π2

2X

)= 0.

(ii) The integral∫Ωs

(π2/2)dxdtdξ is conserved along the flow determined by

(5.4.31).(iii) At the points in (x, t) space where the ray tube collapses, i.e., |Ωs| → 0,the amplitude becomes infinite inversely as the volume |Ωs|.

Proof: From the calculation leading to (5.4.26) and (5.4.27) we get

dπ

ds+ χπ =

1

π

n∑

j=0

∂

∂xj

(π2

2L0qA

0jR

0q

), (5.4.32)

and so the PDE for π can be recast in the divergence free form,

n∑

j=0

∂

∂xj

(π2

2L0qA

0jR

0q

)+

∂

∂ξ

(γ

8π4 +

Σ1

3π3

)≡ Div

(π2

2X

)= 0, (5.4.33)


which proves (i); the proof of (ii) follows form the classical Liouville’s theoremand is similar in spirit to Theorem 5.4.1. As we approach a point where theray tube collapses, volume |Ωs| shrinks to zero; this implies that due to theconservation of

∫Ωs

(π2/2)dxdtdξ, π2 blows up in a neighborhood of this point.Moreover, we see that the square of the amplitude blows up inversely as thevolume |Ωs|.

Remarks 5.4.1:

(i) The manipulations leading to (5.4.33) are valid in regions of smoothness,the boundary of which may contain a caustic point.

(ii) The energy dissipated across shocks, present in the solutions of (5.4.27),can be computed using (5.3.8) or (5.3.25).

5.4.5 Application

Here we apply the results of the foregoing sections to the basic equationsthat describe the propagation of sound waves in a nonuniform atmosphere,and derive the transport equations for the high frequency wave amplitude inthe leading order terms in the expansion. Practically any problem of acousticstakes place in the presence of a gravitational field and as a consequence, theunperturbed state is not uniform. In problems of propagation over large dis-tances in atmosphere or the ocean, these effects may be crucially importantand produce amplifications and refractions of sound waves. The basic equa-tions describing the propagation of sound waves through a stratified fluid maybe expressed in the following form:

∂v

∂t+ (v · ∇x)v + ρ−1∇xρ = −g, (5.4.34)

∂ρ

∂t+ (v · ∇x)ρ+ ρ(∇x · v) = 0, (5.4.35)

∂S

∂t+ (v · ∇x)S = 0, (5.4.36)

where ρ is the fluid density, v = (v1, v2, v3)tr the fluid velocity vector, p =

p(ρ, S) the pressure, S the entropy, t the time, ∇x = (∂/∂x1, ∂/∂x2, ∂/∂x3)is the spatial gradient operator, and g the acceleration due to gravity. Thereference state is characterized by a flow field at rest, v0 = 0, with a spatiallyvarying density and entropy fields, namely ρ0 = ρ0(x) and S0 = S0(x) with

∇xp0 + ρ0g = 0, (5.4.37)

where the subscript zero denotes the unperturbed fluid in equilibrium. Thegoverning system (5.4.34) – (5.4.36) can be cast into the form

∂U

∂t+

3∑

k=1

Ak(U)∂U

∂xk+ F = 0, (5.4.38)


representing a quasilinear hyperbolic system of equations with source termwhich is attributed to the influence of gravity; here U and F are columnvectors defined as U = [v1, v2, v3, ρ, S]tr and F = [g1, g2, g3, 0, 0]tr . The Ak(U)are 5 × 5 matrices with entries Akαβ , 1 ≤ α, β ≤ 5, given by

Ak11 = Ak22 = Ak33 = Ak44 = Ak55 = vk, Aki4 = a2δik/ρ, Aki5 =∂p

∂S

δikρ,

Ak4j = ρδjk , Ak5j = Ak45 = Ak54 = Ak21 = Ak12 = Ak31 = Ak13 = Ak23 = Ak32 = 0,

where 1 ≤ i, j, k ≤ 3, δ is the Kronecker symbol, and a is the sound speedgiven by a2 = (∂p/∂ρ)|S . We look for a small amplitude high frequency asymp-totic solution of (5.4.38) when the length L of the disturbed region is small incomparison with the scale height H of the stratification, defined as a typicalvalue of ρ0 |∇xρ0|−1

, so that ε = L/H 1. In this limit, the perturbationscaused by the wave are of size O(ε) and they depend significantly on the fastvariable ξ = θ(x, t)/ε2, where θ is the phase function to be determined. Thesmall amplitude high frequency solution to (5.4.38) that admits an asymp-totic expansion of the form (5.4.8) where u0 = [0, 0, 0, ρ0(x), S0(x)]tr is theknown background state. Let π be the wave amplitude associated with theright running acoustic wave θ(x, t) = const, propagating with speed a0|∇xθ|.The left and right eigenvectors L and R of Σ3

k=0(∂θ/∂xk)Ak associated with

eigenvalue a0|∇xθ| are given by

L =

[n1, n2, n3,

a0

ρ0,

1

a0ρ0

(∂p

∂S

)

0

]tr, R =

[n1, n2, n3,

a0

ρ0, 0

]tr,

where n = (n1, n2, n3) denotes the unit normal field |∇xθ|−1 ∇xθ to the wavefront. The coefficients Σ1, γ, χ, and h appearing in (5.4.23) and (5.4.24) cannow be easily calculated; in fact, we find that h = 0 and

Σ1 =|∇xθ|

2ε

(1 +

ρ0

a0

∂a

∂ρ

∣∣∣∣0

), γ =

ρ20|∇xθ|a20

∂

∂ρ

(1

ρ

(∂(aρ)

∂ρ

))∣∣∣∣0

,

χ =1

2

(a0∇x · n − 2

(∂a/∂S

∂p/∂S

)

0

(a20n · ∇xρ0 + ρ0n.g

)

−a0

ρ0n · ∇xρ0 − n · ∇xa0

). (5.4.39)

The functions Σ1, γ and χ characterize respectively the genuine nonlinearitycoefficient of Lax, the degree of material nonlinearity, and the variation in waveamplitude due to the wavefront geometry and the varying medium ahead.It may be noticed that although h = 0, the gravity effects enter indirectlythrough its control of ρ0(x) as may be seen in the expression for χ. Thus theevolution equation for the amplitude π describing the propagation of acousticwaves in a stratified medium is given by

dπ

dt+ χπ +

(1

2γπ + Σ1

)ππ,ξ = 0, (5.4.40)

5.5 Singular Ray Expansions 157

with coefficients given by (5.4.39). On substituting the values of Σ1 and γ,given by (5.4.39), one computes the energy dissipated across shocks using(5.3.8).

Remark 5.4.2: The qualitative behavior of the weak solutions of (5.1.6)is quite different from those of (5.4.40); one can check how the solutions of(5.1.6) and (5.4.40), satisfying the same periodic or compact support initialdata, evolve at different times leading to a breakdown. For more details, thereader is referred to ([42] and [43]).

5.5 Singular Ray Expansions

The geometric ray expansion derived in section 5.4 breaks down whenthe signal variations transverse to the direction of wave propagation becomesignificant. This is a common feature in diffraction problems, where ray theorypredicts unbounded behavior near a singular ray.

Following Kluwik and Cox [94], we derive an evolution equation fora density-stratified, nonisentropic flow governed by the hyperbolic system(5.4.38) when diffraction effects are prevalent. In addition to the fast phasevariable, ξ = φ(x, t)/ε2, we introduce other fast variables such as ηn =ψn(x, t)/ε, n = 1, 2, that describe the modulations of the signals in direc-tions transverse to those of the rays. The singular rays are supposed to lie inthe hypersurfaces, ψn(x, t), n = 1, 2. We look for progressive wave solution ofform:

U(x, t) = U0(x) + εU(1)(ξ, η1, η2,x, t) + ε2U(2)(ξ, η1, η2,x, t)

+ε3U(3)(ξ, η1, η2,x, t), ε 1. (5.5.1)

In terms of the new variables the system (5.4.38) can be written as

U,ξ(φ,tI+Akφ,k)+εU,ηn(ψn,tI+Akψn,k)+ε

2(U,t+AkU,k+F) = 0. (5.5.2)

Using (5.5.1) in (5.5.2) along with the Taylor’s series expansion for Ak nearU = U0(x), we obtain at the leading order

U(1)(x, t) = π(ξ, η1, η2,x, t)R

where π is the scalar amplitude which will be determined to the next order,and R is the right null vector of G = φ,tI + Ak

0φ,k. Equations at the next


level of sequential approximations in ε are the following.

G.U(2),ξ = −KnRπ,ηn

− φ,kR.(∇Ak)0Rππ,ξ + ΓRππ,ξ, (5.5.3)

G.U(3),ξ = −(πR),t −Ak

0(πR),k −R.(∇Ak)0U(0),k π − ΓRππ,ξ −KnU

(2),ηn

−φ,kR.(∇Ak)0U(2),ξ π − φ,kU

(2).(∇Ak)0Rππ,ξ −R.(∇F)0π

−ψn,kR.(∇Ak)0Rππ,ηn− 1

2φ,kRR : (∇∇Ak)0Rπ

2π,ξ, (5.5.4)

where Γ = L(R.(∇Ak)0)RΦ,k/(LR) is the quadratic nonlinearity parameter

which is of order O(ε) with L as the left null vector of G, Γ = Γ/ε = O(1),

RR : (∇∇Ak)0 =5∑

n,m=1

RmRn∂2Ak

∂Um∂Un

∣∣∣∣∣U=U0

,

and the matrix Kn is given by

Kn = Iψn,t + Ak0ψn,k. (5.5.5)

The solvability condition for U(2) requires the satisfaction of the following:

LKnR = 0. (5.5.6)

The scalar product of (5.5.4) with L yields the following solvability conditionfor U(3)

dπ

dt+

(Γ +

M

2π

)π∂π

∂ξ+(b.U(2)

) ∂π∂ξ

+

(c.∂U(2)

∂ξ

)π + en.

∂U(2)

∂ηn

+L(R.(∇Ak)0)Rψn,kππ,ηn

2+ χπ +

Γ(L.U(2)π),ξ2

= 0, (5.5.7)

where d/dt = ∂/∂t + aoφ,k∂/∂xk denotes derivative along the ray — thetrajectory of an element of the surface φ(x, t) = constant, and M , b, c, en,and χ are given by

M = φ,kL(RR : (∇∇Ak)0)R/2, b = Lφ,k(∇Ak)0R − ΓL/2,c = L(R.(∇Ak)0)φ,k − ΓL/2, en = LKn/2, (5.5.8)

χ = LAk0(∂R/∂xk) + R.(∇Ak)0U

(0),k + R.(∇F)0/2.

From (5.5.6), it follows that en is orthogonal to R and lies in the 4-dimensionalrow-space of G. Thus, en can be written as

en = βnαGα, α = 1, 2, 3, 4, n = 1, 2, (5.5.9)

where Gα are the linearly independent rows of G. We notice that the pro-jection of ∂U(2)/∂ηn onto the row space of G will be required to construct

5.5 Singular Ray Expansions 159

the term en · (∂U(2)/∂ηn) in the evolution equation of (5.5.7). Integration of(5.5.3) gives,

GU(2) = −(KnR)

∫ ξ ∂π

∂ηndξ − 1

2

(φ,kR.(∇Ak)0 − ΓI

)Rπ2, (5.5.10)

where we assume that U(2) = 0 when U(1) = 0. Differentiating the aboveequation with respect to ηn, we have,

G∂U(2)

∂ηn= −(KnR)

∫ ξ ∂2π

∂ηn∂ηldξ − (φ,kR.(∇Ak)0 − ΓI)Rππ,ηn

.

Since b and c are orthogonal to R, we have b = ωαGα, c = δαGα withωi = δi = ni(

ρoa,so

p,so), i = 1, 2, 3; ω4 = δ4 = ρ−1

o (1 +ρoaoa,so

p,so); thus, the terms

b.U(2), c.∂U(2)/∂ξ, and en.∂U(2)/∂ηn may be written as

b.U(2) = −ω

1

2

(φ,kR.(∇Ak

1)0 − ΓI1

)Rπ2 + (K1nR)

∫ ξ ∂π

∂ηndξ

,

c.∂U(2)

∂ξ= −δ

(φ,kR.(∇Ak

1)0 − ΓI1

)Rππ,ξ + (K1nR)

∂π

∂ηn

,

en.∂U(2)

∂ηn= −βn

(φ,kR.(∇Ak

1)0 − ΓI1

)Rπ

∂π

∂ηn+ (K1nR)

∫ ξ ∂2π

∂ηn∂ηldξ

,

where Ak1 and I1 are 4× 5 matrices obtained from Ak and I, respectively, by

deleting their last row, and K1n = I1ψn,t + Ak1ψn,k. Substituting the above

expressions in the evolution equation (5.5.7), we have,

dπ

dt+(Γ +

γ

2π)π∂π

∂ξ+ χπ + Tnπ

∂π

∂ηn−Wn

∂π

∂ξ

∫ ξ ∂π

∂ηndξ

−Mnl

∫ ξ ∂2π

∂ηn∂ηldξ = 0, (5.5.11)

where Tn, Wn,Mnl and γ are defined as

Tn = −δ(K1nR) − βn(φ,kR.(∇Ak

1)0 − ΓI1

)R +

L(R(∇Ak)0)Rψn,k2

,

Wn = ω(K1nR), Mnl = βn(K1nR),

γ = M− (ω + 2δ)(φ,kR(∇Ak1)o − ΓI1)R.

Equation (5.5.11) can be reduced to an equation similar to the Zabolotskaya-Khokhlov equation on differentiating with respect to ξ which is as follows:

∂

∂ξ

(dπ

dt+(Γ +

γ

2π)π∂π

∂ξ+ χπ +

2∑

n=1

Tnπ∂π

∂ηn−

2∑

n=1

Wn∂π

∂ξ

∫ ξ ∂π

∂ηndξ

)

=2∑

n=1

2∑

l=1

Mnl∂2π

∂ηn∂ηl,


where Γ, the quadratic nonlinearity coefficient may be identified with the Lax’sgenuine nonlinearity parameter and γ, the cubic nonlinearity coefficient, sig-nifies the degree of material nonlinearity; the linear term χπ brings about avariation in the wave amplitude as a result of wave interactions with the vary-ing medium ahead and changes in the geometry of the wavefront as it movesalong the rays. Each of these coefficients including the additional coefficientsTn, Wn and Mnl which bring about the distortion of the signal in the trans-verse direction, is of order O(1) and is a function of the space coordinates. Thenonlinearity parameters lead to distortion of the wave profile and formation ofshock, while the linear term results in growth or decay of the wave amplitudedepending on whether it is negative or positive. When the medium ahead isuniform and the source term is absent the evolution equation reduces to thatof Kluwick and Cox [94] with the coefficients being absolute constant. The lin-ear term vanishes too, if we consider one-dimensional wave propagation in theuniform medium and we get an evolution equation similar to that of Cramerand Sen [43].

Remarks 5.5.1:(i) In the geometric ray theory, the intersection of rays predicts infinite

amplitudes. The geometric singularity is, in fact, associated with the formationof caustics; the breakdown occurs in a thin caustic boundary layer where alocal analysis is appropriate.

(ii) It is easy to incorporate the effects of a small amount of dissipationinto the equations of WNGO, so that the new system is

u,t + Ak(u)u,k + F(u) = ε2(Dij(u)u,j),i ,

where Dij are the n × n viscosity matrices satisfying the stability conditionξiξjL

oqDijR

oq ≥ 0, where ξ ∈ IRn and subscript q refers to the qth eigen-

mode (see Majda and Pego [114]). If one assumes that the matrices involvedin the above equation have Taylor expansion about u = u0, and repeats theweakly nonlinear analysis, it is straightforward to show that the transportequation (5.4.40) for the wave amplitude, in media exhibiting mixed nonlin-earity, gets modified to the (viscous) Burgers equation with cubic nonlinearity(see Sharma et. al ([172])

dπ

dt+ (Σ1 +

γ

2π)ππ,ξ + πχ = µπ,ξξ,

where µ = (LoqDijRoq)/(L

oqR

oq) ≥ 0.

5.6 Resonantly Interacting Waves

A systematic approach to resonantly interacting weakly nonlinear hyper-bolic waves has been proposed by Majda and Rosales [115] and Hunter et

5.6 Resonantly Interacting Waves 161

al. [77]. This approach enables one to analyze situations when many wavescoexist and interact with one another resonantly. In this section we first givean outline of this method, and then use it to a hyperbolic system governingthe one dimensional flow of a nonideal gas.

Let us consider a strictly hyperbolic system in one space dimension, givenby (5.1.1). Let us assume that A and b admit Taylor expansion about u = uo,which is a known constant solution of (5.1.1), such that b(uo) = 0. Letλ1, λ2, . . . , λn be the n-distinct real eigenvalues of A(uo) with correspondingleft and right eigenvectors denoted by L(i) and R(j), 1 ≤ i, j ≤ n, respec-tively, satisfying the normalization condition L(i)R(j) = δij , with δij as theKronecker symbol.

Our object here is to construct formal asymptotic solution of (5.1.1) of theform

u = uo + εu1(x, t,θ) + ε2u2(x, t,θ) + o(ε2), (5.6.1)

satisfying the small amplitude rapidly oscillating initial data

u(x, 0) = uo + εuo1(x, x/ε) + ε2uo2(x, x/ε) + o(ε2). (5.6.2)

Here, ε is a small parameter defined as a ε = τi/τ 1, where τi is thetime scale defined by the input and τ is the time scale characterizing themedium, and uo1 and uo2 are smooth bounded functions, with the remainderterm o(ε2), to be uniform in x as ε→0. It is assumed that in this high frequencylimit, the perturbations caused by the wave are of size O(ε), and they dependsignificantly on the fast variable θ = φ/ε, where θ = (θ1, θ2, . . . , θn) is a vectorof rapidly varying phase functions with θi = φi(x, t)/ε; indeed, φi is the phaseof the ith wave associated with the characteristic speed λi. In (5.6.1), u1 is asmooth bounded function of its arguments. We need to require additionallythat u2 can have at most sublinear growth in θ, i.e., ‖u2‖→0 as ‖θ‖→∞ anduo1(x, x/ε) in (5.6.2) is a periodic smooth function with a compact support andmean zero. In [115], authors show that under these conditions, the expansion(5.6.1) is a uniformly valid approximation for times of order ε−1. Now we use(5.6.2) in (5.6.1), expand A and b in a Taylor series in powers of ε aboutu = uo, replace the partial derivatives ∂/∂X (X being either x or t) by∂/∂X + ε−1

∑ni=1 φi,x∂/∂θi, and equate to zero the coefficients of ε0 and ε1

in the resulting expansions, to get

O(ε0) :

n∑

i=1

(Iφi,t + Aoφi,x)u1,θi= 0, (5.6.3)

O(ε1) :

n∑

i=1

(Iφi,t + Aoφi,x)u2,θi= −u1,t −Aou1,x −∇b|o · u1

−n∑

i=1

φi,x∇A|0 · u1u1,θi, (5.6.4)

where I is the n×n unit matrix and ∇ is the gradient operator with respect tothe dependent variable u; the subscript o refers to the evaluation at u = uo.


A solution of (5.6.3) is

u1 =n∑

i=1

πi(x, t, θi)R(i), (5.6.5)

where θi = φi/ε with φ = x − λi(uo)t. We suppose that the wave ampli-tudes πi and their derivatives with respect to x, t and θ are periodic func-tions of θ, and the wave amplitudes have 0 mean with respect to θ, i.e.,

limT→∞(2T )−1∫ T−T πj(x, t, θ)dθ = 0. We then use (5.6.5) in (5.6.4) and solve

fro u2. To begin with, we expand u2 in the basis of Ao, i.e., u2 =∑mjR

(j),and substitute it in (5.6.4); then on pre-multiplying the resulting equation byL(i), we get

n∑

j=1

(λi − λj)∂mi

∂θj= −∂πi

∂t− λi

∂πi∂x

− µiπi (5.6.6)

−νiπiπ′

i(x, t, θi) −(i)∑

p

Li∇b|0R(p)πp

−(i)∑

p

(i)∑

q

Li · ∇A|o ·R(p)R(q)πpπ′

q(x, t, θq); (i- unsummed),

where π′

q(x, t, θ) = ∂πq/∂θ, µi = Li · ∇b|oR(i), νi = Li · ∇A|oR(i)R(i),∑(i)p

stands for the sum over p with 1 ≤ p ≤ n and p 6= i, and mi are boundedfunctions of θ.

We use the standard properties of periodic functions, and appeal to thesublinearity of u2 in ‖θ‖, which ensures that the expression (5.6.2) does notcontain secular terms; the constancy of θi along the characteristics of theith equation in (5.6.6), and the vanishing asymptotic means of the terms onthe right-hand side of (5.6.6), imply that the wave amplitude πi satisfy thefollowing integro-differential equation (see [115] and [5])

∂πi∂t

+ λi∂πi∂x

+ µiπi + νiπi∂πi∂θi

+∑

i6=j 6=kΛijk lim

T→∞

1

2T

∫ T

−Tπj(θi + (λi − λj)s)π

′

k(θi + (λi − λk)s)ds = 0,

(5.6.7)where Λijk = Li · ∇A|oR(j)R(k) are the interaction coefficients asymmetric inthe indices j and k, and represent the strength of coupling between the jthand kth wave which can produce ith wave through the nonlinear interaction.The coefficients Λiii, which we denote by νi, account for the nonlinear self-interaction, and are nonzero for genuinely nonlinear waves; these coefficientsare indeed zero for linearly degenerate waves. The initial conditions for (5.6.7)are recovered from the initial data (5.6.2). It may be noticed that the coef-ficients µi vanish in the absence of source term in (5.1.1). If all the coupling

5.6 Resonantly Interacting Waves 163

coefficients Λijk(i 6= j 6= k) are zero, or the integral averages in (5.6.7) van-ish, the waves do not resonate, and (5.6.7) reduces to a system of uncoupledBurgers equations.

It may be remarked that for n = 2, there are no integro-differential terms in(5.6.7), i.e., for resonances, the hyperbolic system has at least three equations.The asymptotic equation (5.6.7) can be written in a conservation form, whichremains valid in the weak sense after shocks form (see Cehelsky and Rosales[30]). In gasdynamics, a weakly nonlinear resonant interaction between theacoustic modes and the entropy waves occurs (see [115] and [77]); in morethan one dimension, vorticity plays a role similar to that of entropy. In thisresonance, each acoustic wave interacts with the entropy wave to producethe other sound wave. No new entropy is produced by interactions betweenthe acoustic waves. The resonance results in coupling of the acoustic waveswhich have a dispersive nature; thus, it can turn off the nonlinear steepening ofpressure waves and the shocks which are produced. The numerical calculationsshow that solutions without wave breaking are possible; the interested readermay consult [115] for dispersive coupling due to resonant wave interactions,and [75] for diffraction of interacting waves.

Remarks 5.6.1: It may be remarked that WNGO is not yet a completetheory; many open problems remain in this area. For instance the behavior ofcaustics, singular rays and the case of many phases in multidimensions stillremain essentially open (see [153] and [164]).

Chapter 6

Self-Similar Solutions Involving

Discontinuities

Self-similar solutions are well known and widespread. Self-similarity of thesolutions of PDEs has allowed their reduction to ordinary differential equations(ODEs) which often simplifies the investigation.

Self-similarity means that a solution u(x, t) at a certain instant of time t issimilar to the solution u(x, to) at a certain earlier moment to. In fact, similarityis connected with a change of scales, i.e., u(x, t) = (t/to)

mU(x(t/to)n, to), or

u(x, t) = φ(t)U (x/ψ(t)), where the dimensional scales φ(t) and ψ(t) depend ontime t in the manner indicated, and the dimensionless ratio u/φ is a functionof new dimensionless variable, called the similarity variable, ξ = x/ψ(t). Ingeometry, this type of transformation is called an affine transformation. Theexistence of a function U(ξ) that does not change with time enables us tofind a similarity of the distribution u(x, t) at different moments. The heatconduction equation has a natural symmetry that enables us to transform thedependent variable u by a scaling factor without changing the solution; thesolution of such an equation is

u =E

c(2√πkt)3

exp(−r2/(4kt)), (6.0.1)

where E represents a finite amount of heat supplied initially at a point in aninfinite space filled with a heat conducting medium, c is the specific heat ofthe medium, k its thermal diffusivity, and r the radial distance from the pointat which the heat is supplied instantaneously (see Barenblatt [10]). It maybe noticed from (6.0.1) that there exist temperature and length scales bothdepending on time, i.e., uo(t) = E/(c(kt)3/2) and ro(t) = (kt)1/2 such that

u/uo = U(r/ro), U(ξ) = (8π3/2)−1 exp(−ξ2/4); ξ = r/ro.

Thus, the spatial distribution of temperature, expressed in terms of the vari-able scales uo(t) and ro(t), is independent of time, and can be expressed asa function of the similarity variable ξ = r/ro. The solution of the problemthus reduces to the solution of an ordinary ODE for U(ξ). In fact, the scaleinvariance allows us to scale out the t-dependence by introducing a new de-pendent variable and new spatial variables called similarity variables; the scaleinvariance and similarity method may be applied to other PDEs.

165

166 6. Self-Similar Solutions Involving Discontinuities

Let us consider the following scaling transformations

u∗ = εαu, x∗ = εβx, t∗ = εγt,

where α, β, and γ are constants; in other words, we have defined

u∗(x∗, t∗) = εαu(ε−βx∗, ε−γt∗).

In order to scale out the t-dependence, we should take εγ = t−1, so that t∗ = 1;thus, if we introduce U(x∗) = u∗(x∗, 1), and ξ = x∗, then

u(x, t) = tα/γU(ξ), (6.0.2)

defines the similarity transformation with the similarity variable ξ = x/tβ/r.Generally, it is found that at least one of α, β, and γ may be given an arbitraryvalue; if one takes γ = 1, then (6.0.2) simplifies to

u(x, t) = φ(t)U(x/f(t)), (6.0.3)

where φ(t) = tα and f(t) = tβ . We then substitute (6.0.2) into the given PDEand choose α and β appropriately so that we obtain a differential equation forU , which is independent of t (see McOwen [121]).

In Euler equations of gasdynamics, u(x, t) is the gas velocity; if we carryout this program for this system, we find that since ξ = x/tβ , ξ = constantimplies dx/dt = xβ/t. Thus, it is appropriate to take u = (βx/t)U(ξ), andhence the density, pressure and sound speed can be taken as

ρ = xkΩ(ξ), p = β2xk+2t−2P (ξ), c = (βx/t)C(ξ). (6.0.4)

In other words, the quantities ut/x, ct/x and ρx−k appear constant for anobserver who moves so that ξ = x/tβ is constant; the factor βx/t in u and cis the velocity of such an observer.

The very idea of self-similarity is connected with the study of the structureof PDEs using the notion of symmetry group. This term is used to refer toa group of transformations that maps any solution to another solution of thesystem, leaving it invariant. In Lie theory, such a group depends on continu-ous parameters called continuous group of transformations of the differentialequation/system. Lie initiated the study of such groups of transformationsbased on the infinitesimal properties of the group. Briefly, a Lie group is atopological group in which the group operation and taking the inverses arecontinuous functions, and in which the group manifold has a structure that al-lows calculus to be performed on it. The method of solving ODEs and PDEs,based on invariance under continuous (Lie) groups of transformations, is anatural generalization of the method of self-similar solutions. More informa-tion about the group theoretic method is to be found in [2], [16], [17], [18],[49], [50], [78], [112], [136], [137] and [138].

This chapter includes treatment of self-similar and the basic symmetry

6.1 Waves in Self-Similar Flows 167

(group theoretic) methods for solving Euler equations of gasdynamics involv-ing shocks, characteristics shocks and weak discontinuities together with theirinteractions. A Lie group of transformation is characterized in terms of its in-finitesimal generators which form a Lie algebra; emphasis is placed to discoversymmetries admitted by PDEs, and to construct solutions resulting from sym-metries. The present account is largely based on our papers ([148], [168], and[170]).

6.1 Waves in Self-Similar Flows

In this section, the work of Rogers [152], which examines the closed formanalytical solution for a spherically symmetric blast wave in an atmosphereof varying density, is extended to include in a unified manner the plane andcylindrical cases along with the spherical one. The flow properties behindthese waves are completely characterized and certain observations are notedin respect of their contrasting behavior. Finally, the exact self-similar solutionand the results of interaction theory are used to study the interaction betweena weak discontinuity wave and a blast wave in plane and radially symmetricflows.

The system describing a planar (m = 0), cylindrically symmetric (m = 1)or spherically symmetric (m = 2) adiabatic motion of a perfect gas havingvelocity u, pressure p, and density ρ may be written as

u,t + Au,r + b = 0, (6.1.1)

where u = (u, ρ, p)tr ,A =

u 0 ρ−1

ρ u 0ρa2 0 u

, b = (0,mρu/r,mρa2u/r)tr .

Here a2 = γp/ρ is the speed of sound with γ the specific heat ratio for the gas,r is the linear distance in the planar case (m = 0), or the distance measuredfrom the center in the cylindrical and spherical cases (m = 1, 2), and t denotesthe time.

At time t = 0, an explosion is assumed to take place at a point (m = 2), orline (m = 1), or plane (m = 0) driving ahead of it a blast wave (strong shock)with radius R(t) and velocity s(t) = dR/dt, propagating into a medium thedensity of which decreases as the inverse power of the distance from the sourceof explosion, so that ρ∗ = ρcr

−α, where ρc is the central density before theexplosion and α is a positive real number. An asterisk will be used to indicatequantities evaluated in the quiescent medium immediately ahead of the blastwave.

It should be noted that the matrix A in (6.1.1) has the eigenvalues

λ(1) = u+ a, λ(2) = u, λ(3) = u− a, (6.1.2)


and the corresponding left and right eigenvectors

L(1) = (1, 0, 1/ρa), L(2) = (0, 1,−1/a2), L(3) = (1, 0,−1/ρa),

R(1) = (1, ρ/a, ρa)tr , R(2) = (0, 1, 0)tr , R(3) = (1,−ρ/a,−ρa)tr .(6.1.3)

We now consider a C1 discontinuity wave across the fastest characteristicdetermined by dr/dt = λ(1), and originating from the point (r0, t0) in theregion swept up by the blast wave, and envisage the situation when this C1

discontinuity wave encounters a shock wave at time t = tp > t0. In order tostudy the time evolution of this C1 discontinuity wave, and to determine theinitial amplitudes of the transmitted and reflected waves, if any, we need toknow the solution of the system (6.1.1) in the disturbed region behind theblast wave. To achieve this, we introduce a similarity variable ξ = r/R(t) andseek a solution of the form [152]

u = sf(ξ), ρ = ρ∗(R)h(ξ), p = ρ∗s2g(ξ)/γ, (6.1.4)

where ρ∗ = ρcR−α, s = dR/dt, and

f(1) =2

γ + 1, h(1) =

γ + 1

γ − 1, g(1) =

2γ

γ + 1. (6.1.5)

Since the total energy inside a blast wave remains constant under adiabaticconditions for all times, it turns out that α ≤ m + 3, and the radius of theblast wave is given by

R(t) =

Kt2/(3+m−α), when α < m+ 3R0 exp(Kt), when α = m+ 3

, (6.1.6)

where K is a constant related to the total amount of energy generated by theexplosion, and R0 is the blast wave radius at t = 0.

6.1.1 Self-similar solutions and their asymptotic behavior

In view of equations (6.1.4) and (6.1.6)1, the basic set of equations (6.1.1)can now be expressed in the following matrix formξ − f −1/γh 0−γ (ξ − f)/g 0−1 0 (ξ − f)/h

f ′

g′

h′

=

f(α−m− 1)/2(mγf/ξ) − (m+ 1)

(mf/ξ) − α

,

(6.1.7)where a prime denotes differentiation with respect to ξ. The energy conserva-tion equation provides a first order differential equation which on integrationyields

2 (γf − ξ) g = γ (γ − 1) (ξ − f)hf 2, (6.1.8)

while equations (6.1.7)2,3 can be combined to yield

h = C1g1+m−α

(m+1)(γ−1) ((ξ − f) ξm)1+m−α

(m+1)(γ−1) , (6.1.9)


where the integration constant C1 can be determined by using (6.1.5), as hasbeen done in writing (6.1.8). The compatibility condition for f can be obtainedfrom (6.1.7)1 on using (6.1.8) and (6.1.9) and it has the following form

(ξ/f) f ′ =mγ (1 − γ) (f/ξ)

2+ [(m+ 1) (2γ − 1) − αγ] (f/ξ) − (m+ 1) + α

γ (γ + 1) (f/ξ)2 − 2 (γ + 1) (f/ξ) + 2

.

(6.1.10)It is clear that the system (6.1.7) together with (6.1.5) admits single valuednonsingular solutions provided the matrix on the left of (6.1.7) is nonsingularin the interval (0,1). That is, the functions

φ(ξ) = ξ − f and Ψ(ξ) = (ξ − f) − (g/h)2

do not vanish at any point, ξ, lying in the interval (0,1). Equation (6.1.8)shows that Ψ(ξ) vanishes nowhere on (0,1); for if it does for some ξ (say, ξ1),then f (ξ1) is given by the quadratic equation,

γ (γ + 1) f2 (ξ1) − 2ξ1 (γ + 1) f (ξ1) + 2ξ21 = 0,

but this does not yield a real expression for f (ξ1), and so the assertion isjustified. However, the function φ(ξ) may vanish for some ξ1 ∈ (0, 1), therebyindicating a singularity in the solution at ξ = ξ1; this situation depends onthe values of α. Indeed, it turns out that there exists a critical value αc of αgiven by

αc = 1 + (m (3 − γ) / (γ + 1)) ,

such that no singularity in the solution occurs for α ≤ αc; possible singulari-ties, if any, can occur only for α > αc. On integration, (6.1.10) yields

f = C2ξk1 (γf − ξ)

ω1 ((3 +m− α) ξ/ m (γ − 1) + γ + 1 − f)s1 , (6.1.11)

where

k1 = m (γ − 1) (3 +m− α) / (2 (m− 1 −mγ − γ)) ,

ω1 = (γ − 1) (α−m− 3) / (2 (m− 1 + 2γ − αγ)) ,

s1 =

(m2 + α2 + 2m− 4α+ 5

)γ2 +

(α2 − 3m2 + 2m− 2α− 2αm+ 1

]γ

2 [m (1 − γ) − (γ + 1)) (m− 1 + (2 − α)γ)

+

(4m2 − 2αm+ 2α− 4

)

2 (m (1 − γ) − (γ + 1)) (m− 1 + (2 − α)γ),

and C2 is determined by the boundary conditions (6.1.5). The forms of ω1

and s1 suggest that the solution (6.1.11) is valid as long as α 6= 2+(m−1)/γ.For α = 2 + (m− 1)/γ > αc, integration of (6.1.10) gives

f = C3ξm(1−γ)

2γ (γ f − ξ)γ−1

γ exp

((γ − 1) ξ

2γ (γ f − ξ)

), (6.1.12)


where C3 is determined by (6.1.5). It should be noted that the correspondingsolution for the spherical (m = 2) case obtained by Rogers (see equation (65)of [152]) is in error, as can easily be verified; however, the correct solutionshould be as follows

f = K6x(1−γ)/γ(γf − x)(γ−1)/2γ exp

((γ − 1)x

2γ (γf − x)

),

which is recovered from (6.1.12) with m = 2, since ξ = x and C3 = K6. Theasymptotic behavior of the solutions of (6.1.11) and (6.1.12) near the originare as follows.

Case I. When α = αc = 1 + (m (3 − γ) / (γ + 1)) , the velocity, density andpressure behind plane, cylindrical and spherical waves are given by

f(ξ) =2ξ

γ + 1, h(ξ) =

γ + 1

γ − 1ξm−1, g(ξ) =

2γ

γ + 1ξm+1. (6.1.13)

Equation (6.1.13)1 follows from (6.1.10), while equation (6.1.13)2,3 follow from(6.1.8) and (6.1.9). The solution behind spherical waves found by Kopal etal. [95] and Rogers [152] is recovered from (6.1.13) with m = 2. It shouldbe noted that, unlike the case of spherical waves, in the case of plane andcylindrical waves αc does not vanish for γ. The density behind plane waves,unlike cylindrical and spherical waves, exhibits singular behavior at the origin;indeed, it increases (respectively; decreases) from the shock towards the centerin the case of plane (respectively; spherical) shocks, while it remains constantbehind cylindrical waves. Numerical solutions of equations (6.1.7) satisfying(6.1.5) are exhibited in Figs. 6.1.1 – 6.1.3; the situation under considerationhere is illustrated by the curve VI in Fig. 6.1.1.

Case II. When α < αc, it follows from (6.1.11) that near ξ = 0,

f(ξ) ∼ ξ

γ+

γ + 1

(γ − 1)γξθ,

where θ = ((3 − α) γ +m− 2) / (γ − 1) . Thus the velocity for plane, cylin-drical and spherical waves is nearly a linear function of ξ. The density andpressure distributions near ξ = 0 can be obtained from (6.1.8) and (6.1.9) inthe following form

h(ξ) = O(ξδ), g(ξ) = O(1) as ξ → 0,

where δ = (1 +m− αγ)/(γ − 1). Thus, depending on the value of α, we havethe following cases: (i) when α ≤ (2 +m − γ)/γ, the density vanishes at thecenter. Curves I and II in Fig. 6.1.1 are representative of the correspondingsituations. (ii) When (m + 2 − γ)/γ < α < (m + 1)/γ, the density vanishesat the origin but the density gradient becomes unbounded there as shown bycurve III in Fig. 6.1.1. (iii) When α is equal to (respectively; greater than)


the value (m+ 1)/γ, the density at the origin is finite (respectively; infinite).Thus, the density distribution behind plane, cylindrical and spherical wavesis regular on (0,1) except, perhaps, at the origin, where the behavior could besingular; the pressure, however, remains finite at the origin in this case. Therespective situations are delineated by curves IV and V in Fig. 6.1.1. Whenα = (m + 1)γ, we have the following interesting situation. Equation (6.1.9)yields gh−γ = constant, while equations (6.1.4)2,3 together with (6.1.5) imply

pρ−γ = 4ρ(1−γ)c K(3+m−α)R(αγ−1−m)gh−γ

(γ (3 +m− α)2

)−1

,

which, for α = (m + 1)/γ, yields pρ−γ = constant, and thus in this case theflow behind a plane, cylindrical or spherical shock is isentropic.

Case III. When α > αc, as remarked earlier, a singularity in the solutionappears at ξ = ξ1 ∈ (0, 1), where φ(ξ1) = 0. In this case, it turns out thatthe velocity behind cylindrical and spherical waves tends to infinity near thecenter, whereas it remains finite there for plane waves. Indeed, in sphericalwaves the rise near the center is sharper when compared to the cylindricalwave case. Following a standard procedure, we find from (6.1.11) that

f(ξ) = O(ξ−m(γ−1)/(γ+1)

)as ξ → 0. (6.1.14)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

ξ

ρ /ρ ∗

VIIa

VIIbVIb

VIa

Vb

Va

IVbIVa

IaIbIIaIIbIIIaIIIb

Figure 6.1.1: Density distribution behind plane (m = 0, - - - -) and cylindrical(m = 1, ) shock fronts for γ = 5/3.. Ia: α =0.0, Ib : α =0.0, IIa: α =0.8,IIb : α =0.2, IIIa: α =1.0, IIIb : α =0.4, IVa: α =1.2, IVb : α =0.6, Va: α =1.4,Vb : α =0.75, VIa: α =1.5, VIb : α =1.0, VIIa: α =2.0, VIIb : α =2.0.

It follows from (6.1.8) and (6.1.9) that for α < 2(m + 1)/(γ + 1), andthus for plane and cylindrical waves, α < αc and so the possibility of a singu-larity in the solution on (0,1) is ruled out. However, for spherical waves thissituation includes the possibility that α > αc, thereby indicating the possibil-ity of a singularity in the solution at ξ1 ∈ (0, 1), where the density vanishesbut the density gradient becomes unbounded. It should be noted that forα = 2(m + 1)/(γ + 1) there is no singularity in the density, in the sense


that both the density and its gradient are finite at ξ = ξ1. However, whenα > 2(m + 1)/(γ + 1), the density becomes infinite at ξ1 while its gradientchanges sign indicating that the gas is further compressed after the passage ofthe shock. The pressure distribution behind plane, cylindrical and sphericalwaves can be discussed in a similar manner by exploiting the result obtainedfrom (6.1.8) and (6.1.9) after eliminating the function h(ξ). It turns out thatfor α ≤ 2 (m+ 1) / (γ + 1), as shown by curves I and II in Fig. 6.1.2, andby curves IV and V in Fig. 6.1.3, the pressure distribution behind plane andcylindrical waves is regular on (0,1), while for spherical waves the solutionexhibits a singularity at a point ξ1 ∈ (0, 1), where the gas pressure and itsgradient are finite.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

VIV

IVIII

III

ξ

p/ρ∗S2

0 0.2 0.4 0.6 0.8 1.00

0.5

1

1.5

2

VIII

VII

VI

VIVIIIII

I

ξ

p/ρ∗S2

Figure 6.1.2: Pressure distribution Figure 6.1.3: Pressure distribution

behind plane shock front (m = 0) behind cylindrical shock front

for γ = 5/3. I: α =0.4, II: α =0.75, (m = 1) for γ = 5/3. I: α =0.6,

III: α=0.9, IV: α =1.0, II: α =1.0, III: α =1.2, IV: α =1.4,

V: α =1.5, VI: α =2.0. V: α =1.5, VI: α =1.7,

VII: α =2.0, VIII: α =2.5.

When α > 2(1 + m)/(γ + 1), as shown by curves III, IV and V in Fig.6.1.2, and by curves VI, VII and VIII in Fig. 6.1.3, the pressure at ξ1 is zero,finite or infinite according as α is, respectively, lees than, equal to, or greaterthan m+1; in these cases the pressure gradient is, respectively, infinite, finiteor negatively infinite at ξ1, showing in effect that when α = m + 1, there isno singularity in pressure.

Case IV. When α = m + 3, equation (6.1.6) shows that the system (6.1.1)admits a similarity solution of the type given in (6.1.4) in a region headedby a strong shock of radius R = R0 exp(Kt), propagating into a medium thedensity of which varies according to ρ∗ = ρcR

−(m+3). The solution for f(ξ) inthis case can be found from (6.1.10), for α = m+ 3, in the form

f = K1 (ξm (γf − ξ))−(γ−1)/2

exp(−ξ/f), (6.1.15)


where K1 is determined by (6.1.5)1. The density follows from (6.1.8) and(6.1.9). The asymptotic behavior of the velocity near the origin is the same asthat given by (6.1.14); that is, unlike plane waves where the velocity remainsfinite near the center, the velocity in cylindrical and spherical waves tends to∞; however, the rise near the center is more rapid in spherical waves thanin cylindrical waves. Again, a singularity in the solution occurs at the pointξ1 ∈ (0, 1), which from (6.1.15) is given by

ξ1 =

2 exp((γ − 1)/2)/(γ + 1)(γ+1)/22/[γ+1+m(γ−1)]

,

which shows that for a given γ > 1, an increase in m causes ξ1 to increase.That is, an increase in the wave front curvature causes the singular point tomove toward the wave front. However, an increase in γ has the reverse effect,so that for a given geometry of the wave front, an increase in γ causes ξ1 tomove towards the wave center.

6.1.2 Collision of a C1-wave with a blast wave

We now consider system (6.1.1). Let Λ1 denote the jump in u,r acrossthe C1 discontinuity propagating along the curve determined by dr/dt =λ(1), originating from the point (r0, t0) in the region behind the blast wave,considered in the preceding sections. In view of equations (6.1.4), (6.1.6) and(6.1.13), the medium ahead of this C1 discontinuity wave is characterized bythe solution vector u0 of the form

u0 =

2

γ + 1

r

λt,γ + 1

γ − 1ρcR

−α( rR

)m−1

,2

γ + 1ρcR

−α(R

λt

)2 ( rR

)m+1tr

,

(6.1.16)where λ = (3 + m − α)/2, α = 1 + m(3 − γ)/(γ + 1) and R = Kt1/λ. Let

α(1)(t) be the amplitude of this C1 discontinuity; then Λ1 = α(1)R(1)0 , where

R(1)0 is the vector R(1) in (6.1.3) evaluated in the state u = u0 described by

(6.1.16). In view of (6.1.1), (6.1.2), (6.1.3) and (6.1.16), system (4.3.8) reducesto a single Bernoulli type equation in α(1), which on integration yields

α(1)(t) = [t/t0]−να

(1)0 (t0)

1 +

(α

(1)0 (t0)/Z

)(1 − (t0/t)

ν−1)−1

, (6.1.17)

where α(1)0 is the initial wave amplitude at t = t0

Z = 2(γ − 1)(γ + 1)t0−1 > 0, ν = (m+ 1)(γ + 2Γ) −m+ 5/(2θ),

with Γ2 = 2γ(γ− 1) and θ = (γ+ 1) +m(γ− 1). It should be noted that for aplane (m = 0), or a radially symmetric (m = 1 or 2) flow, ν > 1 for values ofγ > 1. Thus the occurrence of a secondary shock, in the region behind the blast

wave, can take place at a time t = τ > t0, given by τ = t01−Z/|α(1)0 |1/(ν−1),


provided the initial weak discontinuity wave is compressive (α(1)0 < 0) with

its initial amplitude satisfying the inequality α(1)0 < −Z < 0. It is clear that

when the initial weak discontinuity is an expansion wave (α(1)0 > 0), or a

compression wave (α(1)0 < 0) with −Z ≤ α

(1)0 < 0, there is no secondary shock

formation in the self-similar region behind the blast wave, and the C1 wavewill impinge upon the blast wave at some t > t0. However, a compressive C1

wave with α(1)0 < −Z < 0 can meet the blast at, say, t = tp ≡ t, provided

t0 < t < τ ; an equivalent condition for this to occur follows from the foregoingrelations and the fact that r(t) = R(t) in the form

t0

∣∣∣α(1)0

∣∣∣ < 2(ν − 1)/(γ + 1)1 − [R0/r0]

−(ν−1)θ/(Γ−γ+1).

The eigenvalues, the corresponding eigenvectors and the shock velocitys = dR/dt = (γ + 1)R/θt, when evaluated at the collision time t = t, yield

λ(1) = λ(1)(t) = s(2 + Γ)/(γ+ 1), λ(2) = 2s/(γ+1), λ(3) = s(2−Γ)/(γ+ 1),

where s ≡ s(t) = (γ+ 1)R/(θt) with R = R(t). Also, since the pressure aheadof the blast wave (where the variables are designated by an asterisk) is verysmall when compared with the pressure behind, it follows immediately thatthe Lax evolutionary conditions (4.3.22) for a “physical shock” are satisfiedfor the index ` = 2, so that

λ(3) < λ(2) < s < λ(2), λ∗(3) < λ∗(2) < λ∗(1) < s.

In effect, this asserts that when the incident wave with velocity λ(1) encountersthe blast wave, it gives rise to two reflected waves with velocities λ(2) and λ(3),but no transmitted wave. The reflection coefficients α(2), α(3) and the jumpin shock acceleration [[s]] at the collision time t = t can be determined fromthe algebraic system (4.3.23), which can be written in the following form

[[•s]](G −G∗)0 + (∇G)0R

(2)0 α(2)

(s− λ(2)

)2

+ (∇G)0R(3)0 α(3)

(s− λ(3)

)2

= −(∇G)0R(1)0 α(1)

(s− λ(1)

)2

, (6.1.18)

where all the quantities are evaluated in the state u0 at t = t, so that

(G −G∗)0 =

2ρ(γ − 1)−1(γ + 1)

(R/θt

), 1, 2(γ + 1)

(R/θt

)2tr

,

(∇G)0 R(2)0 =

2(R/θt

), 1, 2

(R/θt

)2tr

,

(∇G)0 R(3)0 = Ψ

(Γ − 2) ,−

(R/θt

)−1

,−2(R/θt

)((γ + 1 − Γ)

tr,

(∇G)0 R(1)0 = Ψ

(Γ + 2) ,

(R/θt

)−1

, 2(R/θt

)((γ + 1 − Γ)

tr,


where Ψ = ρ(γ + 1) Γ (γ − 1)−1and ρ = ρcR

−α.The system (6.1.18) yields the following solution at t = t

[[s]] = −Γ(γ + 1) 3γ − 2Γ − 1 (R/θt)[t/t0]−να(1)

0

(Γ + 2)1 +

[α

(1)0 /Z

] [1 −

[t0/t

]ν−1] ,

α(2) = − 4(γ + 1) 3γ − 2Γ − 1 ρ[t/t0

]−να

(1)0

(Γ + 2)Γ(γ − 1)(R/θt

)1 +

[α

(1)0 /Z

] [1 −

[t0/t

]ν−1] , (6.1.19)

α(3) = − (Γ − 2) (3γ − 2Γ − 1)2

(Γ + 2) (Γ + γ − 1)2

[t/t0

]−ν1 +

[α

(1)0 /Z

] [1 −

[t0/t

]ν−1]α(1)

0 .

The coefficients α(2) and α(3) determine the amplitude vectors

Λ(R)2 = α(2)(t)R(2), Λ

(R)3 = α(3)(t)R(3)

of the reflected waves along the characteristic lines with the velocities λ(2) andλ(3), respectively. Equations (6.1.19) demonstrate, as would be expected, thatin the absence of the incident wave, i.e.,

α(1)0 = 0,

the jump in the shock acceleration vanishes and there are no reflected waves.Moreover an increase in the magnitude of the initial discontinuity of the inci-dent wave causes the reflection coefficients and the jump in shock accelerationto increase in magnitude. It is clear from (6.1.19)1 that for γ lying in the

range 1 < γ < 2, the coefficient α(1)0 is always negative, thereby indicating

that after the impact the shock will either accelerate or decelerate, dependingon whether the incident wave is compressive or expansive. This conclusionis in agreement with the observations made by Friedrichs [41], that if theshock front is overtaken by a compression (respectively; expansion) wave, itis accelerated (respectively; decelerated) and consequently the strength of the

shock increases (respectively; decreases). Moreover, since the coefficient α(1)0

in (6.1.19)3 is negative for values of γ in the range 1 < γ < 2, this is indicativeof the situation that the reflected wave along the characteristic dr/dt = u−a isa compression wave whenever the incident wave is expansive, and conversely.This behavior may, perhaps, be attributed to the nonlinear effects. The presentanalysis can really only hint at the manner in which the incident wave inter-acts with the shock and the reflected waves evolve. However, a more elaboratetreatment offering a simultaneous account of the solution in the general waveregion behind the incident and reflected waves together with the evolutionarybehavior of the reflected waves which seem to be responsible for some of theinteresting features of the wave motion that finally develop, offer substantialhope for progress in the understanding of this complex problem.


6.2 Imploding Shocks in a Relaxing Gas

It is well known that in contrast to the self-similar solutions of the firsttype, where the self-similarity of the solution and its representation can beobtained from dimensional analysis, the nature of the similarity in respectof self-similar solutions of the second type is revealed neither by dimensionalanalysis nor by other group properties of the medium, but only by actuallysolving the equations as a nonlinear eigenvalue problem. The collapse of animploding shock wave, is one of the examples of a class of self-similar solutionsof the second type. A theoretical investigation of imploding shock waves in aperfect gas has been presented by Guderley [66], Zeldovich and Raizer [214],and Lazarus [103]; Van Dyke and Guttmann [202] as well as Hafner [68] havederived solutions to the convergent shock problem by means of alternativemethods. Assuming the shock to be infinitely strong, Logan and Perez [112]discussed the self-similar solutions to a problem concerning plane flow of areacting gas with an arbitrary form of the reaction rate.

In flows with imploding shocks, conditions of very high temperature andpressure can be produced near the center (axis) of implosion on account of theself-amplifying nature of imploding shocks. As a result of high temperaturesattained by gases in motion, the effects of nonequilibrium thermodynamics onthe dynamic motion of a converging shock wave can be important. Ever sincethe nonequilibrium effects were recognized to be important in high temper-ature gasdynamics, considerable efforts have been devoted to the restudy ofthe classical problem in the light of these new considerations. Here, we usethe method of Lie group invariance under infinitesimal point transformations([112], [17]) to determine the class of self-similar solutions for unsteady planarand radially symmetric flows of an inviscid relaxing gas involving shocks, anddiscuss the qualitative features of the possible self-similar solutions in a uni-fied manner. The departure from equilibrium is due to vibrational relaxationof a diatomic gas; the rotational and translational modes are assumed to be inlocal thermodynamical equilibrium throughout. The group theoretic methodenables us to characterize completely the state dependent form of the relax-ation rate for which the problem is invariant and admits self-similar solutions.The arbitrary constants, occurring in the expressions for the generators of thelocal Lie group of transformations, give rise to different cases of possible solu-tions with a power law, exponential or logarithmic shock paths. A particularcase of the collapse of an imploding shock is worked out in detail for radiallysymmetric flows. For typical values of the flow parameters, the value of theself-similar exponent is uniquely determined from the condition that the solu-tion of the system of ordinary differential equations describing the self-similarmotion is regular on a regular characteristic passing through the center (axis)of implosion. Indeed, the system of ordinary differential equations is integratediteratively with the similarity exponent so adjusted that the system is made

6.2 Imploding Shocks in a Relaxing Gas 177

regular. Numerical calculations have been performed to determine the valuesof the self-similarity exponent and the profiles of the flow variables behind theshock; for the sake of comparison the similarity exponent is also computedusing Whitham’s rule. This work brings out some interesting features of theinteraction between the relaxing mode and the fluid flow. In the absence ofvibrational relaxation, our results are in good agreement with the values ofthe similarity parameter quoted in [66], [68] and [159].

6.2.1 Basic equations

Assuming that the gas molecules have only one lagging internal mode (i.e.,vibrational relaxation) and the various transport effects are negligible, thebasic equations for an unsteady one dimensional planar (m = 0), cylindrically(m = 1) or spherically (m = 2) symmetric motion of a relaxing gas are [37]

ρ,t + ρu,x + uρ,x +mρu/x = 0, (6.2.1)

u,t + uu,x + ρ−1p,x = 0, (6.2.2)

p,t + up,x + ρa2(u,x +mu/x) + (γ − 1)ρQ = 0, (6.2.3)

σ,t + uσ,x = Q, (6.2.4)

where x is the spatial coordinate being either axial in flows with planar ge-ometry or radial in cylindrically and spherically symmetric flows, t the time,u the particle velocity, ρ the density, p the pressure, σ the vibrational energyand a = (γp/ρ)1/2 the frozen speed of sound with γ being the frozen specificheat ratio of the gas. The quantity Q, which is a known function of p, ρ andσ, denotes the rate of change of vibrational energy, and is given by [163]

Q = σ(p, ρ) − σ /τ(p, ρ) , (6.2.5)

where σ is the equilibrium value of σ defined as σ = σ0(x)+c(p, ρ)R(T−T0(x));here T is the translational temperature, R is the specific gas constant, suffix 0refers to the initial rest conditions, and the quantities τ and c are respectivelythe relaxation time and the ratio of vibrational specific heat to the specificgas constant. The equation of state is taken to be of the form

p = ρRT. (6.2.6)

Now, we consider the motion of a shock front, x = X(t), propagating into aninhomogeneous medium specified by

u0 ≡ 0, p0 ≡ constant, ρ0 = ρ0(x), σ0 = σ0(x). (6.2.7)

Let V be the shock velocity; then the usual Rankine Hugoniot jump conditionsacross the shock front may be written as

u(X(t), t) = 2γ+1

ρ0(X(t))V 2−γp0ρ0(X(t))V , ρ(X(t), t) =

(γ+1)ρ20(X(t))V 2

(γ−1)ρ0(X(t))V 2+2γp0,

p(X(t), t) = 2ρ0(X(t))V 2−(γ−1)p0γ+1 , σ(X(t), t) = σ0(X(t)).

(6.2.8)


6.2.2 Similarity analysis by invariance groups

Since the system (6.2.1) – (6.2.4) is a set of quasilinear hyperbolic partialdifferential equations, it is hard, in general, to determine a solution withoutapproximations. Here, we assume that there exists a solution of (6.2.1) –(6.2.4) subject to (6.2.8) along a family of curves, called similarity curves,for which the set of partial differential equations reduce to a set of ordinarydifferential equations, and also assume that the shock trajectory is embeddedin the family of similarity curves; this type of solution is called a similaritysolution.

In order to determine a similarity solution and the similarity curves, weseek a one parameter infinitesimal group of transformations

t∗ = t+ εψ(x, t, ρ, u, p, σ), x∗ = x+ εχ(x, t, ρ, u, p, σ),

ρ∗ = ρ+ εS(x, t, ρ, u, p, σ), u∗ = u+ εU(x, t, ρ, u, p, σ), (6.2.9)

p∗ = p+ εP (x, t, ρ, u, p, σ), σ∗ = σ + εE(x, t, ρ, u, p, σ),

where the generators ψ, χ, S, U, P and E are to be determined in such a waythat the partial differential equations (6.2.1) – (6.2.4), together with the con-ditions (6.2.7) and (6.2.8), are invariant with respect to the transformations(6.2.9); the entity ε is so small that its square and higher powers may beneglected. The existence of such a group allows the number of independentvariables in the problem to be reduced by one, and thereby allowing the system(6.2.1) – (6.2.4) to be replaced by a system of ordinary differential equations.

In the sequel, we shall use summation convention and, therefore, introducethe notation

x1 = t, x2 = x, u1 = ρ, u2 = u, u3 = p, u4 = σ, pij = ∂ui/∂xj ,

where i = 1, 2, 3, 4 and j = 1, 2. The system (6.2.1) – (6.2.4), which can berepresented as

Fk(xj , ui, pij) = 0, k = 1, 2, 3, 4,

is said to be constantly conformally invariant under the infinitesimal group(6.2.9), if there exist constants αrs(r, s = 1, 2, 3, 4) such that for all smoothsurfaces, ui = ui(xj), we have

LFk = αkrFr, (6.2.10)

where L is the Lie derivative in the direction of the extended vector field

L = ξjx∂

∂xj+ ξiu

∂

∂ui+ ξipj

∂

∂pij,

with ξ1x = ψ, ξ2x = χ, ξ1u = S, ξ2u = U, ξ3u = P, ξ4u = E, and

ξipj=∂ξiu∂xj

+∂ξiu∂uk

pkj −∂ξ`x∂xj

pi` −∂ξ`x∂un

pi`pnj , ` = 1, 2, n = 1, 2, 3, 4 (6.2.11)


being the generators of the derivative transformation.

Equation (6.2.10) implies

∂Fk∂xj

ξjx +∂Fk∂ui

ξiu +∂Fk∂pij

ξipj= αksFs, k = 1, 2, 3, 4.

Substitution of ξipjfrom (6.2.11) into the last equation gives a polynomial in

the pij . Setting the coefficients of pij and pijpkl to zero gives a system of first

order, linear partial differential equations in the generators ψ, χ, S, U, P andE. This system, which is called the system of determining equations of thegroup, can be solved to find the invariance group (6.2.9).

If the above program is carried out for the the system of partial differentialequations (6.2.1) – (6.2.4), we obtain the most general group under which thesystem is invariant; indeed, the invariance of the continuity equation (6.2.1)yields the following system of determining equations

S,t + uS,x + ρU,x +m

x

(ρU + uS − ρuχ

x

)= α13

(γ − 1)ρQ+

mγpu

x

+ α11mρu

x− α14Q,

S,ρ − ψ,t − uψ,x = α11, S,u − ρψ,x = α12, S,p = α13,

S,σ = α14, U − χ,t + uS,ρ − uχ,x + ρU,ρ = α11u, (6.2.12)

S + uS,u + ρ(U,u − χ,x) = α11ρ+ α12u+ α13γp,

ρU,p + uS,p = α12/ρ+ α13u, ρU,σ + uS,σ = α14u.

In a similar way, invariance of the momentum equation (6.2.2) yields

U,t + uU,x + ρ−1P,x = α21mρu/x+ α23 (γ − 1)ρQ +mγpu/x − α24Q,

U,ρ = α21, U,u − ψ,t − uψ,x = α22, U,p − ρ−1ψ,x = α23, U,σ = α24,

U − χ,t + u(U,u − χ,x) + ρ−1P,u = α21ρ+ α22u+ α23γp, (6.2.13)

uU,ρ + ρ−1P,ρ = α21u, −S/ρ2 + uU,p + ρ−1(P,p − χ,x) = α22ρ−1 + α23u,

uU,σ + ρ−1P,σ = α24u.

Next, invariance of the energy equation (6.2.3) yields

P,t + uP,x + γpU,x +mγ(uP + pU − puχ/x)/x+ (γ − 1)SQ

+ (γ − 1) SQ,ρ + PQ,p +EQ,σ ρ = α31mρu/x

+ α33 (γ − 1)ρQ +mγpu/x− α34Q,

P,ρ = α31, P,u − γpψ,x = α32, P,p − ψ,t − uψ,x = α33,

P,σ = α34, uP,ρ + γpU,ρ = α31u, (6.2.14)

γP + uP,u + γp(U,u − χ,x) = α31ρ+ α32u+ α33γp,

U − χ,t + uP,p − uχ,x + γpU,p = α32/ρ+ α33u, uP,σ + γpU,σ = α34u.


Finally, invariance of the rate equation (6.2.4) yields

E,t + uE,x − (SQ,ρ + PQ,p +EQ,σ) = α41mρu/x

+α43 (γ − 1)ρQ +mγpu/x− α44Q,

E,ρ = α41, E,u = α42, E,p = α43,

E,σ − uψ,x − ψ,t = α44, uE,ρ = α41u, (6.2.15)

uE,u = α41ρ+ α42u+ α43γp, uE,p = α42/ρ+ α43u,

U − χ,t + uE,σ − uχ,x = α44u, ψ = ψ(x, t), χ = χ(x, t).

From (6.2.12)5,9, (6.2.13)2,5,9 and (6.2.14)2,5,6 we have U,σ = P,σ = U,ρ =P,ρ = α21 = α31 = α24 = α34 = 0; consequently, derivatives of (6.2.13)4 and(6.2.13)6 with respect to ρ and p yield U,pρ = ψ,x = U,p = α23 = 0. Thus, wehave from (6.2.12)4,8 and (6.2.15)4,8 that S,u = E,u = α12 = α42 = 0.

Next, differentiating (6.2.12)7 with respect to p and σ and using (6.2.12)4,we get S,p = S,σ = α13 = α14 = 0. Similarly, using (6.2.14)7 and (6.2.15)7, wehave S = S(x, t, ρ), U = U(x, t, u), P = P (x, t, p), E = E(x, t, σ) and αij = 0for i 6= j. Finally, equations (6.2.12)2, (6.2.13)3, (6.2.14)4 and (6.2.15)5 leadto

S,ρρ = U,uu = P,pp = E,σσ = 0,

which, in turn, imply

S = (α11 + ψ,t)ρ+ S1(x, t), U = (α22 + ψ,t)u+ U1(x, t),P = (α33 + ψ,t)p+ P1(x, t), E = (α44 + ψ,t)σ +E1(x, t), ψ = ψ(t),

(6.2.16)where S1, U1, P1 and E1 are arbitrary functions of x and t only. Substitu-tion of (6.2.16) into (6.2.12)7, (6.2.13)1,6, (6.2.14)1,7, and (6.2.15)1 yields onsimplification

S1 ≡ P1 ≡ 0, E1 ≡ d, U1 ≡

0, if m = 1, 2k1, if m = 0

,

ψ = at+ b, χ =

(2a+ α22)x, if m = 1, 2(2a+ α22)x+ k1t+ c, if m = 0

, (6.2.17)

SQ,ρ + PQ,p +EQ,σ = α44Q, α33 = 2α22 + α11 + 2a, α44 = 2α22 + a,

where a, b, c, d and k1 are integration constants. Using equations (6.2.17)1,2,3,4,8in equations (6.2.16) and (6.2.17)6, we obtain

S = (α11 + a)ρ, U =

(α22 + a)u, if m = 1, 2(α22 + a)u+ k1, if m = 0

,

P = (2α22 + α11 + 3a)p, E = 2(α22 + a)σ + d, (6.2.18)

ψ = at+ b, χ =

(α22 + 2a)x, if m = 1, 2(α22 + 2a)x+ k1t+ c, if m = 0

,

SQ,ρ + PQ,p +EQ,σ = (2α22 + a)Q.

Thus, the generators of the local Lie group of transformations, involving ar-bitrary constants a, b, c, d, k1, α11 and α22, are known.


6.2.3 Self-similar solutions and constraints

The arbitrary constants, occurring in the expressions for the generators ofthe local Lie group of transformations, give rise to several cases of possiblesolutions, which we discuss below.

Case I. When a 6= 0 and α22 + 2a 6= 0, the change of variables from (x, t) to(x, t) defined as

x =

x, if m = 1, 2

x+ c (α22 + 2a)−1 , if m = 0; t = t+ b/a,

does not alter the equations (6.2.1) – (6.2.4); thus, rewriting the set of equa-tions (6.2.18) in terms of the new variables x and t, and then suppressing thetilde sign, we have

S = (α11 + a)ρ, U =

(α22 + a)u, if m = 1, 2(α22 + a)u+ k1, if m = 0

,

P = (2α22 + α11 + 3a)p, E = 2(α22 + a)σ + d, ψ = at, (6.2.19)

χ =

(α22 + 2a)x, if m = 1, 2(α22 + 2a)x+ k1t, if m = 0

,

SQ,ρ + PQ,p +EQ,σ = (2α22 + a)Q.

The similarity variable and the form of similarity solutions for ρ, u, p, σ and Qreadily follow from the invariant surface condition, ui(x

∗, t∗) = u∗i (x, t), whichyields

ψρ,t + χρ,x = S, ψu,t + χu,x = U,

ψp,t + χp,x = P, ψσ,t + χσ,x = E.

The last set of equations together with (6.2.19)7 yields on integration thefollowing forms of the flow variables

ρ = t(1+α11/a)S(ξ), u =

t(δ−1)U(ξ), if m = 1, 2

t(δ−1)U(ξ) − k∗, if m = 0,

p = t(2δ−1+α11/a)P (ξ), σ = t2(δ−1)E(ξ) − d∗, Q = p(2δ−3)a

α11+(2δ−1)a q(η, ζ) ,(6.2.20)

where k∗ = k1/(δ − 1)a, d∗ = d/2(δ − 1)a and q is an arbitrary function of ηand ζ

η = ρp− α11+a

α11+(2δ−1)a , ζ = p (σ + d∗)−α11+(2δ−1)a

2(δ−1)a , δ = (α22 + 2a)/a. (6.2.21)

The functions S, U , P and E depend only on dimensionless form of the simi-larity variable ξ, which is determined as

ξ =

x/(Atδ), if m = 1, 2

x/(Atδ) + k∗t(1−δ)/A, if m = 0,, (6.2.22)


where A is a dimensional constant, whose dimensions are obtained by thesimilarity exponent δ. Since the shock must be a similarity curve, it may benormalized to be at ξ = 1. The shock path and the shock velocity V are, then,given by

X =

Atδ, if m = 1, 2,

Att(δ−1) − k∗/A

. if m = 0.

, (6.2.23)

V =

δX/t , if m = 1, 2Aδt(δ−1) − k∗

, if m = 0.

(6.2.24)

Equation (6.2.20)5, together with (6.2.21), yields the general form of Q forwhich the self-similar solutions exist. At the shock, we have the followingconditions on the functions S, U , P and E

ρ|ξ=1 = t(1+α11/a)S(1), u|ξ=1 =

t(δ−1)U(1), if m = 1, 2

t(δ−1)U(1) − k∗, if m = 0,

p|ξ=1 = t(2δ−1+α11/a)P (1), σ|ξ=1 = t2(δ−1)E(1) − d∗.(6.2.25)

Equations (6.2.25), in view of the invariance of jump conditions (6.2.8), suggestthe following forms of ρ0(x), σ0(x)

ρ0(x) ≡ ρc(x/x0)θ, σ0(x) ≡ σ0(x) ≡ σc(x/x0)

Γ + σc0, (6.2.26)

and the following conditions on the functions S, U , P and E at the shock

U(1) = 2δAγ+1

ρ0V2−γp0ρ0V 2 , E(1) =

σc (A/x0)

Γ , if σ0 is varying0 , if σ0 is constant

,

P (1) =(2ρ0V 2−(γ−1)p0)ρcδ

2A2+θ

(γ+1)ρ0V 2xθ0

, S(1) = (γ+1)ρ0V2ρcA

θ

(γ−1)ρ0V 2+2γp0xθ0,

(6.2.27)together with

k∗ = 0, d∗ =

−(σc + σc0), if σ0 is constant−σc0, if σ0 is varying

, (6.2.28)

where θ = (α11 + a) /δa,

Γ =

0, if σ0 is constant2(δ − 1)/δ, if σ0 is varying

, (6.2.29)

and ρc, x0, σc and σc0 are some reference constants associated with themedium.

Equations (6.2.27) show that the necessary condition for the existence ofa similarity solution for shocks of arbitrary strength is that ρ0(X(t))V 2 mustbe a constant, which implies that θ and δ are not independent, but rather

θδ + 2(δ − 1) = 0. (6.2.30)


However, for strong shocks, V a0, constancy of ρ0V2 or equivalently the

condition (6.2.30) is not required, and thus, we have

S(1) = (γ+1)ρcAθ

(γ−1)xθ0

, U(1) = 2δAγ+1 , P (1) = 2δ2ρcA

θ+2

(γ+1)xθ0

,

E(1) =

σc (A/x0)

Γ, if σ0 is varying

0 , if σ0 is constant,

(6.2.31)

together with the equations defined in (6.2.26), (6.2.28) and (6.2.29).Using (6.2.27) or (6.2.31), we rewrite the equations (6.2.20), (6.2.21),

(6.2.22), (6.2.23) and (6.2.24) as

ρ = ρ0(X(t))S∗(ξ), u = V U∗(ξ),p = ρ0(X(t))V 2P ∗(ξ), σ = V 2E∗(ξ) − d∗,

Q = p2δ−32δ−2 q(η, ζ), η = ρp

−θδ(2+θ)δ−2 , ζ = p (σ + d∗)−

(2+θ)δ−22(δ−1) ,

ξ = x/Atδ , X = Atδ , V = δX/t,

(6.2.32)

where S∗(ξ) = xθ0S(ξ)/ρcAθ, U∗(ξ) = U(ξ)/(δA), P ∗(ξ) = xθ0P (ξ)/(ρcδ

2Aθ+2)and E∗ = E(ξ)/δ2A2.

We may note that this case leads to a class of similarity solutions where theshock path is given by a power law (6.2.32)9. The form of Q, given by (6.2.5),should be consistent with equation (6.2.32)5 in order to have a similaritysolution; thus, if the quantities τ and c are assumed to be of the form

c = c∗pβ1ρβ2 , τ = τ∗pβ3ρβ4 , (6.2.33)

with β1, β2, β3, β4, c∗ and τ∗ being constants associated with the reference

medium, it is required that the following relations hold

β1 θδ + 2(δ − 1) = −β2θδ, β3 θδ + 2(δ − 1) = 1 − β4θδ, (6.2.34)

We, thus find that the requirement of a self-similar flow pattern poses thesimilarity conditions (6.2.29) and (6.2.34) on the parameters β1, β2, β3, β4, θ,Γand δ. It may, however, be noted that there are no constraints, like (6.2.34),in the absence of relaxation. Substituting (6.2.32) in the governing equations(6.2.1) – (6.2.4) and using (6.2.5), (6.2.26), (6.2.28), (6.2.29), (6.2.33) and(6.2.34) we obtain the following system of ordinary differential equations inS∗, U∗, P ∗ and E∗ which on suppressing the asterisk sign becomes,

(U − ξ)S′ + SU ′ + θS +mSU

ξ= 0,

(U − ξ)U ′ +1

SP ′ +

(δ − 1)U

δ= 0, (6.2.35)

(U − ξ)P ′ + γPU ′ + θP +2(δ − 1)P

δ+mγPU

ξ+ (γ − 1)SQ∗ = 0,

(U − ξ)E′ + 2(δ − 1)E/δ −Q∗ = 0.


where a prime denotes differentiation with respect to the independent variableξ and

Q∗=

(e0δ

2β1P β1Sβ2

(P

S− p0

ρ0V 2

)−E

)/(τ0δ

2β3+1P β3Sβ4),

if σ0 is constant

(e0δ

2β1P β1Sβ2

(P

S− p0

ρ0V 2

)−E +

σcδ2

)/(τ0δ

2β3+1P β3Sβ4),

if σ0 is varying

with σc, τ0 and e0 as dimensionless parameters defined as

σc = σcAΓ−2x−Γ

0 , τ0 = τ∗x−θ(β3+β4)0 A2β3+θ(β3+β4)ρ(β3+β4)

c ,

e0 = c∗x−θ(β1+β2)0 A2β1+θ(β1+β2)ρβ1+β2

c .

In view of (6.2.30), the term p0/ρ0V2 in the expression of Q∗, is either a

constant or can be neglected, depending on whether the shock is of arbitrarystrength or of infinite strength respectively. Indeed, for a shock of arbitrarystrength, the jump conditions become

U(1) = 2γ+1

ρ0V2−γp0ρ0V 2 , S(1) = (γ+1)ρ0V

2

(γ−1)ρ0V 2+2γp0, δθ + 2(δ − 1) = 0,

P (1) =(2ρ0V 2−(γ−1)p0)

(γ+1)ρ0V 2 , E(1) =

σc/δ

2, if σ0 is varying0, if σ0 is constant

.

(6.2.36)However, for an infinitely strong shock, the jump conditions are

S(1) = (γ+1)(γ−1) , U(1) = 2

γ+1 , P (1) = 2γ+1 ,

E(1) =

σc/δ

2, if σ0 is varying0, if σ0 is constant.

.(6.2.37)

Thus, the system (6.2.35) is to be solved subjected to the jump conditions(6.2.36) and (6.2.37) depending on whether the shock is of arbitrary strengthor an infinite strength.

Case II. When a = 0 and α22 6= 0, the change of variables from (x, t) to(x, t) defined as

x = x+ c/α22, t = t,

leaves the basic equations (6.2.1) – (6.2.4) unchanged. The similarity variableand the form of similarity solutions for the flow variables readily follow from(6.2.18), and can be expressed in the following forms on suppressing the tilde


sign

ρ = ρ0(X(t))S(ξ), u = V U(ξ), p = ρ0V2P (ξ), σ = V 2E(ξ) − d∗,

Q = p2+θ2 q(η, ζ), η = ρp−

θ2+θ , ζ = p (σ + d∗)−

2+θ2 ,

ρ0(x) = ρc(x/x0)θ, σ0(x) = σc(x/x0)

Γ + σc0,

Γ =

0, if σ0 is constant2, if σ0 is varying

, d∗ =

−(σc + σc0), if σ0 is constant−σc0, if σ0 is varying

,

(6.2.38)where the dimensionless similarity variable ξ, the shock location X and theshock velocity V are given by

ξ = (x/x0) exp(−δt/A), X = x0 exp(δt/A), V = (δx0/A) exp(δt/A), (6.2.39)

with A as a dimensional constant. Here, we note that this case leads to a classof similarity solutions with an exponential shock path given by (6.2.39)2. If therelaxation rate is given by (6.2.5) and (6.2.33), the requirement of a self-similarflow pattern in this case poses the following similarity conditions

β1 = − β2θ

2 + θ, β3 = − β4θ

2 + θ. (6.2.40)

Thus, substituting (6.2.38) in the system of equations (6.2.1) – (6.2.4) andusing (6.2.5), (6.2.33), (6.2.39) and (6.2.40), we obtain the following set ofordinary differential equations

(U − ξ)S′ + SU ′ + θS + mSUξ = 0,

(U − ξ)U ′ + 1S P

′ + δU = 0,

(U − ξ)P ′ + γPU ′ + (θ + 2)P + mγPUξ + (γ − 1)SQ∗ = 0,

(U − ξ)E′ + 2E −Q∗ = 0,

(6.2.41)

where Q∗ is the same as in (6.2.35) with

σc = σcA2/x2

0, τ0 = τ∗ρ(β3+β4)c x2β3

0 A−(2β3+1), e0 = c∗x2β1

0 ρβ1+β2c A−2β1 .

The system (6.2.41) is to be solved subject to the constraints (6.2.40) and con-ditions for S(ξ), U(ξ), P (ξ) and E(ξ) at ξ = 1, which are given by (6.2.36)1,2,4,5together with θ = −2 for a shock of arbitrary strength; however for a shockof infinite strength, the corresponding conditions are given by (6.2.37).

Case III. Here, we consider a situation when a 6= 0 and α22 + 2a = 0.The analysis reveals that this situation can not arise in a radially symmetric(m = 1, 2) flow as it does not allow for the existence of a similarity solution insuch a flow configuration. However, this situation can occur in a plane (m = 0)flow where the change of variables from (x, t) to (x, t) defined as

x = x, t = t+ b/a,

does not alter the basic equations (6.2.1) – (6.2.4); consequently, equations


(6.2.18) imply the following forms of the flow and similarity variables.

ρ = ρ0(X(t))S(ξ), u = V U(ξ),

p = ρ0V2P (ξ), σ = V 2E(ξ) + d∗,

Q = p−3

θδ−2 q(η, ζ), η = ρpδθ

2−δθ , ζ = p (σ − d∗)θδ−2

2 , (6.2.42)

ξ = (x − x0δln(t/A))/x0, X = x0δln(t/A), V = δx0/t,

ρ0(x) = ρcexp(θx/x0), σ0(x) = σcexp(Γx/x0) + σc0,

where Γ =

0 if σ0 is constant−2/δ if σ0 is varying

, d∗ =

σc + σc0, if σ0 is constantσc0, if σ0 is varying

and A is a dimensional constant. It may be noted that this case leads to aclass of similarity solutions with logarithmic shock path given by (6.2.42)9. Inaddition, for the relaxation rate given by (6.2.5) and (6.2.33), the similarityconsiderations imply the following constraints

β1 =β2θδ

2 − δθ, β3 =

β4θδ − 1

2 − θδ. (6.2.43)

Substituting (6.2.42) in equations (6.2.1) – (6.2.4) and using (6.2.5), (6.2.33)and (6.2.43), we obtain the following set of ordinary differential equations

(U − 1)S′ + SU ′ + δθS = 0,(U − 1)U ′ + SP ′ − U/δ = 0,(U − 1)P ′ + γPU ′ + (θ − 2/δ)P + (γ − 1)SQ∗ = 0,(U − 1)ξE′ − 2E/δ −Q∗ = 0,

(6.2.44)

where Q∗ is the same as in (6.2.35) with σc, τ0 and e0 given by

σc = σcA2/x2

0, τ0 = τ∗ρ(β3+β4)c A−θδ(β3+β4)x2β3

0 , e0 = c∗ρβ1+β2c A−θδ(β1+β2)x2β1

0 .

It is interesting to note that for a nonrelaxing gas (Q = 0) with θ = 1, thesimilarity variable, the shock path and the form of similarity solutions givenby (6.2.42) together with similarity equations (6.2.44) are precisely the sameas those employed by Hayes [69] in analyzing the self-similar propagation ofa shock wave in an exponential medium.

For a shock of arbitrary strength, the jump conditions at ξ = 0 are

U(0) = 2γ+1

ρ0V2−γp0ρ0V 2 , S(0) = (γ+1)ρ0V

2

(γ−1)ρ0V 2+2γp0, θδ = 2,

P (0) =(2ρ0V 2−(γ−1)p0)

(γ+1)ρ0V 2 , E(0) =

σc/δ


(6.2.45)However, for an infinitely strong shock, the jump conditions at ξ = 0, become

S(0) = (γ+1)(γ−1) , U(0) = 2

γ+1 , P (0) = 2γ+1 ,

E(0) =

σc/δ


(6.2.46)


The system (6.2.44) is to be solved subject to the constraints (6.2.43) and theconditions (6.2.45) or (6.2.46) depending on whether the shock is of arbitrarystrength or of infinite strength.

Case IV. When a = α22 = 0, the situation is similar to the preceding casein the sense that it does not allow for the existence of a self-similar solution ina radially symmetric flow. However, the plane flow of a relaxing gas involvinga shock wave moving at constant speed admits a self-similar solution; andthe corresponding similarity variable and the flow variables have the followingform

ρ = ρ0S(ξ), u = V U(ξ), p = ρ0V2P (ξ), σ = V 2E(ξ) + d∗t,

Q = q(η, ζ), ξ = (x− δx0t/A)/x0, X = x0(1 + δt/A),V = δx0/A, ρ0(x) = ρc exp(θ(x − x0)/x0),

σ0(x) =

σc(x− x0)/x0 + σc0, if σ0 is varyingσc0, if σ0 is constant

,

(6.2.47)

where η = ρ/p, ζ = pd exp(−σ) and d∗ =

(σcδx0/A), if σ0 is varying,0, if σ0 is constant.

We note that the shock velocity in this case is constant and the density andthe vibrational energy ahead of the shock are varying according to the law(6.2.47)9,10. For the relaxation rate given by (6.2.5) and (6.2.33), the similarityconstraints in this case for θ 6= 0 turn out to be

β1 + β2 = 0, and β3 + β4 = 0. (6.2.48)

It may, however, be noted that when θ = 0, there are no constraints like(6.2.48).

Substituting (6.2.47) in the basic set of equations (6.2.1) – (6.2.4) form = 0, and using (6.2.5), (6.2.33) and (6.2.48), we obtain

(U − 1)S′ + SU ′ + θS = 0,(U − 1)U ′ + P ′/S = 0,(U − 1)P ′ + γPU ′ + θP + (γ − 1)SQ∗ = 0,(U − 1)E′ −Q∗ = 0,

(6.2.49)

where Q∗ is the same as in (6.2.35) with

σc = σc0A2/x2

0, e0 = c∗ρβ1+β2c x2β1

0 A−2β1 , τ0 = τ∗ρβ3+β4c x2β3

0 A−(2β3+1).

The condition for E at ξ = 1 is given by E(1) = σc/δ2, while the functions

U, S and P at ξ = 1 are given by (6.2.36)1,2,4 or (6.2.37)1,2,3 depending onwhether the shock is of arbitrary strength with a constant density ahead or ofinfinite strength, respectively. In the absence of relaxation, (Q = 0), and for aconstant ambient density, (θ = 0) it follows from equations (6.2.49) that theflow variables, namely the velocity, pressure and density behind the shock areconstant.


6.2.4 Imploding shocks

Here, we consider the problem of an imploding shock for which V a0

in the neighbourhood of implosion, and assume σ0 to be a constant. For theproblem of a converging shock collapsing to the center (axis), the origin oftime t is taken to be the instant at which the shock reaches the center (axis)so that t ≤ 0 in (6.2.35). In this regard, we modify slightly the definition ofthe similarity variable by setting

X = A(−t)δ , ξ = x/A(−t)δ , (6.2.50)

so that the intervals of the variables are −∞ < t ≤ 0, X ≤ x < ∞ and1 ≤ ξ <∞. Thus, the system (6.2.35) is to be solved subject to the similarityconditions (6.2.34) and the jump conditions at the shock

S(1) = (γ + 1)/(γ − 1), U(1) = 2/(γ + 1), P (1) = 2/(γ + 1), E(1) = 0,

Q∗ = e0δ2β1P 1+β1S1−β2 −E/τ0δ2β3+1P β3Sβ4,(6.2.51)

and the boundary conditions at ∞. At the instant of collapse t = 0, thegas velocity, pressure, density and the sound speed at any finite radius x arebounded. But with t = 0 and finite x, ξ = ∞. In order for the quantitiesu = (δX/t)U(ξ), p = ρ0(X)V 2P (ξ), ρ = ρ0(X)S(ξ), σ = V 2E(ξ) andc2 = γ(δX/t)2(P (ξ)/S(ξ)) to be bounded when t = 0 and x is finite, theentities U, P/S and E must vanish. Thus, we have the following boundaryconditions at ξ = ∞,

U(∞) = 0, γP (∞)/S(∞) = 0, E(∞) = 0. (6.2.52)

In the matrix notation the equations (6.2.35) can be written as

CW ′ = B, (6.2.53)

where W = (U, S, P,E)tr , and the matrix C and the column vector B can beread off by inspection of equations (6.2.35). It may be noted that the system(6.2.35) has an unknown parameter δ, which is not obtainable from an energybalance or the dimensional considerations as is the case of diverging shockwaves driven by release of energy at a plane or an axis or a point of implosion;indeed, it is computed only by solving a nonlinear eigenvalue problem fora system of ordinary differential equations. For the implosion problem, therange of similarity variable is 1 ≤ ξ < ∞. System (6.2.53) can be solved forthe derivatives U ′, S′, P ′ and E′ in the following form:

U ′ = ∆1/∆, S′ = ∆2/∆, P ′ = ∆3/∆, E′ = ∆4/∆, (6.2.54)

where ∆, which is the determinant of the system, is given by

∆ = (U − ξ)2(U − ξ)2 − γP/S

,


and ∆k, k = 1, 2, 3, 4 are the determinants obtained from ∆ by replacing thekth column by the column vector B, and are given by

∆1 =

(ξ − U)U

(δ − 1)

δ+

(θ +

2(δ − 1)

δ

)P

S+mγPU

Sξ

+ (γ − 1)e0δ

2β1P β1+1Sβ2−1 −E

τ0δ(2β3+1)P β3Sβ4

(U − ξ)2,

∆2 = −S∆1 + (θS +mSU/ξ)∆ / (U − ξ) ,∆3 = −γP∆1 + θP + 2(δ − 1)P/δ +mγPU/ξ

+(γ − 1)e0δ

2β1P β1+1Sβ2−1 −E

τ0δ(2β3+1)P β3Sβ4−1

/(U − ξ),

∆4 = −∆

2(δ − 1)E − e0δ

2β1P β1+1Sβ2−1 −E

τ0δ2β3P β3Sβ4

1

δ(U − ξ).

It may be noted that U < ξ in the interval [1,∞), whilst ∆ is positive at ξ = 1and negative at ξ = ∞ indicating thereby that there exists a ξ ∈ [1,∞) atwhich ∆ vanishes, and consequently the solutions become singular. In orderto get a nonsingular solution of (6.2.35) in the interval [1,∞), we choose theexponent δ such that ∆ vanishes only at the points where the determinant∆1 is zero too; it can be checked that at points where ∆ and ∆1 vanish, thedeterminants ∆2,∆3 and ∆4 also vanish simultaneously. To find the exponentδ in such a manner, we introduce the variable Z,

Z(ξ) = (U(ξ) − ξ)2 − γP (ξ)/S(ξ), (6.2.55)

which, in view of (6.2.54), implies

Z ′ =2(U − ξ)(∆1 − ∆) − γ∆3/S + γP∆2/S

2/∆. (6.2.56)

Equations (6.2.54), in view of (6.2.56), become

dU

dZ=

∆1

∆5,

dS

dZ=

∆2

∆5,

dP

dZ=

∆3

∆5,

dE

dZ=

∆4

∆5, (6.2.57)

where

∆5 = 2(U − ξ)(∆1 − ∆) − γ∆3/S + γP∆2/S2

with ξ given by (6.2.55) in the form

ξ = U + Z + γP/S1/2.

6.2.5 Numerical results and discussion

We integrate the equations (6.2.57) from the shock Z = Z(1) to the sin-gular point Z = 0, by choosing a trial value of δ, and compute the values of


U, S, P,E and ∆1 at Z = 0; the value of δ is corrected by successive approx-imations in such a way that for this value, the determinant ∆1 vanishes atZ = 0. The computation has been performed by choosing e0 = 1, τ0 = 0.1,β2 = 0, β3 = −1 and γ = 1.4. The corresponding values of β1 and β4 arecomputed from (6.2.34). The values of δ, obtained from the numerical cal-culations and CCW approximation, [210], for different values of θ, τ0, γ andm are given in Table 1. The relaxation effects enter through the parameterτ0 while the effects of density stratification become important through theparameter θ. The fact that δ is always less than 1 shows that the shock waveis continuously accelerated; indeed, the shock velocity V becomes infinite asX → 0, but less rapidly than X−1. We find that an increase in any of theparameter γ, θ,m or τ−1

0 , causes the similarity exponent δ to decrease, andconsequently an increase in the shock velocity, as the shock approaches thecenter (axis) of implosion.

Analytical expressions of the dimensionless flow variables at the instantof collapse of the shock wave at t = 0, X = 0 (t = 0, x 6= 0 correspondto ξ = ∞), where conditions (6.2.52) hold, can easily be obtained from theequations (6.2.54) in the following form.

U ∼ ξδ−1

δ , S ∼ ξθ, P ∼ ξθ+2(δ−1)/δ, E ∼ ξ2(δ−1)/δ .

The above relations imply that both the velocity U and the vibrationalenergy E tend to zero, while the gas pressure P remains bounded (respec-tively; unbounded) at the instant of collapse of the shock wave if δθ+2(δ−1)is negative (respectively; positive). However, the density S at the time of col-lapse is infinite; this is in contrast to the corresponding situation of uniforminitial density, where S remains bounded. Numerical integration of equations(6.2.57) for 1 ≤ ξ < ∞ has been carried out, and the values of the flow vari-ables before collapse and at the instant of collapse are depicted in Figs. (6.2.1– 6.2.3). The numerical solutions in the neighborhood of ξ = ∞ are in confor-mity with the above mentioned asymptotic results. The typical flow profilesshow that the pressure and density increase behind the shock wave, while thevelocity decreases; this is because a gas particle passing through the shock issubjected to a shock compression; indeed, this increase in pressure and densitybehind the shock may also be attributed to the geometrical convergence orarea contraction of the shock wave.

This increase in pressure and density, and decrease in particle velocity arefurther reinforced by an increase in the ambient density exponent θ. In con-trast to the density profiles, which exhibit monotonic variations, the pressureprofiles in cylindrically and spherically symmetric flow configurations exhibitnonmonotonic variation for values of θ < θc, where θc = 2(1 − δ)/δ. Indeed,for θ ≤ θc, the pressure profiles attain a maximum value, which appears tomove towards the point (axis) of implosion as the shock collapses. Howeverfor θ > θc, the pressure variations are monotonic like density variations, andthe profiles steepen as the shock converges; the steepening rate of pressureand density profiles enhances with an increase in θ. However, a decrease in τ0


counteracts, to some extent, the steepening rate and results in a falling off ofthe particle velocity, density and pressure relative to what they would be inthe absence of relaxation.

As might be expected, the differences between planar and nonplanar flowconfigurations are considerable. For plane waves, the flow distribution is rel-atively less influenced by the interaction between the relaxation and gasdy-namic phenomena as compared to cylindrical and spherical waves; however,for imploding shocks, where the changes in flow variables may be attributedto the geometrical convergence or area contraction of the shock waves, thevariations that result from relaxing gasdynamics are noteworthy. A study ofthe vibrational energy and temperature profiles behind the shock indicatessizeable reductions in these quantities. It is noted that a decrease in τ0 causesan increase in temperature in the region behind the shock as compared towhat it would be in the absence of relaxation.

Table 1 : Similarity exponent δ for plane and radially symmetric flow of a relaxinggas with e0 = 1., β1 = β2 = 0, β3 = −1 and for different values of the ambientdensity exponent θ, the specific heat gas constant γ and relaxation time τ0.

m γ θ τ0 computed delta δ Whitham’s rule β4

2 1.4 0.5 0.1 0.6648 0.6659 1.99182 1.4 1.0 0.1 0.6158 0.6213 1.37622 1.4 2.0 0.1 0.5446 0.5480 1.08181 1.4 0.5 0.1 0.7670 0.7665 2.39261 1.4 1.0 0.1 0.7044 0.7080 1.58041 1.4 2.0 0.1 0.6121 0.6144 1.18310 1.4 0.5 0.1 0.9042 0.9028 2.78800 1.4 1.5 0.1 0.7525 0.7559 1.44740 1.4 2.0 0.1 0.6980 0.6991 1.28362 5/3 1.0 0.1 0.5575 0.5928 1.20631 5/3 1.0 0.1 0.6505 0.6842 1.46270 5/3 1.0 0.1 0.7730 0.8090 1.70642 1.4 1.0 1.0 0.6257 0.6214 1.40191 1.4 1.0 1.0 0.7140 0.7081 1.59950 1.4 1.0 1.0 0.8309 0.8229 1.79652 5/3 1.0 1.0 0.5919 0.5928 1.31061 5/3 1.0 1.0 0.6855 0.6842 1.54120 5/3 1.0 1.0 0.8141 0.8090 1.7717

In an implosion problem, the self-similar region decreases with time inproportion to the radius of the shock front; the effective boundary of theself-similar region is then considered to be at some constant value of ξ = ξ1.Thus, the mass, M, per unit area (m=0), or length (m=1) or volume (m=2)contained in this region is proportional to

M ∝∫ x1

X

ρxm dx = ρ0X(1+m)

∫ ξ1

1

S(ξ)ξm dξ.


Figure 6.2.1: (a) Pressure, (b) velocity, (c) density, (d) temperature, (e) vibrational

energy. Flow pattern; m = 0, γ = 1.4, β2 = 0, β3 = −l, Γ = 0 and e0 = 1.

In nonrelaxing gases (i.e., τ0 = ∞), where there are no constraints such as(6.2.37), we have obtained the exponent δ for different values of γ and θ forplane and radially symmetric shocks, in a quiescent perfect gas with initial

http://www.crcnetbase.com/action/showImage?doi=10.1201/9781439836910-c6&iName=master.img-504.png&w=324&h=463


density which is either constant or decreasing towards the axis of implosionaccording to a power law; the results which are given in Tables 2, 3, 4 comparewell with those obtained by Whitham [210], Guderley [66], Hafner [68], andSakurai [159].

Figure 6.2.2: (a) Pressure, (b) velocity, (c) density, (d) temperature, (e) vibrational

energy. Flow pattern; m = 1, γ = 1.4, β2 = 0, β3 = −l, Γ = 0 and e0 = 1.





Figure 6.2.3: (a) Pressure, (b) velocity, (c) density, (d) temperature, (e) vibrationalenergy. Flow pattern; m = 2, γ = 1.4, β2 = 0, β3 = −l, Γ = 0 and e0 = 1.

As the above integral with respect to ξ from 1 to ξ1 is a constant, the masscontent in this region with a finite radius is finite, and tends to zero as X → 0,i.e., M ∼ X1+m+θ → 0 as X → 0. Thus, the density variation ahead of theshock causes the total mass to decrease more rapidly with time as comparedto the corresponding case in which the initial density is uniform.





Table 2 : Similarity exponent δ for a spherically symmetric flow of an ideal gas forvarious values of the heat exponent γ and the ambient density exponent θ.

θ γ computed delta δ Whitham’s rule Hafner Guderley0.0 1.2 0.7571 0.7540 0.7571 0.75710.0 1.4 0.7172 0.7173 0.7172 0.71720.0 5/3 0.6884 0.6893 0.6884 0.68830.0 2.0 0.6670 0.6667 0.66700.0 3.0 0.6364 0.6295 0.63640.1 1.4 0.7067 0.70640.5 1.2 0.7122 0.70540.5 1.4 0.6683 0.66590.5 5/3 0.6373 0.63741.0 1.2 0.6730 0.66261.0 5/3 0.5947 0.59282.0 1.2 0.6073 0.5909 0.60732.0 1.4 0.5588 0.5481 0.55882.0 5/3 0.5264 0.5200 0.52642.0 2.0 0.5034 0.5000 0.50342.0 3.0 0.4711 0.4707 0.47111.0 1.4 0.6267 0.62141.2 1.4 0.6112 0.60521.5 1.4 0.5906 0.5824

Table 3 : Similarity exponent δ for a cylindrically symmetric flow of an ideal gasfor various values of the heat exponent γ and the ambient density exponent θ.

γ θ computed delta δ Whitham’s rule Hafner Guderley1.2 0. 0.8612 0.8598 0.8621 0.86121.4 0. 0.8353 0.8354 0.83535/3 0. 0.8156 0.8160 0.8156 0.81572.0 0. 0.8001 0.8000 0.80013.0 0. 0.7757 0.7727 0.77575/3 .5 0.7454 0.74432.0 .5 0.7273 0.72735/3 1. 0.6884 0.68422.0 1. 0.6689 0.66671.1 2. 0.7178 0.6983 0.71781.2 2. 0.6718 0.6540 0.67181.4 2. 0.6283 0.6144 0.62835/3 2. 0.5994 0.5891 0.59942.0 2. 0.5790 0.5714 0.57903.0 2. 0.5500 0.5464 0.55021.4 .3 0.7941 0.79261.4 .5 0.7694 0.76651.4 .8 0.7356 0.7303


Table 4 : Similarity exponent δ for a plane flow of an ideal gas for various valuesof the heat exponent γ and the ambient density exponent θ.

θ γ computed delta δ Whitham’s rule Hafner Sakurai.3 1.4 0.9408 0.9393.4 1.4 0.9231 0.9207.5 1.4 0.9062 0.9028 0.9062 0.9062.6 1.4 0.8901 0.8856.7 1.4 0.8746 0.8691.8 1.4 0.8598 0.8531.9 1.4 0.8455 0.83771 1.4 0.8318 0.8229 0.8319 0.8318.5 1.1 0.9335 0.9305 0.9335.5 1.2 0.9195 0.9162 0.9196 0.9195.5 5/3 0.8976 0.8944 0.8976 0.8976.5 2.0 0.8918 0.8889 0.8918.5 3.0 0.8844 0.8819 0.88441 1.1 0.8783 0.8700 0.87831 1.2 0.8544 0.8453 0.8544 0.85441 5/3 0.8174 0.8090 0.8174 0.81741 2.0 0.8078 0.8000 0.80781 3.0 0.7954 0.7887 0.79542 1.1 0.7881 0.7699 0.78812 1.2 0.7514 0.7321 0.7514 0.75142 1.4 0.7177 0.6991 0.7177 0.71772 5/3 0.6966 0.6793 0.6966 0.69662 2.0 0.6826 0.6667 0.68262 3.0 0.6648 0.6511 0.6648

6.3 Exact Solutions of Euler Equations via Lie Group

Analysis

For nonlinear systems involving discontinuities such as shocks, we do notnormally have the luxury of complete exact solutions, and for analytical workhave to rely on some approximate analytical or numerical methods which maybe useful to set the scene and provide useful information towards our under-standing of the complex physical phenomenon involved. One of the most pow-erful methods used to determine particular solutions to PDEs is based uponthe study of their invariance with respect to one parameter Lie group of pointtransformations (see [2], [17], [18], [78], [136], [138], and [151]). Indeed, with thehelp of symmetry generators of these equations, one can construct similarityvariables which can reduce these equations to ordinary differential equations(ODEs); in some cases, it is possible to solve these ODEs exactly. Besides these

6.3 Exact Solutions of Euler Equations via Lie Group Analysis 197

similarity solutions, the symmetries admitted by given PDEs enable us to lookfor appropriate canonical variables which transform the original system to anequivalent one whose simple solutions provide nontrivial solutions of the orig-inal system (see Refs. [48], [49], [134], and [135]). Using this procedure, Amesand Donato [1] obtained solutions for the problem of elastic-plastic deforma-tion generated by a torque, and analyzed the evolution of weak discontinuityin a state characterized by invariant solutions. Donato and Ruggeri [50] usedthis procedure to study similarity solutions for the system of a monoatomicgas, within the context of the theory of extended thermodynamics, assumingspherical symmetry. In this section, we use this approach to characterize aclass of solutions of the basic equations governing the one dimensional planarand radially symmetric flows of an adiabatic gas involving shock waves. Sincethe system involves only two independent variables, we need two commutingLie vector fields, which are constructed by taking a linear combination of theinfinitesimal operators of the Lie point symmetries admitted by the system athand. It is interesting to note that one of the special exact solutions obtainedin this manner is the well known solution to the blast wave problem studiedin the theory of explosion in the gaseous media, studied in section 6.1 (see [1],[50], [96], [129], and [148]).

6.3.1 Symmetry group analysis

As in [48], [49], [134], and [135], let us assume that the system of N non-linear partial differential equations

FR

(x, t, u,

∂u

∂x,∂u

∂t

)= 0, R = 1, 2, · · · , N, (6.3.1)

involving two independent variables x, t and the unknown vector u(x, t), whereu(x, t) = (u1(x, t), u2(x, t), · · · , uN (x, t)) ∈ RN , admits s− parameter Liegroup of transformations with infinitesimal operators

ζi = Xi(x, t, u)∂

∂x+ Ti(x, t, u)

∂

∂t+

N∑

j=1

Uij(x, t, u)∂

∂uj, i = 1, 2, · · · , N

(6.3.2)such that there exist r(≤ s) infinitesimal generators ζ1, ζ2, · · · , ζr that forma solvable Lie algebra. Let us now construct generators

Y1 =

r∑

k=1

αkζk, Y2 =

r∑

k=1

βkζk

where αk and βk are constants, to be determined, such that [Y1, Y2] = 0.Now we introduce the canonical variables τ , ξ and ν = (ν1, ν2, ..., νN ) ∈ RN

related to the infinitesimal generator Y1, defined by Y1τ = 1, Y1ξ = 0, andY1νi = 0, i = 1, 2, · · · , N . In terms of these canonical variables, the infinitesi-mal operator Y1 reduces to Y1 = ∂/∂τ , i.e., it corresponds to a translation in


the variable τ only; consequently, owing to the invariance, the system (6.3.1)must assume the form

FR

(ξ, ν,

∂ν

∂ξ,∂ν

∂τ

)= 0, R = 1, 2, · · · , N. (6.3.3)

In terms of these new variables, the operator Y2 can be written as

Y2 = (Y2ξ)∂

∂ξ+ (Y2τ)

∂

∂τ+

N∑

j=1

(Y2νj)∂

∂νj, (6.3.4)

where we require that the condition Y2ξ 6= 0 holds; thus, it is possible tointroduce new canonical variables η, τ ∗ and w = (w1, w2, · · · , wN ) ∈ RN

defined by Y2η = 1, Y2τ∗ = 0, and Y2wi = 0, i = 1, 2, · · · , N which transform

(6.3.3) to the form

FR

(w,

∂w

∂η,∂w

∂τ∗

)= 0, R = 1, 2, · · · , N. (6.3.5)

The resulting system (6.3.5) is an autonomous system associated with (6.3.1);in order to illustrate the method, outlined as above, we consider the systemof Euler equations of ideal gasdynamics in the next section.

6.3.2 Euler equations of ideal gas dynamics

The equations governing the one dimensional unsteady planar and radi-ally symmetric flows of an adiabatic index γ in the absence of viscosity, heatconduction and body forces can be written in the form [210]

ρt + uρx + ρux +mρu

x= 0,

ut + uux + ρ−1px = 0, (6.3.6)

pt + upx + γpxmγpu

x= 0,

where t is the time, x the spatial coordinate being either axial in flows withplanar (m = 0) geometry or radial in cylindrically (m = 1) and spherically(m = 2) symmetric flows. The state variable u denotes the gas velocity, pthe pressure, and ρ the density. By a straightforward analysis, it is found thatthe Lie groups of point transformations that leave the system (6.3.6) invariantconstitute a 4-dimensional Lie algebra generated by the following infinitesimaloperators:

ζ1 = ρ∂

∂ρ+ p

∂

∂p, ζ2 = t

∂

∂t− u

∂

∂u− 2p

∂

∂p,

ζ3 = x∂

∂x+ u

∂

∂u+ 2p

∂

∂p, ζ4 =

∂

∂t.


In order to construct generators Y1, Y2 such that [Y1, Y2] = 0, let

Y1 = α1ζ1 + α2ζ2 + α3ζ3 + α4ζ4,

= (α2t+ α4)∂

∂t+ α3x

∂

∂x+ (α3 − α2)u

∂

∂u

+(α1 − 2α2 + 2α3)p∂

∂p,

Y2 = β1ζ1 + β2ζ2 + β3ζ3 + β4ζ4,

= (β2t+ β4)∂

∂t+ β3x

∂

∂x+ (β3 − β2)u

∂

∂u

+(β1 − 2β2 + 2β3)p∂

∂p,

where α2β4 − α4β2 = 0 and α1, α3, β1, β3 are arbitrary constants. Sincethe system is invariant under the group generated by the generator Y1, weintroduce canonical variables τ , ξ, R, U and P such that Y1τ = 1, Y1ξ = 0,Y1R = 0, Y1U = 0 and Y1P = 0. This implies that when α2, α3 6= 0, we have

τ = (1/α2) log(α2t+ α4), ξ = (α2t+ α4)x−α2/α3 , (6.3.7)

R = ρx−α1/α3 , U = ux(α2−α3)/α3 , P = px2α2−α1−2α3)/α3 .

In terms of these new variables, Y2 becomes

Y2 =β2

α2

∂

∂τ+β2α3 − β3α2

α3ξ∂

∂ξ+β1α3 − β3α1

α3R∂

∂R+α2β3 − α3β2

α3U

∂

∂U

+(α3β1 − α1β3) + 2(α2β3 − β2α3)

α3P∂

∂P.

Now, we introduce canonical variables τ , ξ, R, U and P such that Y2τ = 0,Y2ξ = 1, Y2R = 0, Y2U = 0 and Y2P = 0; thus, the corresponding character-istic conditions yield

ξ = (α3/A) log(ξ), τ = τ − (β2/α2)ξ,

R = Rξ(α1β3−α3β1)/A, U = U ξ, P = P ξ2+((α1β3−α3β1)/A),(6.3.8)

where A = α3β2 − α2β3 6= 0. In view of (6.3.7) and (6.3.8), we are led to thefollowing transformations

τ = (−β3/A) log(α2t+ α4)x−β2/β3, ξ = (α3/A) log(α2t+ α4)x

−α2/α3,ρ = R(ξ, τ)xL(α2t+ α4)

K , u = U(ξ, τ)x(α2t+ α4)−1, (6.3.9)

p = P (ξ, τ)xL+2(α2t+ α4)K−2,


where L = (α1β2 − α2β1)/A, K = (α3β1 − α1β3)/A with β3 6= 0, and R, U ,P are arbitrary functions of ξ and τ . Using (6.3.9) in (6.3.6) we get

(α2α3 − Uα2)∂R

∂ξ+(β2U − β3α2)

∂R

∂τ+ β2R

∂R

∂τ− α2R

∂U

∂ξ

+(Kα2 + LU + U +mU)AR = 0,

(α2α3 − Uα2)∂U

∂ξ+(β2U − β3α2)

∂U

∂τ+β2

R

∂P

∂τ− α2

R

∂P

∂ξ

+(U2 − α2U)A+ (L+ 2)APR−1 = 0, (6.3.10)

(α2α3 − Uα2)∂P

∂ξ+(β2U − β3α2)

∂P

∂τ+ β2γP

∂U

∂τ− α2γP

∂U

∂ξ

+(Kα2 − 2α2 + LU + 2U +mγU + γU)AP = 0,

where A is same as in (6.3.8). The above equations can be solved completely(6.3.10)1,3 have the closed form solutions as

R(ξ, τ) = R1(η) exp−((Kα2 + (L+ 1 +m)U)/(α2(α3 − U)))Aξ, (6.3.11)

P (ξ, τ) = P1(η) exp−((Kα2 − 2α2 + (L+ 2 + γ + γm)U)/(α2(α3 − U)))Aξ,

where

η = τ − β2U − α2β3

α2(α3 − U)ξ =

1

α3 − Ulog(x(α2t+ α4)

−U/α2

),

and R1(η) is an arbitrary function of η. Using (6.3.11) into (6.3.10)2, we getthe compatibility conditions for U and P1(η) as

U = 2α2/(γ + 1 +mγ −m) or U = α2, (6.3.12)

and P ′1(η)+(α1 +2α3 +mU −U)P1(η)+(α3 −U)U(U −α2)R1(η) = 0. Thus,

in view of (6.3.9), (6.3.11) and (6.3.12), the solution of the system (6.3.6) canbe expressed as follows.

Case-Ia. When U = α2 6= α3, the solution of the system (6.3.6) takes theform

ρ = R1(η)x(α1+(m+1)α2)/(α3−α2)(α2t+ α4)

−(α1+(m+1)α3)/(α3−α2),

u = α2x/(α2t+ α4), p = C(α2t+ α4)−(m+1)γ ,

(6.3.13)

where C is an arbitrary constant, R1(η) is an arbitrary function of η, and

η = (1/α3 − α2) log(x/(α2t+ α4)).

Case-Ib. When U = 2α2/Γ 6= α3, where Γ = γ + 1 + mγ − m 6= 0, thesolution of the system (6.3.6) takes the form

ρ = R1(η)xA1(α2t+ α4)

−A2 , u = (2α2/Γ)x/(α2t+ α4),

p =C −A3

∫R1(η) exp(A4η)dη

(α2t+ α4)

−2(m+1)γ/Γ,(6.3.14)


where C is an arbitrary constant, R1(η) is an arbitrary function of η, and

η =Γ

Γα3 − 2α2log(x(α2t+ α4)

−2/Γ),

A1 =Γα1 + 2(m+ 1)α2

Γα3 − 2α2, A2 =

2(α1 + (m+ 1)α3

Γα3 − 2α2,

A3 =2α2

2(m+ 1)(1 − γ)(Γα3 − 2α2)

Γ3, A4 =

Γ(α1 + 2α3) + 2(m− 1)α2

Γ.

Case-II. Let U ≡ α3. Then (6.3.10)1,3 imply that

R(ξ, η) = R1(ξ) exp−(Kα2 + (L+ 1 +m)U)τ,P (ξ, η) = P1(ξ) exp−(Kα2 − 2α2 + (L+ 2 +mγ + γ)U)τ, (6.3.15)

where τ and ξ are the same as defined in (6.3.9), and R1(ξ) and P1(ξ) arearbitrary functions of ξ. Moreover, on using (6.3.15) into (6.3.10)2, we get thecompatibility conditions for U and P1(ξ) as

U = α3 = 2α2/Γ, (6.3.16)

P ′1(ξ) +

(β1 + 2β3 +

2(m− 1)β2

Γ

)P1(ξ)

+2α2(γ − 1)(m+ 1)(α3β2 − α2β3)

Γ2R1(ξ) = 0.

Thus, in view of the equations (6.3.9), (6.3.15) and (6.3.16), the solution ofthe system (6.3.6) can be written as

ρ = R1(ξ)xB1 (α2t+ α4)

−B2 u = (2α2/Γ)x/(α2t+ α4), (6.3.17)

p =

C −B3

∫R1(ξ) exp(B4ξ)dξ

(α2t+ α4)

−2(m+1)γ/Γ,

where C is an arbitrary constant, R1(ξ) is an arbitrary function of ξ, and

ξ =Γ

Γβ3 − 2β2log(x(α2t+ α4)

−2/Γ),

B1 =Γβ1 + 2(m+ 1)β2

Γβ3 − 2β2, B2 =

2(β1 + (m+ 1)β3

Γβ3 − 2β2,

B3 =2α2

2(m+ 1)(1 − γ)(Γβ3 − 2β2)

Γ3, B4 =

Γ(β1 + 2β3) + 2(m− 1)β2

Γ.

It may be remarked that the solution (6.3.13), (respectively, (6.3.14)) involvesarbitrary parameters, α1, α2 and α3 with α2 6= α3 (respectively, α3 6= 2α2/Γ),whereas the solution (6.3.17) depends on the parameters, α2, β1, β2 and β3.In fact, solution (6.3.17) is exactly the same as (6.3.14) i.e., the constants B1,B2, B3 and B4 and ξ are exactly the same as A1, A2, A3, A4 and η when α1,α2 and α3 are replaced by β1, β2 and β3.


6.3.3 Solution with shocks

As is well known, a shock wave may be initiated in the flow region, and onceit is formed, it will propagate by separating the portions of the continuous re-gion. At shock, the correct generalized solution satisfies the Rankine-Hugoniotjump conditions. Let x = X(t) be the shock location in the x− t plane prop-agating in to the medium where ρ = ρ0(x), u =≡ 0 and p = p0 = constant.If the shock speed V = dX/dt is very large compared with the sound speeda0 =

√γp0/ρ0(x), and the medium behind the shock is given by the solution

(6.3.13) or (6.3.14), then at the shock front the following relations hold [210]:

ρ =γ + 1

γ − 1ρ0(X(t)), u =

2

γ + 1V, p =

2

γ + 1ρ0(X(t))V 2. (6.3.18)

I. Let the medium behind the shock be represented by the solution (6.3.13).Then equations (6.3.18) imply

R1(ηs)X(t)

h

α1+(m+1)α2α3−α2

i

(α2t+ α4)

h

α1+(m+1)α3α3−α2

i

= γ+1γ−1ρ0(X(t)),

α2X(t)α2t+α4

= 2γ+1V, C(α2t+ α4)

−(m+1)γ = 2γ+1ρ0(X(t))V 2,

(6.3.19)

where C is an arbitrary constant and ηs = (α3−α2)−1 log

(X(t)(α2t+ α4)

−1).

From (6.3.19)2, the shock speed V (t) can be written as V = ((γ +1)α2/2)X(t)/(α2t+ α4), implying that

X(t) = X0(T/T0)(γ+1)/2, (6.3.20)

where T = α2t+α4 and T0 = α2t0 +α4, with X0 and t0 being related with theposition and time of the shock. Thus, on using (6.3.20) in equations (6.3.19)1,3we find that R1(ηs) and ρ(X(t)) must have the following forms:

R1(ηs) = 2CR10

(γ−1)α22

(TT0

) (1−γ)(α1−(m+3)α2+2(m+2)α3)

2(α3−α2)

,

ρ0(X(t)) = ρc(X(t)/X0)2(1−2γ−mγ)/(γ+1),

(6.3.21)

where ρc = 2CT 2−mγ−γ0 /(γ + 1)α2

2X20 and R10 = X−B1

0 TB20 with

B1 = α1 + (m− 1)α2 + 2α3/α3 − α2,B2 = α1 − (2 −mγ − γ)α2 + (3 −mγ − γα3)/α3 − α2.

In view of (6.3.20), ηs can be written as

ηs =1

α3 − α2log

(X0

T0

(T

T0

)(γ−1)/2),

and hence R1(ηs) = R10 exp [−α1 + (m+ 3)α2 − 2(m+ 2)α3] ηs, where

R10 =2C

(γ − 1)α22

X2(m+1)0 T

−(m+1)(γ+1)0 .


Thus, for a shock, X(t) = X0(T/T0)(γ+1)/2, propagating into a nonuniform

region ρ(x) = ρc(x/X0)2(1−2γ−mγ)/(γ+1), u = 0, p = p0, the downstream flow

given by (6.3.13) takes the form

ρ = γ+1γ−1ρc

(xX0

)−2(m+2) (TT0

)m+3

, u = α2xT ,

p = γ+12 ρc

(α2X0

T0

)2 (TT0

)−(m+1)γ

.(6.3.22)

II. Similarly, when the medium behind the shock is represented by (6.3.14),conditions (6.3.18) imply that the speed of such a shock is given by

X(t) = X0

(α2t+ α4

α2t0 + α4

)δ, (6.3.23)

where δ = (γ + 1)/Γ, and the following conditions for R1(ηs) and ρ0(x) musthold:

A3

∫R1(ηs) exp(A4ηs)dηs +A5R1(ηs)(α2t+ α4)

A6 − C = 0,

ρ0(X(t)) =γ − 1

γ + 1XA1

0 (α2t0 + α4)δA1−A2

(α2t+ α4

α2t0 + α4

)δA1−A2

,

where A3 is same as in (6.3.14), ηs = ΓΓα3−2α2

log(X0(α2t+ α4)

(γ−1)/Γ)

and

A5 =2(γ − 1)α2

2X2+A1

0

Γ2(α2t0 + α4)(2+A1)(δA1−A−2), A6 = δA1 −A2 +

2(m+ γ)

Γ.

The solution of the above integral equation exits when C = 0, and it can beexpressed as

R1(η) ∝ exp−(α1 + (1 −m)α3 + (4m/Γ)α2)η. (6.3.24)

Thus, in view of (6.3.24), equations (6.3.14), which describe the flow down-stream from the shock x = X(t), yield

ρ =γ + 1

γ − 1ρc

(x

X0

)m−1(α2t+ α4

α2t0 + α4

)−4m/Γ

, u =2α2

Γ

x

α2t+ α4,

p =

(2(γ + 1)ρcα2X0

Γ2(α2t0 + α4)

)2(x

X0

)m+1(α2t+ α4

α2t0 + α4

)−2(m+1)(γ+1)/Γ

, (6.3.25)

ρ0(X(t)) = ρc

(α2t+ α4

α2t0 + α4

)(mγ−3m−γ−1)/Γ

, X(t) = X0

(α2t+ α4

α2t0 + α4

)(γ+1)/Γ

,

where Γ is same as in (6.3.14). It is interesting to note that the above solu-tion (6.3.25) is exactly the same as the one obtained in the literature, usingdifferent approaches, for describing a blast wave (strong shock) propagatinginto a medium the density of which varies according to a power law of thedistance measured from the source of explosion (see Subsection 6.1.1, [129],and Chapter 2 in [96]).

Chapter 7

Kinematics of a Shock of Arbitrary

Strength

The study of hyperbolic system of equations and the associated problem of de-termining the motion of a shock of arbitrary strength has received considerableattention in the literature. The determination of the shock motion requires thecalculation of the flow in the region behind the shock. If the shock is weak,Friedrich’s theory [55] offers a solution to this problem. Other methods bywhich the rise of entropy across the shock can be accounted for approximatelyhave been given by Pillow [143], Meyer [124], and Lighthill [109]. Approximateanalytical solutions to the problem of decay of a plane shock wave have beenproposed by Ardavan-Rhad [6] and Sharma et. al [179]. Other methods whichare generally employed for solving this problem in case of strong shock wavesbelong to the so-called shock expansion theory; the reader is referred to thework carried out by Sirovich and his co-workers ([33] and [187]). A simplerule proposed by Whitham [210], called the characteristic rule or the CCWapproximation, determines the motion of a shock without explicitly calculat-ing the flow behind for a large, though restricted class of problems, with goodaccuracy. Although these methods have been developed in an ad hoc manner,they yield remarkably accurate results; see for example, the papers on shockpropagation problems involving diffraction [40], refraction [29], focusing ([71],[28]), and stability [161].

The work presented in this chapter derives motivation from the study relat-ing to an intrinsic description of shock wave propagation carried out by Nunzi-ato and Walsh [131], Chen [32], Wright [212], and Sharma and Radha [169]. Arigorous mathematical approach which can be used for describing kinematicsof a shock of arbitrary strength has been proposed by Maslov [119], withinthe context of a weak shock, by using the theory of generalized functions toderive an infinite system of identities that hold on the shock front. Ravindranand Prasad [150], following Maslov [119], have derived a pair of compatibilityconditions for the derivatives of flow variables that hold on the shock frontpropagating into a nonisentropic gas region. In an attempt to describe wavemotion, Best [15] also derived a sequence of transport equations, which holdon the shock front. However, his approach is based on CCW approximationand admits the fact that the application of a characteristic equation at theshock is somewhat ad hoc.

Different methods for studying shock kinematics have been proposed by

205

206 7. Kinematics of a Shock of Arbitrary Strength

Grinfeld [65], Anile and Russo [4], and Fu and Scott [56] who use the theoryof bicharacteristics (rays) to derive identity relationships that hold along therays. Using a procedure based on the kinematics of one dimensional motion,Sharma and Radha [169] studied the behavior of 1-dimensional shock wavesin inhomogeneous atmosphere by considering an infinite system of transportequations for the variation of coupling terms Π(k) and providing a naturalclosure on the infinite hierarchy by setting Π(k) equal to zero for the largest kretained. This truncation criterion yields a good approximation of the infinitehierarchy; (see [4], [11], and [169]).

The present Chapter, which is largely based on our papers [147] and [174],makes use of the singular surface theory to study the evolutionary behaviorof shocks in an unsteady flow of an ideal gas, and of bores in shallow waterequations by considering a sequence of transport equations for the variationof jumps in flow variables and their space derivatives across the wave front.

7.1 Shock Wave through an Ideal Gas in 3-Space Dimen-

sions

In this section, we consider a shock wave propagating into an ideal gas re-gion with varying density, velocity and pressure fields, and derive a transportequation for the shock strength. This equation shows that the evolution ofthe shock depends on coupling to the rearward flow; as a matter of fact, thissingle transport equation is inadequate to reveal the complete evolutionarybehavior of a shock of arbitrary strength because the coupling term in thisequation remains undetermined. We, therefore, need to examine certain as-pects in more detail in subsequent sections, where the shock strength equationturns out to be just one of the infinite sequence of compatibility conditionsthat hold on a shock front. This enables us to determine the coupling termapproximately using a truncation criterion, the accuracy of which has beentested by comparing the results with the Guderley’s similarity solution.

As pointed out by Wright [212], it is found that the shock front can de-velop discontinuities in slope and Mach number (or, shock strength), whichare referred to as shock-shocks [210]; intuitively, it appears that shock-shocksmight only persist in the case of intersecting shocks, and that otherwise theunderlying shock surface would tend to modify its curvature in such a waythat shock-shocks are eliminated. Here, we specialize the results for the two-dimensional shock motion and notice that our exact equations bear a struc-tural resemblance to those of geometrical shock dynamics [210]; indeed, itis shown that the shock ray tube area and the shock Mach number are re-lated according to a differential relation which arises in a natural way fromthe analysis. It is interesting to note that our relations, in the weak shock

7.1 Shock Wave through an Ideal Gas in 3-Space Dimensions 207

limit, coincide exactly with the corresponding equations of geometrical shockdynamics. The implications of the lowest order and first order truncation ap-proximations are discussed when the shock strength varies from the acousticlimit to the limit of geometrical shock dynamics. The lowest order trunca-tion approximation yields the results of geometrical acoustics. The nonlinearbreaking of wave motion for disturbances propagating on weak shocks is dis-cussed and the time taken for the onset of a shock-shock is determined. Inthe strong shock limit, the results for the implosion problem are comparedwith Guderley’s similarity solution. The first order truncation approximationin the weak shock limit leads to the governing equation for the amplitude ofan acceleration wave, which is in full agreement with the results obtained in[205]. In the weak shock limit, the asymptotic decay formulae for the shockand rearward precursor disturbances are found to be in full agreement withthe earlier results. In passing, we remark that one could have derived an equiv-alent system of transport equations for the variation of the jumps in densityor velocity and their respective derivatives, as they have the required depen-dence through basic equations and jump conditions. In fact, the process ofconverting transport equations for the shock strength and coupling terms, de-fined respectively in terms of the jump in pressure and its derivatives, intoan equivalent system of equations involving the jumps in density or velocityand their respective derivatives is quite straightforward. We have carried outthe computation of the exponent for the implosion problem using the secondorder truncation approximation, and the approximate values are found to beclose to the numerical results. However, there are still open questions relat-ing to (i) the dependence of truncation on the choice of the forms of shockstrength and coupling terms, and the search for an appropriate choice whichcould provide desired accuracy with a minimum number of ordinary differen-tial equations obtained after employing the truncation criterion and (ii) therate of convergence of the underlying truncation approximation and the esti-mates of error bounds of the solution at a particular level of approximation.To tackle these problems some serious efforts are presently underway; if theunderlying approach turns out to be sufficiently general, it will be very use-ful in dealing with still unresolved problems of shock propagation (e.g., Von- Neumann paradox in the field of Mach reflection). Finally, we refer to analternative approach proposed by Maslov [119], and show that the compati-bility conditions derived by using this approach coincide exactly with thoseobtained by using the singular surface theory.

We use a fixed cartesian coordinate system xi (i = 1, 2, 3) so that in regionsadjacent to the shock wave, the conservation laws for mass, momentum andenergy may be expressed in the form

∂tρ+ uiρ,i + ρui,i = 0; ∂tui + ujui,j + ρ−1p,i = 0; ∂tp+ uip,i + γpui,i = 0,

(7.1.1)where the range of Latin indices is 1,2,3 and the summation convention onrepeated indices is implied; ∂t and a comma followed by an index i denote,respectively, the partial differentiation with respect to time t and space co-


ordinate xi ; ρ, ui, p and γ are respectively the gas density, velocity, pres-sure and specific heat ratio. The shock is propagating through a given stateui = ui0(t, ~x), p = p0(t, ~x), ρ = ρ0(t, ~x) where ~x = (x1, x2, x3). Let the shocksurface at any time t and position ~x be given by S(t, ~x) = 0. Across a shocksurface, the jump in a physical quantity is denoted by [f ]=f−−f+, where f+

and f− are values of f immediately ahead of and just behind the shock surfacerespectively. If G is the normal speed of the shock surface with ~n = (n1, n2, n3)as its unit normal, then

G = − (∂tS) (S,jS,j)−1/2

; ni = (S,i) (S,jS,j)−1/2

. (7.1.2)

Across the shock wave, the conservation laws of mass, momentum and energyyield the following jump relations

[ui] =zs(a

+0 )2/γU

ni , [ρ] = 2ρ+

0 zs 2γ + (γ − 1)zs−1 , zs = [p]/p0 ,(7.1.3)

where U = G − ui+0 ni, is the normal shock speed relative to the fluid, and itis given by

U2 = (a+0 )2 1 + (γ + 1)zs/2γ , (7.1.4)

with a+0 = (γp+

0 /ρ+0 )1/2 as the equilibrium sound speed. In addition, we have

the following compatibility relations [201]

[∂tf ] = d[f ]/dt−G [nif,i] , [f,i] = ni[njf,j ] + ∂i[f ] , (7.1.5)

where d/dt and ∂i denote respectively the temporal rate of change followingthe shock and the spatial rate of change along the shock surface defined as

df/dt = ∂tf + (U + ui+0 ni) njf,j , ∂if = f,i − ninjf,j . (7.1.6)

It immediately follows from (7.1.6)2 that

ni ∂if = 0, ∂ini = ni,i. (7.1.7)

Taking jump in equations (7.1.1) across the shock and using (7.1.5) and (7.1.7)and the fact that jump in ui is parallel to ni, we get

∂[ρ]/∂τ +[ui]ni − U

[nkρ,k] +

([ρ] + ρ+

0

) ni[nju

i,j ] + ∂i[u

i]

+ [ui](ρ0,i)+ + [ρ](ui0,i)

+ = 0, (7.1.8)

∂[ui]/∂τ +[uj ]nj − U

[nku

i,k] +

([ρ] + ρ+

0

)−1ni[njp,j ] + ∂i[p]

− [ρ](p0,i)+/ρ+

0

+ [uj ](ui0,j)

+ = 0, (7.1.9)

∂[p]/∂τ +[ui]ni − U

[nkp,k] + γ

([p] + p+

0

)ni[nju

i,j ] + ∂i[u

i]

+ [ui](p0,i)+ + γ[p](ui0,i)

+ = 0, (7.1.10)

where ∂/∂τ = d/dt+ ui+0 ∂i ; ui0 = ui0 − ninjuj0 .


Eliminating [njui,j ] between (7.1.8) and (7.1.9) and then using (7.1.3) and

(7.1.5), we obtain the following nonlinear equation for the shock strength zs:

∂zs/∂τ + k1 ni,i = I − k2 [nip,i] , (7.1.11)

where

k1 = 2γUzs(1 + zs)(2γ + (γ − 1)zs) /Θ,k2 = (γ + 1)Uzs(2γ + (γ − 1)zs) /Θp+

0 ,

I = k3(∂ρ+0 /∂τ) + k4(∂p

+0 /∂τ) + k5ni(p0,i)

+ + k6ninj(ui0,j)

+ + k7(ui0,i)

+,

k3 =2γU2zs(1 + zs)

/Θp+

0 ,

k4 = −zs (3γ + (2γ − 1)zs)(2γ + (γ + 1)zs) /Θp+0 ,

k5 =2(γ + 1)z2

sU/Θp+

0 ,

k6 = −2γ(1 + zs)zs(2γ + (γ + 1)zs) /Θ,k7 = −γzs(2γ + (γ − 1)zs)(2γ + (γ + 1)zs) /Θ,Θ = 8γ2 + (9γ2 + γ)zs + (2γ2 + γ − 1)z2

s .

The growth and decay behavior of a shock wave propagating into an inhomo-geneous medium is governed by the shock strength equation (7.1.11). The termni,i in (7.1.11) is equal to the twice the mean curvature of the shock surface.The inhomogeneous term I in (7.1.11) is a known function of zs, ~n, ~x and t;it may be noted that this inhomogeneous term vanishes for the homogeneousand rest conditions ahead of the shock. If we differentiate equations (7.1.2),we obtain on using the definitions of ∂/∂τ and ∂i the following kinematicrelation

∂ni/∂τ = −nj(∂iuj+0 ) − ∂iU. (7.1.12)

Equation (7.1.12), on using (7.1.4), can be rewritten

∂ni∂τ

+(γ + 1)(a+

0 )2

4γU∂izs = − U

2p+0

∂ip+0 +

U

2ρ+0

∂iρ+0 − nj ∂iu

j+0 , (7.1.13)

where the right-hand side is a known function of zs, ~n, ~x and t. From thedefinition of ∂/∂τ , it follows that the instantaneous motion of the shock surfaceis described by the equation

∂xi/∂τ = ui+0 + Uni . (7.1.14)

It may be noted that the shock rays are described by equations (7.1.13) and(7.1.14). If the shock is propagating into a region at rest, equation (7.1.14)shows that the shock rays are also normal and they form a family of orthogonaltrajectories to the successive positions of the shock surface. However, equation(7.1.13) shows that the shock rays bend around in response to the gradientsin zs, p

+0 , ρ

+0 and ui+0 .

Equation (7.1.11), (7.1.13) and (7.1.14), which form a coupled system of


partial differential equations in the unknowns zs, ~x and ~n can be used to calcu-late successive positions of the shock front, its orientation, and the distributionof shock strength zs on it, provided the term [nip,i] in (7.1.11) is made known.This term, which is the jump in the pressure gradient along the normal to theshock surface, provides a coupling of the shock with the rearward flow; indeed,it represents the effect of disturbances or induced discontinuities that catch-upwith the shock from behind. This coupling term rather depends on the pasthistory of the flow. Thus, the equations (7.1.11), (7.1.13) and (7.1.14) showthat the evolutionary behavior of the shock strength at any time t depends notonly on the distribution of shock strength zs on the shock surface, but it alsodepends on the inhomogenities present in the medium ahead of the shock, thenormal and the mean curvature of the shock surface, and the jump in pressuregradient along normal to the shock wave at time t. This coupling term canbe estimated using a natural truncation criterion; however, we assume for themoment that this coupling term is known exactly.

7.1.1 Wave propagation on the shock

Let θ and φ be the angles which the normal ~n to the shock surface makeswith the x1 and x3 axes respectively. Then the unit normal ni and its diver-gence can be expressed as

~n = (cosθ sinφ, sin θ sinφ, cosφ), ni,i = sinφ (∂θ/∂η2) − (∂φ/∂η1),(7.1.15)

where

∂/∂η1 = − cos θ cosφ ∂/∂x1 − sin θ cosφ ∂/∂x2 + sinφ ∂/∂x3,∂/∂η2 = − sin θ ∂/∂x1 + cos θ ∂/∂x2,

(7.1.16)

represent spatial rates of change along two mutually orthogonal directionstangential to the shock surface. In view of (7.1.15) and (7.1.16), the shockstrength equation (7.1.11) and the shock ray equations (7.1.13) and (7.1.14)become

∂zs∂τ

+ k1

(sinφ

∂θ

∂η2− ∂φ

∂η1

)= F1(zs, θ, φ, ~x(~η), τ),

∂θ

∂τ+

Γ

sinφ

∂zs∂η2

= F2(zs, θ, φ, ~x(~η), τ),

∂φ

∂τ− Γ

∂zs∂η1

= F3(zs, θ, φ, ~x(~η), τ), (7.1.17)

∂x1

∂τ= u1+

0 + U cos θ sinφ,

∂x2

∂τ= u2+

0 + U sin θ sinφ,

∂x3

∂τ= u3+

0 + U cosφ,


where

F1 = I − k1[nip,i] ; Γ = (γ + 1)(a+0 )2/4γU,

F2 = − U

2 sinφ

(1

p+0

∂p+0

∂η2− 1

ρ+0

∂ρ+0

∂η2

)− cos θ

∂u1+0

∂η2

− sin θ∂u2+

0

∂η2− cotφ

∂u3+0

∂η2,

F3 =U

2

(1

p+0

∂p+0

∂η1− 1

ρ+0

∂ρ+0

∂η1

)cos θ sinφ

∂u1+0

∂η1

+ sin θ sinφ∂u2+

0

∂η1+ cosφ

∂u3+0

∂η1.

It is interesting to note that equations (7.1.17) form a hyperbolic system ofsix coupled partial differential equations in the unknowns zs, ~n and ~x, andtherefore represent a wave motion for disturbances propagating on the shock.Indeed, equations (7.1.17) together with the initial values of θ, φ, zs and ~x,specified as functions of η1 and η2 at τ = 0, allow us to determine the successivepositions xi of the shock front, the inclinations θ and φ of its normal, and thedistribution of strength zs on it. The wave motion on the shock surface itselfis brought out by studying the characteristic surfaces of equations (7.1.17),which can be rewritten in the form

∂WI

∂τ+AµIJ

∂WJ

∂ηµ= BI , I, J = 1, 2, . . . , 6; µ = 1, 2, (7.1.18)

where W = (zs, θ, φ, x1, x2, x3)tr , and the matrices AµIJ and BI , which are

known functions of η1, η2, and τ , can be read off by inspection of (7.1.17);here the summation on J and µ is implied.

Let ψ(η1, η2, τ) = 0 be the equation for the characteristic surface of(7.1.17). If ξµ denotes the unit normal to the characteristic surface in itsdirection of propagation, and if V (> 0) denotes its speed then

ξµ = −V (∂ψ/∂ηµ) /(∂ψ/∂τ) , (7.1.19)

and it can be easily shown that all possible speeds of propagation satisfy thecharacteristic condition

det |AµIJξµ − V δIJ | = 0, (7.1.20)

where δIJ is the Kronecker function.Equation (7.1.20) has six roots; the root V=0, which is of multiplicity

four, corresponds to the requirement that the derivatives of zs, θ, φ, etc. arecontinuous on the shock surface. The other two nonzero characteristic roots,V = ±c1/2, where

c =γ(γ + 1)zs(1 + zs) 1 + (γ − 1)zs/2γ (a+

0 )2

8γ2 + (9γ2 + γ)zs + (2γ2 + γ − 1)z2s

, (7.1.21)


correspond to wave propagation on the shock surface. It is evident from(7.1.21) that the speed V of the wave propagation is independent of the di-rection ξµ.

7.1.2 Shock-shocks

Let us consider the characteristic surface ψ(η1, η2, τ) = 0, which is propa-gating with the speed V = c1/2, say. Then across ψ = 0, the variables zs, θ, φ,etc. are continuous but discontinuities in their derivatives are permitted. Thejump in an entity f across ψ = 0 is denoted by [[f ]], where the brackets standfor the quantity enclosed immediately behind the wave front ψ = 0 minus itsvalue just ahead of ψ = 0, which we shall denote by f∗. The first and secondorder geometric and kinematic compatibility conditions, which hold across ψ= 0, are

[[∂f

∂τ

]]= −V f ,

[[∂f

∂ηµ

]]= ξµf ,

[[∂2f

∂ηµ∂τ

]]= −V ξµf − V

∂f

∂ηµ+ ξµ

df

dτ+ f

dξµdτ

, (7.1.22)

[[∂2f

∂ηµ∂ηδ

]]= ξµξδf + ξδ

∂f

∂ηµ+ ξµ

∂f

∂ηδ+ f

∂ξµ∂ηδ

,

where

f ≡[[ξµ

∂f

∂ηµ

]], f ≡

[[ξµξδ

∂2f

∂ηµ∂ηδ

]],

d

dτ≡ ∂

∂τ+ V ξµ

∂

∂ηµ,

∂[[f ]]

∂ηµ≡[[∂f

∂ηµ

]]− ξµ

[[ξδ∂f

∂ηδ

]]and

dξµdτ

= −(∂V

∂ηµ

)

W=W∗

.

Forming jumps across ψ=0, in equations (7.1.17), we obtain on using(7.1.22)1,2 that

θ = (Γ∗/c1/2∗ sinφ∗) ξ2 z , φ = −(Γ∗/c

1/2∗ ) ξ1 z , xi = 0. (7.1.23)

Following a standard procedure [167], if we differentiate equations (7.1.17)with respect to ηµ, form jumps across ψ = 0 in the resulting equations, makeuse of the compatibility conditions (7.1.22) and the relations (7.1.23), andthen eliminate quantities with overhead bars, we obtain the following Riccatitype equation in z on ψ = 0 :

dz

dτ+

d(lnΛ1/2)

dτ+ P

z +Q z2 = 0, (7.1.24)


where

Λ = c1/2∗ /k1∗

(sinφ∗)ξ22 ,

P = −V∗K +V∗2

cot φ∗ ξ

22

(∂φ

∂ξ

)

∗− cot φ∗ ξ2

(∂φ

∂η2

)

∗

−ξ1 cosφ∗

(∂θ

∂η2

)

∗+

(∂ lnk1

∂ξ

∣∣∣∣zs

)

∗+

(∂ lnk1

∂zs

∣∣∣∣η

)

∗

(∂zs∂ξ

)

∗

− ξ2k1∗

sinφ∗

(∂F1

∂θ

)

∗+

ξ1k1∗

(∂F1

∂φ

)

∗+ Γ∗cotφ∗ ξ1 ξ2

(∂zs∂η2

)

∗

+2

(∂ lnΓ

∂zs

∣∣∣∣η

)

∗

(∂zs∂ξ

)

∗− sinφ∗ ξ2

(∂F2

∂zs

∣∣∣∣η

)

∗+

(∂ lnΓ

∂ξ

∣∣∣∣zs

)

∗

+ξ1Γ∗

(∂F3

∂zs

)

∗

+

1

2

sinφ∗

(∂k1

∂zs

)

∗

(∂θ

∂η2

)

∗−(∂k1

∂zs

)

∗

(∂φ

∂η1

)

∗

−ξ22(∂F2

∂θ

)

∗+ ξ1ξ2 sinφ∗

(∂F2

∂φ

)

∗+ξ1ξ2sin

φ∗

(∂F3

∂θ

)

∗

−ξ21(∂F3

∂φ

)

∗−(∂F1

∂zs

)

∗

,

Q =V∗2

(∂lnk1

∂zs

∣∣∣∣η

)

∗− c

3/2∗

2k1∗U∗,

where K = −(1/2)(∂ξµ/∂ηµ) is the mean curvature of the characteristic sur-face ψ = 0, and (∂f/∂ξ) ≡ ξµ(∂f/∂ηµ)∗ is the derivative of f in the directionnormal to the characteristic surface on its upstream side.

Equation (7.1.24) is well known in the theory of acceleration waves; athorough treatment of the properties of the solution of such an equation,within a general framework, is given in Section 4.2. Equation (7.1.24) can beintegrated to yield

z =z(0) (Λ/Λ0)

−1/2exp(−q(τ))1 + Λ

1/20 z(0) I1(τ)

, (7.1.25)

where q(τ) =∫ τ0 P (t) dt, I1(τ) =

∫ τ0 (Q/Λ1/2)exp(−q(τ ′)) dτ ′, and z(0) indi-

cates the value of z at τ = 0. Equation (7.1.25) shows that if q(τ) and I1(τ)are continuous on [0,∞) and have finite limits as τ → ∞ and if sgnz(0) = −sgnI1(τ), then z will remain finite provided |z(0)| < ζ, where

ζ = (Λ1/20 I1(∞))−1. But if |z(0)| > ζ then there will exist a finite time τ ∗,

given by I1(τ∗) = −(Λ

1/20 z(0))−1, such that z → ∞ as τ → τ∗; this signifies

the appearance of a shock wave at an instant τ ∗. This shock discontinuity inthe wave motion on the original gasdynamic shock has been referred to as“shock-shock,” which corresponds physically to the formation of Mach stemsin Mach reflection [210].


7.1.3 Two-dimensional configuration

Assuming the state ahead of the shock to be uniform and at rest, and set-ting φ = π/2 and ∂/∂x3 = 0 in equations (7.1.17), the shock wave propagationin x1, x2 plane can be described by the following equations

∂θ

∂η+U

k1

∂zs∂σ

= − γ + 1

2γ(1 + zs)p0

(∂p

∂n

)−,

∂θ

∂σ+

(γ + 1)a20

4γU2

∂zs∂η

= 0, (7.1.26)

∂x1

∂σ= cosθ,

∂x2

∂σ= sinθ,

where dσ = Udτ is the distance traveled by shock in time dτ along the shockray, dη ≡ dη2 is the distance measured along the shock front, and the deriva-tive (∂p/∂n)− refers to the instantaneous value of the space derivative of pbehind the shock along its normal. The successive shock positions and therays, described by the family of curves σ= constant and η= constant, re-spectively, form an orthogonal coordinate system (σ, η), which is exactly thesame as the orthogonal coordinate system (α, β) introduced in the theory ofgeometrical shock dynamics. Thus the line elements along the two family ofcurves are

dσ = Mdα and dη = Adβ, (7.1.27)

where A(α, β) is the ray-tube area along the shock ray and M = U/a0 is theshock Mach number, which in view of (7.1.4) can be expressed as

M = 1 + (γ + 1)zs/2γ1/2. (7.1.28)

From the definition of ray derivative, and the fact that (niA),i = 0, the en-tities A, ni and U are related as ∂A/∂τ = AUni,i. This relation for the two-dimensional problem, under consideration, reduces to ∂A/∂σ = A(∂θ/∂η),which, in turn, assumes the following form when expressed in the (α, β) coor-dinates

∂θ/∂β = M−1(∂A/∂α). (7.1.29)

In view of (7.1.27) and (7.1.28), the shock strength equation (7.1.26)1 in the(α, β) coordinates becomes

∂θ

∂β+A b(M)

M2 − 1

∂M

∂α= Aω

(∂p

∂n

)−, (7.1.30)

where ω = −(γ + 1)22ρ0a20(2γM

2 − γ + 1)−1, and

b(M) =(γ + 1)2(2γ − 1)(M 2 − 1)2 + (9γ + 1)M2 − 5γ + 3

(2γM2 − γ + 1)(2 + (γ − 1)M2).


If we use (7.1.29) in (7.1.30), we obtain the following differential equationconnecting the variation of A and M along the rays:

1

A

∂A

∂α+Mb(M)

M2 − 1

∂M

∂α= ωM

(∂p

∂n

)−. (7.1.31)

Similarly, equations (7.1.26)2 – (7.1.26)4, in terms of the (α, β) coordinatesand the Mach number M , become

(∂θ/∂α) = −(1/A) (∂M/∂β), (7.1.32)

(∂x1/∂α) = Mcosθ, (7.1.33)

(∂x2/∂α) = Msinθ. (7.1.34)

As mentioned earlier, the coupling term (∂p/∂n)− in the above equations,which is determined in the following section using a truncation criterion, isassumed to be known for the moment. Thus, equations (7.1.30) – (7.1.34),which form a complete set of equations to determine A,M, θ, x1 and x2, de-scribe the shock motion taking into account the effect of acoustic disturbancesbehind the shock.

It is interesting to note that our equations (7.1.29) and (7.1.32) to (7.1.34)coincide exactly with those of geometrical shock dynamics. Thus, the basicequations of geometrical shock dynamics, which form an open system thatrequires an additional relation to complete it, are exact; the approximationlies only in the introduction of a relation between A and M , which serves toprovide an approximate closure on the basic equations and determines themotion of a shock with good accuracy [210]. However, the present treatmentleads to an equation (7.1.31), connecting the coupling term (∂p/∂n)− withthe variations of A and M along the rays. It may be noticed that for a weakshock (zs 1), b(M) → 4, and consequently, the corresponding weak waveapproximation of (7.1.30), which is obtained by neglecting terms of the sizeO(z2

s) and O(zs(∂p/∂n)−), coincides exactly with that of geometrical shockdynamics. It may be remarked that an area-Mach number relation similar to(7.1.31) for the case of shock propagation in an ideal gas with nonuniformpropagation ahead has been considered previously by [29] and [161].

7.1.4 Transport equations for coupling terms

As noted earlier, it is possible to obtain an analytical description of thecomplete growth and decay history of a shock of arbitrary strength, providedwe know the coupling term [nip,i]; to achieve this goal, we proceed as follows.In order to make algebraic calculations less cumbersome, we consider the stateahead of the shock to be uniform and at rest. The transport equation (7.1.11)for the shock strength, then, becomes

∂zs/∂τ + k1ni,i = −k2(nip,i)−, (7.1.35)


where k1 and k2 are same as in (7.1.11) with ρ+0 = ρ0= constant, p+

0 = p0 =constant, and u+

0 = 0.

When equations (7.1.3), (7.1.13) and (7.1.35) are combined and the result-ing equation is used in (7.1.9), we obtain that

(njui,j)

− = ε1ni(njp,j)− + ε2ninj,j + ε3∂izs, (7.1.36)

where

ε1 = (8γ + 3(γ + 1)zs)(2γ + (γ − 1)zs) /2ρ0UΘ,

ε2 = −zsa

20(1 + zs)(4γ + (γ + 1)zs)

/UΘ,

ε3 = a20(4γ + (γ − 3)zs) 2γU(2γ + (γ − 1)zs)−1

,

with Θ being the same as in (7.1.11). Similarly, equations (7.1.8) and (7.1.9),in view of (7.1.3), (7.1.35) and (7.1.36), yield

(njρ,j)− = ε∗1 (njp,j)

− + ε∗2 nj,j , (7.1.37)

where

ε∗1 =3(γ2 − 1)z2

s + (10γ2 − 6γ)zs + 16γ2

2γ2U2a40(2γ + (γ − 1)zs)

2Θ−1

,

ε∗2 = 4γ(γ2 − 1)ρ0U2z3s

a20(2γ + (γ − 1)zs)

2Θ−1

.

The second order compatibility relations, which we need later, are as follows[201]

[∂tf,i] = ni∂[njf,j ]

∂τ+ [njf,j ]

∂ni∂τ

+∂(∂i[f ])

∂τ− Uni[nlnkf,lk]

+U(∂jni)(∂j [f ]), (7.1.38)

[f,ij ] = (∂i∂j [f ]) + (∂inj)[nkf,k] + nj(∂i[nkf,k]) + ni(∂j [nkf,k])

−ni(∂jnk)(∂k [f ]) + ninj [nlnkf,lk]. (7.1.39)

Now we differentiate equations (7.1.1) with respect to xk, take jump in theresulting equations across the shock, multiply these equations by nk, and usethe compatibility conditions (7.1.5), (7.1.38), and (7.1.39) to obtain

∂(niρ,i)−

∂τ+[ui]ni − U

(nlnmρ,lm)− + (ρ0 + [ρ]) ni(nlnku

i,lk)

−

+nk∂(∂k[ρ])

∂τ+ (nlu

i,l)

−ni(nkρ,k)

− + ∂i[ρ]

+ (ρ0 + [ρ])∂i(nlu

i,l)

−

− (∂inl)(∂l[ui])

+ (nlρ,l)− ni(nkui,k)− + ∂i[u

i]

+[ui](∂i(nlρ,l)

− − (∂inl)(∂l[ρ])

= 0, (7.1.40)


∂(nlui,l)

−

∂τ+[uj ]nj − U

(nlnmu

i,lm)− + (ρ0 + [ρ])

−1ni(nlnkp,lk)

−

+nk∂(∂k[u

i])

∂τ+ (ρ0 + [ρ])

−1∂i(nlp,l)

− − (∂inl)(∂l[p])

+[uj ]∂j(nlu

i,l)

− − (∂jnl)(∂l[ui])

+ (nluj,l)

−nj(nku

i,k)

− + ∂j [ui]

− (ρ0 + [ρ])−2

(nlρ,l)− ni(nkp,k)− +∂i[p]

= 0, (7.1.41)

∂(nip,i)−

∂τ+[ui]ni − U

(nlnmp,lm)− + γ (p0 + [p])ni(nlnku

i,lk)

−

+nk∂(∂k[p])

∂τ+ [ui]

∂i(nlp,l)

− − (∂inl)(∂l[p])

+γ (p0 + [p])∂i(nlu

i,l)

− − (∂inl)(∂l[ui])

+(nlui,l)

−ni(nkp,k)

− + ∂i[p]

+γ(nlp,l)−ni(nku

i,k)

− + ∂i[ui]

= 0. (7.1.42)

If we eliminate (nlnkui,lk)

− between (7.1.41) and (7.1.42), substitute in the

resulting equation the value of ∂(njui,j)

−/∂τ , which is obtained from (7.1.36),and make use of (7.1.3), (7.1.35), (7.1.36) and (7.1.37), we obtain the followingtransport equation for the coupling term Π(1) = (nip,i)

−

∂Π(1)

∂τ+ φ1Π(2) + φ2Π

2(1) + φ3Π(1) + φ4 = 0, (7.1.43)

where Π(2), defined as Π(2) = (ninjp,ij)−, depends on the second order space

derivatives of pressure behind the shock along its normal, and the quantitiesφ1, φ2, φ3, φ4, which are defined on the shock, have the following form

φ1 = (γ + 1)zsa20λ/2γU,

λ = (2γ + (γ − 1)zs)a20

2γρ0a

20U(1 + zs)ε1 + (2γ + (γ − 1)zs)a

20

−1,

φ2 = λ

(γ + 1)ε1 +

2γρ0U(1 + zs)

(2γ + (γ − 1)zs)

(−k2

dε1dzs

+ ε21

− (2γ + (γ − 1)zs)2ε∗1

(2γ + (γ + 1)zs)2ρ20

),

φ3 = λ

zsa

20

U− 2γρ0U(1 + zs)

(2γ + (γ − 1)zs)

(k1dε1dzs

+ k2dε2dzs

+(2γ + (γ − 1)zs)

2ε∗2(2γ + (γ + 1)zs)2ρ2

0

− 2ε1ε2

)+ γp0(1 + zs)ε1 + (γ + 1)ε2

ni,i ,


φ4 = λ

γp0(1 + zs)

dε3dzs

+ ε3p0 +(γ + 1)a2

0p0

4γU

+2γρ0U(1 + zs)

(2γ + (γ − 1)zs)

((γ + 1)2a4

0ε216γ2U3

+(γ + 1)a2

0ε34γU

+(4γ + (γ + 1)zs)a

40

4γ2U3

(ε3 +

(γ + 1)a20

4γU

))(∂izs

)(∂izs

)

+λ

γp0(1 + zs)ε3 −

(γ + 1)a20ρ0(1 + zs)ε2

2(2γ + (γ − 1)zs)

(∂i∂izs

)

+λ

γp0(1 + zs)ε2 +

2γρ0U(1 + zs)

(2γ + (γ − 1)zs)

(ε22 − k1

dε2dzs

)(ni,i)

2

−λzs(1 + zs)a

20p0

U+

2γρ0U2(1 + zs)ε2

(2γ + (γ − 1)zs)

ni,jnj,i .

We note from (7.1.43) that the behavior of the coupling term Π(1) dependson another coupling term Π(2), which is again unknown. For its determination,we repeat the above procedure; and proceeding in this way, we obtain thefollowing infinite set of transport equations for the coupling terms Π(k) =(ni1ni2 · · ·nikp,i1i2...ik)−; k = 1, 2, 3, ...

∂Π(k)

∂τ+ φkΠ(k+1) + Φk(ρ0, p0, ~n, zs,Π(1),Π(2), · · · ,Π(k)) = 0, (7.1.44)

where Φk are known functions of their arguments and the quantities φk aredefined on the shock front.

The infinite set of equations (7.1.11), (7.1.13), (7.1.14), (7.1.43) and(7.1.44) with k = 2, 3, · · · , and ui0 ≡ 0, p+

0 = p0, ρ+0 = ρ0, supplemented

by the initial conditions

~x = ~x0, ~n = ~n0, zs = zs0, Π(k) = Π0(k), k ≥ 1 at τ = 0, (7.1.45)

can be conveniently used in the investigation of the shock location, orientation,strength and the values of the coupling terms Π(k) at any time t. It may benoted that the above infinite system of equations is an open system; in orderto provide a natural closure on the infinite hierarchy of equations, we setΠ(k+1)= 0 in (7.1.44) for the largest k retained. The truncated system, whichis closed, can be regarded as a good approximation of the infinite hierarchyof the system governing shock propagation ([169], [4]).

7.1.5 The lowest order approximation

The lowest order truncation approximation of the infinite hierarchial sys-tem of transport equations leads to the shock strength equation obtainedfrom (7.1.11) by setting (nip,i)

− = 0. Let us consider the case of a weak shock(zs 1) propagating into a constant state at rest. It then follows from (7.1.3)


that

[ui] = (zsa0/γ)ni +O(z2s ),

U = a01 + (γ + 1)zs/4γ+O(z2s ),

[ρ] = ρ0zs/γ +O(z2s ),

k1 = zsa0/2 +O(z2s ), (7.1.46)

k2 = zs(γ + 1)/4ρ0a0 +O(z2s ),

Γ = (γ + 1)a01− (γ + 1)zs/4γ/4γ +O(z2s ).

In the lowest order truncation approximation, the basic equations (7.1.26), onusing (7.1.46), reduce to the following system in the weak shock limit

∂zs∂τ

+zsa0

2

∂θ

∂η= 0,

∂θ

∂τ+

(γ + 1)a0

4γ

∂zs∂η

= 0, (7.1.47)

∂x1

∂τ= a0

1 +

(γ + 1)zs4γ

cosθ,

∂x2

∂τ= a0

1 +

(γ + 1)zs4γ

sinθ.

These equations represent a wave motion for disturbances propagating on theweak shock; the speeds with which the disturbances propagate turn out tobe V = ± a0(γ + 1)zs/8γ1/2. These waves carry the changes of the shockshape and the shock strength along the weak shock. We consider waves withV > 0, running into a constant state and carrying the initial discontinuity inthe derivative (∂zs/∂η). Analogous to the prototype of three-dimensional wavemotion, discussed in the preceding section, these waves break in a typical non-linear fashion; indeed, it turns out that the derivative (∂zs/∂η) at any time τon the wave front is given by ∂zs/∂η = (∂zs/∂η)01+Λ(∂zs/∂η)0τ−1, where(∂zs/∂η)0 is the value of ∂zs/∂η at τ = 0, and Λ = (a0/2)((γ + 1)/8γ)1/2.Thus, whenever (∂zs/∂η)0 < 0, the wave motion develops a shock-shock aftera finite time τs = −(Λ(∂zs/∂η)0)

−1. As noted earlier, in this lowest ordertruncation approximation, the basic equations (7.1.26), in the weak shocklimit, coincide exactly with the corresponding equations of geometrical shockdynamics.

For a three-dimensional geometrical configuration, we have ni,i = −2K,where K is the mean curvature of the shock surface, which for a planar(m = 0), cylindrically (m = 1) or spherically (m = 2) symmetric flow isgiven by K = −m/2X, where X is the radius of the wave front at time t.Consequently, the shock strength equation (7.1.35), in the lowest order ap-proximation, becomes

Mb(M)

M2 − 1

∂M

∂X+m

X= 0, (7.1.48)


where M = U/a0 is the shock Mach number, b(M) is the same as in (7.1.30),and the distance traversed by the shock along its normal is dX = Udτ . Forweak shocks (M → 1, b → 4) equation (7.1.48) yields on integration M =1 + (M0 − 1)(X/X0)

−m/2, M |X=X0 = M0, which is the exact result ofgeometrical acoustics for weak pulses. However, for an infinitely strong shock(M → ∞, b→ (γ+1)(2γ−1)/γ(γ−1)), equation (7.1.48) yields on integrationM = M0(X/X0)

−β∗ , where β∗ = mγ(γ − 1)/ (γ + 1)(2γ − 1). Thus, inthe lowest order approximation, the values of the exponent β∗ for cylindrical(m = 1) and spherical (m = 2) shocks for γ = 1.4 turn out to be 0.130 and0.259 respectively, whilst the corresponding numerical values obtained fromthe Guderley’s similarity solution for imploding shocks are respectively 0.197and 0.394 (see [66]). The relative error in the values of the exponent β∗ suggestthat one should take into account higher approximations.

7.1.6 First order approximation

To assess the error involved in the lowest order approximation and alsoto give some confidence in the validity of the proposed truncation criterion,we consider the first order (k = 1) approximation; to this approximation,the motion of a shock running into a constant state at rest is described bysetting Π(2) = 0 in (7.1.43). Thus, to this approximation, the set of governingequations is the following closed system

∂zs∂τ

+ k1ni,i = −k2Π(1),

∂ni∂τ

+(γ + 1)a2

0

4γU∂izs = 0, (7.1.49)

∂xi∂τ

= Uni,

∂Π(1)

∂τ+ φ2Π

2(1) + φ3Π(1) + φ4 = 0,

which together with the initial conditions (7.1.45) for k =1 can be integratednumerically to determine the shock motion.

However, if the shock amplitude is weak (zs 1), equations (7.1.49) reduceto

∂zs∂X

+(γ + 1)zs

4ρ0a20

Π(1) +zs2ni,i = 0,

∂ni∂X

+γ + 1

4γ∂izs = 0, (7.1.50)

∂ni∂X

−

1 +(γ + 1)zs

4γ

n,i = 0,

∂Π(1)

∂X+γ + 1

2ρ0a20

Π2(1) +

1

2ni,iΠ(1) = 0,


where the distance traversed by the weak wave along its normal is dX = a0 dτ .It is interesting to note that equation (7.1.50)4, which governs the amplitudeof a weak discontinuity or an acceleration wave, is in full agreement with theresults obtained in [205].

Equations (7.1.50)1 and (7.1.50)4, together with the initial conditions(7.1.45), yield on integration

zs = zs0J(X) exp(−I(X)), (7.1.51)

Π(1) = Π0(1)J(X)

1 +

γ + 1

2ρ0a20

Π0(1)J1(X)

−1

, (7.1.52)

where zs0 and Π0(1) are the values of zs and Π(1) at X = X0,

I(X) =γ + 1

4ρ0a20

∫ X

X0

Π(1)(t) dt, J(X) = exp

(∫ X

X0

K(t) dt

),

and J1(X) =∫XX0J(t) dt, with K as the mean curvature of the weak wave.

For a planar (m = 0), cylindrically (m = 1) or spherically (m = 2) symmetricflow, equations (7.1.51) and (7.1.52) yield

zs = zs0(X/X0)−m/2

1 +

γ + 1

2ρ0a20

Π0(1)J1(X)

−1/2

, (7.1.53)

Π(1) = Π0(1)(X/X0)

−m/2

1 +γ + 1

2ρ0a20

Π0(1)J1(X)

−1

, (7.1.54)

where

J1(X) =

X −X0, (plane)2X0(X/X0)

1/2 − 1, (cylindrical)X0 ln(X/X0), (spherical).

It can be seen from (7.1.53) and (7.1.54) that for Π0(1) ≥ 0, both the shock

strength zs and the precursor disturbance Π(1) eventually decay to zero. In-deed, the shock wave decays like

zs =

X−1/2, (plane)X−3/4, (cylindrical) as X → ∞,

X−1(lnX)−1/2, (spherical),

while the precursor disturbance Π(1) decays like

Π(1) =

X−1, (plane and cylindrical)(X lnX)−1, (spherical),

as X → ∞.

However, if Π0(1) < 0, both the shock strength and the precursor disturbance

behind the shock grow without bound after a finite running length X∗ givenby the solution of J1(X∗) = −2ρ0a

20/(γ + 1)Π0

(1). The occurrence of this


unbounded behavior of the shock strength, or equivalently that of the shockspeed, can be attributed to the quasilinear hyperbolic nature of this governingsystem of equations whose solutions generally exist only in a finite interval.

For a strong shock (M 1), we have the following asymptotic formulaefor the functions k1, k2, φ1, φ2, φ3 and φ4 in (7.1.49)

k1 ∼ 4(γ − 1)γ2M3a0/(2γ − 1)(γ + 1)2,k2 ∼ γ(γ − 1)M/ρ0a0(2γ − 1),φ2 ∼ 3(γ + 1)(5γ2 − 9γ + 2)/4ρ0a0M(2γ − 1)(5γ − 1), (7.1.55)

φ3 ∼ γa0M(14γ2 − 41γ + 11)ni,i (γ + 1)(2γ − 1)(5γ − 1)−1 ni,i,

φ4 ∼ (3γ4 + 8γ3 − 48γ2 + 36γ − 7)ρ0a30

4γ2(5γ − 1)(γ − 1)2M

(∂izs

)(∂izs

)

+3(γ − 1)ρ0a

30M

(5γ − 1)(γ + 1)

(∂i∂izs

)

− 4γ(γ2 − 4γ + 1)ρ0a30M

3

(γ − 1)(5γ − 1)(γ + 1)2ni,jnj,i + γ(2γ − 1)−1(ni,i)

2.

Making use of (7.1.55) in equations (7.1.49)1 and (7.1.49)4, and then elimi-nating Π(1) from the resulting equations, we obtain the following transportequation for the shock Mach number M for plane (m = 0), cylindrical (m = 1)and spherical (m = 2) wave fronts

∂2M

∂X2− (10γ2 − 21γ + 5)

(γ − 1)(5γ − 1)M

(∂M

∂X

)2

− mγ(3γ + 1)

(γ + 1)(5γ − 1)X

(∂M

∂X

)

−2mγ2(2m+ γ + 1)M

(γ + 1)2(5γ − 1)X2= 0.

This equation has the solution

M = B∗X−β2/(1−BXδ)1/q2 , (7.1.56)

where B and B∗ are integration constants, β2 = (q1−δ)/2q2, δ = (q21 −4q2q3),q1 = 1+mγ(3γ+1)(5γ2 +4γ− 1)−1, q2 = (5γ2 − 15γ+4)/(5γ2− 6γ+1) andq3 = 2mγ21 + 2m(γ+ 1)−1/(5γ2 + 4γ− 1). It then follows by using (7.1.55)and (7.1.56) in (7.1.49)4 that

Π(1) = −4ρoa20(2γ − 1)M2

(γ2 − 1)X

γ(γ − 1)m

(γ + 1)(2γ − 1)+Bβ1X

δ − β2

1 −BXδ

, (7.1.57)

where β1 = (q1+δ)/2q2,. The integration constants B and B∗ can be obtainedfrom (7.1.56) and (7.1.57) by using the initial conditions M = M0,Π(1) = Π0

(1)

at X = X0.It may be noticed that for γ lying in the interval 1 < γ < 2, we have

q1 > 0, δ > 0, β2 > 0, while β1 < 0, q2 < 0. Thus, for converging waves,equation (7.1.56) yields the propagation law M ∼ B∗ X−β2 as X → 0,

7.2 An Alternative Approach Using the Theory of Distributions 223

where the values of the exponent β2 for cylindrical (m = 1) and spherical(m = 2) shocks for γ = 1.4 turn out to be respectively 0.228 and 0.437,which are close to the numerical values obtained from the Guderley’s similaritysolution for imploding shocks.

7.2 An Alternative Approach Using the Theory of Dis-

tributions

In the preceding section, we have used the theory of singular surfaces toderive an infinite system of transport equations along the shock rays, whichenable us to investigate the dynamical coupling between shock fronts andthe flow behind them. A different method, using the theory of generalizedfunctions, has been proposed by Maslov [119] to arrive at an infinite systemof compatibility conditions that hold on the shock front. Although, the infinitesystem of equations arising from this method appear to be quite different fromthe system which is found by using the singular surface theory, we show herein this section that the two seemingly different approaches yield the samesystem of equations.

The basic equations (7.1.1) can be written in the following conservationforms in R4:

∂tρ+ (ρui),i = 0,

∂t(ρui) + (ρuiuj),j + p,i = 0, (7.2.1)

∂tE +Qi,i = 0,

where E = (γ−1)−1p+ρuiui/2 and Qi = uiγ(γ − 1)−1p+ ρujuj/2

. Let Ω,

defined by an equation of the form S(t, ~x) = 0, be a smooth three-dimensionalsurface in R4 on which a piecewise smooth solution of (7.2.1) undergoes adiscontinuity of first kind. At any fixed time t, the surface Ω is a smooth 2-dimensional surface, called a shock surface or a shock front, and it is denotedby Ωt. Unlike Maslov, we consider a general situation where |∇S| need not beunity.

Any piecewise smooth vector solution f(t, ~x) ≡ (ρ, ~u, p)tr of (7.2.1), whichsuffers a discontinuity of first kind on Ω, can be represented in the form

f(t, ~x) = f0(t, ~x) + θ (S(t, ~x)) f1(t, ~x), (7.2.2)

where f0, f1, S ∈ C∞(R4), and θ is the Heaviside function

θ(τ) =

1, τ > 00, τ ≤ 0.

Here, f0 refers to the known state ahead of the shock, whereas f0 + f1 refers


to the state behind the shock. To the piecewise smooth solution f , we assigna set of functions fαβ (α = 0, 1 and β = 0, 1, 2, · · · , ) defined on Ω and havingthe interpretation of derivatives along the normal ~n = (n1, n2, n3) to the shockfront Ωt at time t, i. e.,

fαβ|Ω =

(ni

∂

∂xi

)βfα

∣∣∣∣∣Ω

. (7.2.3)

It is assumed that the functions fαβ |Ω have been extended in a smooth mannerfrom Ω to the space R4 so that they are constant along a normal to Ωt. Thus,we have

ni (∂fαβ/∂xi)|Ω = 0. (7.2.4)

Let the generalized function h(t, ~x), representable as a finite sum of functionsh(r) be defined as

h(t, ~x) =N∑

r=1

h(r), h(r) = e(r)m∂f (r)

∂xm; m = 1, 2, 3, 4 (r − unsummed)

(7.2.5)

where f (r) is a function of ρ, ui and p, which can be represented in the form

(7.2.2) with f(r)α as functions in C∞(R4) and f

(r)αβ having the definitions out-

lined as in (7.2.3) and (7.2.4); e(r) are constant vectors in R4 having com-

ponents e(r)m ; x4 stands for time t; and the derivatives are to be interpreted

in the generalized sense. We assign to h a set of functions hαβ , α = 0, 1 andβ = −1, 0, 1, 2, 3, . . . such that

h1,−1|Ω =

N∑

r=1

f(r)10 e

(r)m

∂S

∂xm

∣∣∣∣∣Ω

, (7.2.6)

hαβ|Ω =

N∑

r=1

(f

(r)α,β+1e

(r)m

∂S

∂xm+ e(r)m

∂f(r)αβ

∂xm

)∣∣∣∣∣Ω

, α = 0, 1;β = 0, 1, 2, · · ·

(7.2.7)

Now, if h =

N∑

r=1

e(r)m∂f (r)

∂xm= 0 with the usual summation convention on m,

then it has been shown by Maslov [119] that

h1,−1|Ω = 0, hα =

N∑

r=1

e(r)m∂f

(r)α

∂xm= 0, α = 0, 1. (7.2.8)

Further, since h = 0, all derivatives of the functions hα vanish on Ω, i.e.,

(ni

∂

∂xi

)βhα

∣∣∣∣∣Ω

= 0, α = 0, 1, β = 0, 1, 2, . . . (7.2.9)


Maslov has inferred from (7.2.9) that hαβ |Ω = 0, which, in fact, may not betrue for all β as has been pointed out in [150]. However his relation (7.2.9) isabsolutely right on Ω, and it shall be exploited here in the present context.One can easily check that for β = 0, the relation (7.2.9) yields

hα0|Ω = 0, α = 0, 1, (7.2.10)

while for β = 1, it yields

hα1|Ω +

N∑

r=1

1

|∇xS|

(f

(r)α1 e

(r)m ni

∂2S

∂xixm+ e(r)m ni

∂2f(r)α0

∂xixm

)∣∣∣∣∣Ω

= 0, (7.2.11)

where ∇x is the spatial gradient operator.Let h(t, ~x) be a generalized function which is the result of substituting

(7.2.2) into the left-hand side of the system (7.2.1). Thus, equation (7.2.8)1,together with (7.2.6), yields

ρ10 (∂tS) + (ρui)10S,i = 0,

(ρui)10 (∂tS) + (ρuiuj)10S,j + p10S,i = 0, (7.2.12)

E10 (∂tS) + (Qi)10S,i = 0,

where

(ρui)10 = (ρ00 + ρ10)ui10 + ρ10u

i00,

(ρuiuj)10 = (ρ00 + ρ10)ui10u

j00 + uj10(u

i00 + ui10)

+ ρ10u

i00u

j00,

E10 = (γ − 1)−1p10 +(ρ00 + ρ10)(2u

i10u

i00 + ui10u

i10) + ρ10u

i00u

i00

/2,

Qi10 = γ(γ − 1)−1p10u

i00 + ui10(p00 + p10)

+uj00u

j00

(ρ10(u

i00 + ui10

)

+ρ00ui10

)+ (ρ00 + ρ10)(u

i00 + ui10)(u

j10u

j10 + 2uj10u

j00)/2.

Eliminating ρ10 and ui10 from equations (7.2.12)1 to (7.2.12)3, we obtain theequation of motion of the shock front in the following form

∂tS + ui00S,i + U |∇xS| = 0, on Ω (7.2.13)

whereU2|Ω =

a200 (1 + (γ + 1)z/2γ)

|Ω, (7.2.14)

with z = (p10/p00) as the shock strength and a200|Ω = γ(p00/ρ00)|Ω as the

square of sound speed. Consequently, we get from (7.2.12) and (7.2.13), thefollowing Rankine-Hugoniot jump conditions across the shock front

ρ10|Ω = 2ρ00|Ωzs2γ + (γ − 1)zs−1, ui10|Ω = zsa200|Ω/γU |Ωni, (7.2.15)

where zs = (p10/p00)|Ω, and ni = S,i/|∇xS| is the normal vector to the shock


surface, and ρ10|Ω, ui10|Ω and p10|Ω are the jumps across the shock front inρ, ui and p respectively. It is evident from (7.1.13) that the normal velocity,G, of the shock front is given by

G = ui00ni + U. (7.2.16)

The transport equation describing the behavior of shock strength zs can beobtained from equations (7.2.10)1 and (7.2.7), which imply the following setof equations on Ω

∂tρ00 + ρ01 (∂tS) + (ρui)00,i + (ρui)01S,i = 0,∂t(ρu

i)00 + (ρui)01 (∂tS) + (ρuiuj)00,j + (ρuiuj)01S,j + p00,i + p01S,i = 0,∂tE00 +E01 (∂tS) +Qi00,i +Qi01S,i = 0,

(7.2.17)while (7.2.10)2 implies the following set of equations on Ω

∂tρ10 + ρ11 (∂tS) + (ρui)10,i + (ρui)11S,i = 0,∂t(ρu

i)10 + (ρui)11 (∂tS) + (ρuiuj)10,j + (ρuiuj)11S,j + p10,i + p11S,i = 0,∂tE10 +E11 (∂tS) +Qi10,i +Qi11S,i = 0,

(7.2.18)where

(ρui)00 = ρ00ui00; (ρui)01 = ρ00u

i01 + ρ01u

i00,

(ρuiuj)00 = ρ00ui00u

j00; (ρuiuj)01 = ρ00(u

i00u

j01 + ui01u

j00) + ρ01u

i00u

j00,

E00 = (γ − 1)−1p00 + ρ00ui00u

i00/2,

E01 = (γ − 1)−1p01 + ui00(ρ00ui01 + ρ01u

i00/2),

Qi00 =γ(γ − 1)−1p00 + ρ00u

j00u

j00/2

ui00,

Qi01 = γ(γ − 1)−1(p00ui01 + p01u

i00) + ρ00u

i00u

j00u

j01 + (ρ01u

i00

+ρ00ui01)u

j00u

j00/2,

(ρui)11 = ρ11(ui00 + ui10) + ui11(ρ00 + ρ10) + ρ10u

i01 + ρ01u

i10,

(ρuiuj)11 = (ρui)11(uj00 + uj10) + uj11

(ρui)00 + (ρui)10

+ (ρui)10u

j01

+(ρui)01uj10,

E11 = (γ − 1)−1p11 + (ρuiui)11/2,

Qi11 = γ(γ − 1)−1p11(u

i00 + ui10) + ui11(p00 + p10) + p10u

i01 + p01u

i10

+(ρuiuj)10u

j01 + (ρuiuj)11(u

j00 + uj10) + uj11

((ρuiuj)00

+(ρuiuj)10)

+ (ρuiuj)01uj10

/2;

here, the quantities (ρui)10, (ρuiuj)10, E10 and Qi10 are same as defined in

equations (7.2.12).Eliminating ∂tρ10 between (7.2.18)1 and (7.2.18)2, we get an equation in-

volving ∂tui10; this together with (7.2.18)1 can be used in (7.2.18)3 to yield an


equation involving ∂tp10. When the ui11-eliminant of the resulting equationsinvolving ∂tu

i10 and ∂tp10 is combined with equations (7.2.15) and (7.2.17),

we obtain the following transport equation for the shock strength zs

∂zs∂τ

∣∣∣∣Ω

+ k1 ni,i = I |Ω − k2(p11|Ω)|∇xS|, (7.2.19)

where ∂ /∂τ = ∂ /∂t+ (ui00 + Uni)∂ /∂xi,

I = k3(∂ρ00/∂τ) + k4(∂p00/∂τ) + k7ui00,i +

(k8u

i01ni + k5p01

)|∇xS|,

k8 = −γ(γ + 1)z(2 + z)(2γ + (γ + 1)z)/Θ,

with k1, k2, k3, k4, k6 and k7 being the same as in (7.1.11); here p+0 , ρ+

0 , anda+0 have the same meaning as p00|Ω, ρ00|Ω and a00|Ω, respectively.

We note the following connection between the entities that appear in thepreceding section and the present section

[p] = p10|Ω, ui+0 = ui00|Ω, ρ+0 = ρ00|Ω, p+

0 = p00|Ω, a+0 = a00|Ω,

[nip,i] = ni (∂ip1)|Ω = ni (p10,i + p11S,i) |Ω = |∇xS|p11|Ω, (7.2.20)

(p0,i)+

= (p0,i)|Ω =(p00,i + p01n|∇xS|

)∣∣Ω,

(ui0,j

)+=(ui0,j

)∣∣Ω

=(ui00,j + ui01nj |∇xS|

)∣∣Ω.

In view of (7.2.20), it then follows that (7.2.19) is exactly the same as (7.1.11).The bicharacteristics (or, rays) corresponding to the Hamilton-Jacobi equation(7.2.13) are given by

dxidt

= Uni + ui00,dS,idt

= H,i (7.2.21)

where H ≡ ∂tS + (Uni + ui00)S,i. It follows from (7.2.13), (7.2.14) and thedefinition of ∂/∂τ – derivative in (7.2.19), that the shock ray equations (7.2.21)are precisely the same as (7.1.13) and (7.1.14). In order to derive the transportequation for p11- the jump in the pressure gradient along the normal to shockfront, we need to exploit (7.2.11). For the sake of simplicity, we consider thestate ahead of the shock to be uniform and at rest. Thus we have from (7.2.3),(7.2.9) and (7.2.12)

p0 = p00, ρ0 = ρ00, u0 = u00,p0α = ρ0α = u0α = 0, α = 1, 2, · · · . (7.2.22)

In view of (7.2.7) and (7.2.22), the equation (7.2.11) for α = 1, yields the


following set of equations on Ω

|∇xS|∂tρ11 + ρ12 (∂tS) + (ρui)11,i + (ρui)12S,i

+ ni (∂tρ10),i

+ ρ11ni (∂tS),i + (ρui)11njS,ij = 0, (7.2.23)

|∇xS|∂t(ρu

i)11 + (ρui)12 (∂tS) + (ρuiuj)11,j + (ρuiuj)12S,j

+ p11,i +p12S,i + nj(∂t(ρu

i)10),j

+ (ρui)11nj (∂tS),j

+(ρuiuj)11nkS,kj = 0, (7.2.24)

|∇xS|(∂tE11 +E12 (∂tS) +Qi11,i +Qi12S,i

)+ ni (∂tE10),i

+E11ni (∂tS),i +Qi11njS,ij = 0, (7.2.25)

where

(ρui)12 = ρ12ui10 + ui12(ρ00 + ρ10) + 2ρ11u

i11,

(ρuiuj)11 = (ρ00 + ρ10)(ui12u

j10 + ui10u

j12 + 2ui11u

j11)

+2ρ11(ui11u

j10 + ui10u

j11) + ρ12u

i10u

j10,

E12 = (γ − 1)−1p12 + (ρuiui)12/2,

Qi12 = γ(γ − 1)−1p12u

i10 + ui12(p00 + p10) + 2p11u

i11

+(ρ00 + ρ10)uj10u

j10u

i12 + 2uj10u

j12u

i10 + 4uj10u

j11u

i11

/2

+ρ11uj10

(uj10u

i11 + 2uj11u

i10

)+ ρ12u

i10u

j10u

j10/2,

with (ρui)11, (ρuiuj)11, E11 and Qi11 being the same as defined in (75); in view

of these definitions and (7.2.18), equations (7.2.23) to (7.2.25) can be rewrittenas

|∇xS|∂tρ11 + ρ12 ∂tS + (ρ00 + ρ10)(u

i11,i + ui12S,i) + ρ11(u

i10,i + ui11S,i)

+ui10(ρ11,i + ρ12S,i) + ui11(ρ10,i + ρ11S,i)

+ ni (∂tρ10),i

+ ρ11ni (∂tS),i + ui10ρ11njS,ij + (ρ00 + ρ10)ui11njS,ij = 0, (7.2.26)

|∇xS|∂tu

i11 + ui12 ∂tS + uj10(u

i11,j + ui12S,j) + uj11(u

i10,j + ui11S,j)

+(ρ00 + ρ10)−1(p11,i + p12S,i) − ρ11(ρ00 + ρ10)

−2(p10,i + p11S,i)

+ nj(∂tu

i10

),j

+ ui11nj (∂tS),j + uk10ui11njS,kj (7.2.27)

+ (ρ00 + ρ10)−1p11njS,ij = 0,

|∇xS|∂tp11 + p12 ∂tS + ui10(p11,i + p12S,i) + ui11(p10,i + p11S,i)

+ γp11(ui10,i + ui11S,i) +γ(p00 + p10)(u

i11,i + ui12S,i)

+ ni (∂tp10),i

+ p11ni (∂tS),i + ui10p11njS,ij + γ(p00 + p10)ui11njS,ij = 0. (7.2.28)

According to conditions (7.2.22), equation (7.2.19) assumes the following form

∂z

∂τ

∣∣∣∣Ω

+ k1 ni,i = −k2(p11|Ω)|∇xS|. (7.2.29)

7.3 Kinematics of a Bore over a Sloping Beach 229

In view of (7.2.15), (7.2.22) and (7.2.29), equations (7.2.18)1,2 imply

ui11|Ω = ε1ni(p11|Ω) +(ε2ninj,j + ε3 ∂izs

)/|∇xS|, (7.2.30)

ρ11|Ω = ε∗1 (p11|Ω) + ε∗2nj,j/|∇xS| (7.2.31)

where ε1, ε2, ε3, ε∗1 and ε∗2 are same as in (7.1.36) and (7.1.37).

It may be noticed that the space derivative of f satisfies

f,i|Ω = ∂i (f |Ω) . (7.2.32)

Thus, if we eliminate ui12 between (7.2.27) and (7.2.28) and substitute in theresulting equation the values of ui11 and ρ11 given by (7.2.30) and (7.2.31), weobtain the following transport equation for p11 involving a coupling term p12

∂p11

∂τ

∣∣∣∣Ω

+(φ1(p12|Ω) + φ2

(p211

∣∣Ω

))|∇xS|+χ1p11|Ω+(φ4/|∇xS| = 0, (7.2.33)

where

χ1 = λ

γp0(1 + zs)ε1 + (γ + 1)ε2 +

zsa20

U− 2γρ0U(1 + zs)

(2γ + (γ − 1)zs)

(k1dε1dzs

+k2dε2dzs

+(2γ + (γ − 1)zs)

2ε∗2(2γ + (γ + 1)zs)2ρ2

0

− 2ε1ε2

)ni,i +

(γ + 1)a200

4γUnjzs,j

+φ1

|∇xS|ninjS,ij ,

with φ1, φ2, φ4 and λ being the same as in (7.1.43). One can easily verify that

ε1 =1

γ(1 + zs)

2γ + (γ − 1)zs

2ρ0U+ k2

,

ε2 = − 1

γ(1 + zs)

zs(1 + zs)a

200

U− k1

∂i|∇xS| = nkS,ik,

ni,j = S,ij − ninkS,kj /|∇xS|.

(7.2.34)

Further, it may be noticed that

Π(1) = p11|∇xS|, Π(2) = p12|∇xS| + p11ninjS,ij .

Consequently, equation (7.1.43) can be rewritten as

∂p11

∂τ

∣∣∣∣Ω

+ φ1p12|∇xS| + φ2p211|∇xS| + φ3p11 + φ4/|∇xS| = 0, (7.2.35)

where φ3 = φ3 + (φ1/|∇xS|)ninjS,ij + ((γ + 1)a200/4γU)nizs,i.

In view of (7.2.34), it follows that χ1 = φ3. Consequently, the transportequation (7.2.33) is identical with (7.1.43) or (7.2.35), which we have beenseeking to establish.


7.3 Kinematics of a Bore over a Sloping Beach

In contrast to the gasdynamic case, considered in section 7.1, where thedetermination of the motion of a shock of arbitrary strength is highly com-plicated because of the entropy variations in the flow, the hydraulic analogyhas no counterpart to entropy variations; and hence the flow behind hydraulicjumps or bores can be investigated with relatively greater ease as comparedto the gasdynamic case. In this connection the work of Peregrine [142], whichanalyzes the problem of water wave interactions in the surf zone using thefinite amplitude shallow water equations may be mentioned. The shallow wa-ter equations are a set of hyperbolic equations which approximate the fullfree surface gravity flow problem with viscosity and surface tension effectsneglected [141]; these equations, being quasilinear and hyperbolic, admit dis-continuous solutions which in water are called bores. The motion of a boremoving over a sloping beach through still water has been studied by Keller etal. [90], while an asymptotic analysis of its approach to the shoreline has beenperformed by Ho and Meyer [72]. An application of the weighted average flux(WAF) method for the computation of the global solution to an initial bound-ary value problem for the unsteady two dimensional shallow water equationshas been given by Toro [199]. Using an approach based mainly on the theory ofstructure of the nonsingular Euler-Poisson-Darboux equation, an interestingstudy on the shoreward travel of a bore into a water at rest on a sloping beachhas been carried out by Ho and Meyer [72]. They clarified the role of shore-singularity and used it to provide an understanding of the way the solutionforgets its initial conditions as the bore approaches the shore. In their workthey introduced a monotonicity assumption concerning the seaward boundarywhich enabled them to determine an asymptotic first order approximation forbore development near the shore, and the results for relatively weak boreswith an initial bore strength M = 1.15 were shown to coincide with thoseobtained by Keller et al. [90].

In this section, we study the propagation of bore of arbitrary strength as itapproaches the shoreline over a sloping beach, and investigate the dynamicalcoupling between the bore and the rearward flow by considering an infinitesystem of transport equations for the variations of the bore strength M andthe jump in the space derivatives of the flow variables across the bore. The firsttransport equation from an infinite hierarchical system derived in this sectionbears a close structural resemblance with the evolutionary equation obtainedby Ho and Meyer [72] for the bore strength M , which in the limit of vanishingwave strength ratio equals the result obtained by using the characteristic rule.

As the wave strength ratio in the evolutionary equation of Ho and Meyer[72] remains an unknown, their equations are unable to reveal the completehistory of the evolutionary behavior of a bore of arbitrary strength. Thisis accomplished in the present study by examining the dynamical coupling


between the bore and the rearward flow, which is investigated by consideringan infinite system of transport equations for the variations of the bore strengthM and the jumps in the space derivatives of the flow variables across the bore.

The implications of the first three truncation approximations are discussedfor the case when the bore strength varies from a weak to the strong limit.The results of the present investigation are compared with the approximateresults obtained by using the characteristic rule (see Whitham [210]).

7.3.1 Basic equations

The equations describing one dimensional flow of water in terms of the socalled shallow water approximation can be written in the form (see [84], [80],and [141]).

∂tV + A(∂xV) + B = 0, (7.3.1)

where V = (c, u)tr , and B = (0,−2c0(dc0/dx))tr are the column vectors, while

A is the 2 × 2 matrix having the components A11 = u, A12 = c/2, A21 = 2cand A22 = u. Here, u is the x component of velocity and c =

√gh is the

speed of propagation of a surface disturbance with g the acceleration due togravity, and h the depth of the water; h0(x) is the undisturbed water depthand c0 =

√gh0 is the corresponding surface wave propagation speed. The

eigenvalues of A are λ1,2 = u± c, and the corresponding left eigenvectors arefound to be the row vectors L1,2 = (1,±1/2). Thus, the system of equations(7.3.1) possesses two families of characteristics, dx/dt = u ± c, representingwaves propagating in the ±x direction with the characteristic speeds u ± c.These waves carrying an increase in elevation break leading to the occurrenceof a discontinuity known as “bore” in the flow region.

We consider the motion of a right running bore traveling in the regionx ≤ x0 toward the shoreline at x = x0. It is assumed that the water depth isuniform in x < 0 and that the bore is initially moving with constant speed inthat region; the beach, which starts at x = 0 and has the shoreline at x = x0,has slope h′0(x) for x lying in the interval 0 ≤ x ≤ x0.

The bore conditions, derived from the conservation form of (7.3.1), can bewritten in a convenient form (see [210], [90], and [141])

[u] = 2Mc0(M2 − 1)/

√2M2 − 1, [c] = c0

(√2M2 − 1 − 1

),

[h] = 2h0

(M2 − 1

), U = Mc0

√2M2 − 1,

(7.3.2)

where dX/dt = U is the bore velocity being X(t) as the location of bore,the parameter M = U/c may be regarded as the bore strength, and [f ] =f − f0 denotes the jump in f across the bore, with f and f0 the values off immediately behind and ahead of the bore respectively; here the entity frepresents any of the physical variables u, c and h. As the bore forms on afamily of positive characteristics, we have u0 + c0 < U < u+ c. In the sequel,we shall make use of the well known compatibility condition

d [f ]

dx=

1

U[∂tf ] + [∂xf ] , (7.3.3)


where d/dx denotes the displacement derivative following the bore; indeed, iffx and ft are discontinuous at the bore x = X(t), but continuous everywhereelse, then using the chain rule for the total time derivative of f along the boreand forming the jumps in usual way, one obtains equation (7.3.3). It is evidentthat equation (7.3.3) expresses the condition for the discontinuity [f ] in thefunction f(x, t) to persist during the motion of the bore x = X(t); in particularif f is continuous across the bore, so that [f ] = 0, condition (7.3.3) reducesto [∂tf ] = −U [∂xf ], which shows that there is a discontinuity in the timederivative ft at points of the bore whenever there is a discontinuity in the spacederivative fx, provided the velocity of the bore is different from zero. We shallsee in what follows that when the dynamical conditions of compatibility, whichare the jump relations resulting from the basic equations (7.3.1), are combinedwith the above compatibility condition (7.3.3), important information can beobtained regarding the evolutionary behavior of the bore.

Taking jump in the system (7.3.1) across the bore and using (7.3.3), weget

Ud [V]

dx+ Q [∂xV] + [A] (∂xV)0 + [B] = 0,

where Q = [A] + A0 − UI with I the unit 2×2 matrix. Since the matrix Q isnonsingular, we can define P = Q−1 and obtain

PUd [V]

dx+ P [A] (∂xV)0 + P [B] + [∂xV] = 0. (7.3.4)

Since the quantities [V], [A], [P], U and [B] are expressible in terms of thebore strength M and the flow quantities upstream from the bore, equation(7.3.4), on multiplying by the left eigenvector L1, yields the following transportequation for the bores of arbitrary strength (see Appendix – A)

α1(x,M)dM

dx+ β1(x,M) + Π1 = 0, (7.3.5)

where α1 and β1 are known functions of x and M , and the entity

Π1 = L1 [∂xV] = [∂x(c+ u/2)] ,

which provides a coupling of the bore with the flow behind the bore, maybe regarded as the amplitude of induced disturbances that overtake the borefrom behind. Since the coupling term Π1 is an unknown function, equation(7.3.5) is unable to reveal the complete history of the evolutionary behaviorof a bore of arbitrary strength. We, therefore, need to obtain a transportequation for the coupling term Π1. To achieve this goal, we differentiate thevector equation (7.3.1) with respect to x, take the jumps across the bore,use the compatibility condition (7.3.3) and then multiply the resulting vector


equation by the nonsingular matrix P to obtain

PUd[∂xV]/dx+ [∂xV] · [∇VA] + (∂xV)0 · [∇VA]

+[∂xV] · (∇VA)0([∂xV] + (∂xV)0)((∂xV) · (∇VA))0[∂xV]

+[∂xV] · ([∇VB] + (∇VB0) + (∂xV)0 · [∇VB]

+[A](∂2xV)0 + [∂2

xV] = 0, (7.3.6)

where ∇V denotes gradient operator with respect to the elements of V, and∂2x denotes second order partial derivative with respect to x.

Equation (7.3.6), after pre-multiplying by the left eigenvector L1 and us-ing (7.3.2), (7.3.4) and (7.3.5) yields the following transport equation for thecoupling term Π1

α2(x,M)dΠ1

dx+ β2(x,M,Π1) + Π2 = 0, (7.3.7)

where α2 and β2 are known functions of x, M and Π1, and

Π2 = L1[∂2xV]

=[∂2x(c+ u/2)

],

is another coupling term which is again unknown. We, thus, note that thebehavior of the coupling term Π1 depends on another coupling term Π2 whosedetermination calls for the repetition of the above procedure; subsequently,we obtain the following transport equation for Π2, which in turn involves thecoupling term Π3 = L1

[∂3xV]

=[∂3x(c+ u/2)

]

α3(x,M,Π1)dΠ2

dx+ β3(x,M,Π1,Π2) + Π3 = 0, (7.3.8)

where α3 and β3 are known functions of x, M , Π1 and Π2. Proceeding in thisway, we obtain an infinite set of transport equations for the coupling termsΠn, n = 2, 3, 4, . . . , behind the bore

αn(x,M,Π1,Π2, · · · ,Πn−2)dΠn−1

dx+ βn(x,M,Π1,Π2, · · · ,Πn−1) + Πn = 0,

(7.3.9)where αn and βn are known functions of x, M, Π1, Π2 · · · , Πn−1, and

Πn = L1 [∂nxV] = [∂nx (c+ u/2)] .

This infinite system of transport equations (7.3.5), (7.3.7), (7.3.8) and (7.3.9)together with dX/dt = U , supplemented by the initial conditions

x = x0, M = M0, Πn = Πn0, n ≥ 1 at t = 0 (7.3.10)

can be conveniently used in the investigation of the bore location X(t), thebore strength M and the amplitudes Πn of the induced disturbances at anytime t. It may be noted that the above infinite system of equations is anopen system; in order to provide a natural closure on the infinite hierarchy ofequations we set Πn = 0 in (7.3.9) for the largest n retained. The truncatedsystem of n+1 equations involving unknowns X,M,Π1,Π2 · · · , and Πn−1 canbe regarded as a good approximation of the infinite hierarchy of equations (seeAnile and Russo [3]).


7.3.2 Lowest order approximation

The lowest order truncation approximation of the infinite hierarchical sys-tem of transport equations leads to the bore strength equation obtained from(7.3.5) by setting Π1 = 0; this corresponds to the situation when there is nodynamical coupling between the bore and the rearward flow. The resultingequation, in view of (7.3.2), can be written as (see Appendix – A)

dM

dx+G(M)

c0

dc0dx

= 0, (7.3.11)

where

G(M) =(M2−1)

(2M2−1)(2M4 + 5M3 +M2−1) + (4M2−1)

√2M2−1

M (8M6 + 12M5 − 6M4 − 11M3 + 3M2 + 3M − 1).

Here the range of M is 1 < M < ∞; small values of M − 1 correspond toweak bores and large values of M correspond to strong bores. For weak andstrong bores, equation (7.3.11), together with (7.3.2), yields

M − 1 ∝ c−5/20 , [h] ∝ c

−1/20 , for M − 1 1,

M ∝ c−1/20 , [h] ∝ c0, for M 1.

(7.3.12)

It is quite interesting to note that the above asymptotic results are identicalwith those predicted by the characteristic rule. At this juncture, it is worth-while to consider the evolutionary behavior of bores using the characteristicrule, and compare the results with the approximation procedure employedhere.

The characteristic rule, proposed by Whitham, consists in applying thedifferential relation which is valid along a positive characteristic to the flowquantities just behind the bore. The equations in system (7.3.1) may be com-bined to obtain a differential equation along positive characteristics as

Du

Dx+ 2

Dc

Dx− 2c0u+ c

Dc0Dx

= 0, (7.3.13)

where D/Dx = ∂x+(u+c)−1∂t denotes the derivative along the characteristicDxDt = u + c. When equation (7.3.13) is applied along the bore and boreconditions (7.3.2) are used, we obtain the following evolutionary equation forthe bore (see [210] and [90])

DM

Dx+F (M)

c0

Dc0Dx

= 0, (7.3.14)

where

F (M) =(M − 1)(M2 − 0.5)(M4 + 3M3 +M2 − 1.5M − 1)

2(M + 1)(M − 0.5)2(M3 +M2 −M − 0.5).


It is interesting to note that equation (11), obtained by using lowest or-der approximation, bears a close resemblance with the equation (7.3.14), ob-tained by using the characteristic rule, except for the functional forms ofG(M) and F (M) and the difference in the differential operators D/Dx andd/dx. However, it is remarkable that in the weak or strong bore limit both thefunctions F (M) and G(M) exhibit the same asymptotic behavior, i.e., bothG,F → M/2 or 5(M − 1)/2 according as M → 1 or ∞, respectively. For thesake of comparison of the functions F and G in the entire range 1 ≤M <∞,Fig.7.3.1 depicts their variations versus M−1. It is evidently clear from thecurves that except for a small range of M , i.e, 1 < M < 5, the evolutionarybehavior of the bores described by the lowest order approximation is almostidentical with that predicted by the characteristic rule.

Figure 7.3.1: Functions F (M) and G(M), which appear in the characteristic rule(Eq.(7.3.14)) and the lowest order truncation approximation (Eq.(7.3.11)), versusM−1.

In view of the relation c0 =√gh0, it is evident from the asymptotic results

(7.3.12) that for weak bores, the bore height, [h], increases as h0 decreases,whereas for strong bores, [h] decreases with h0. It follows from (7.3.2), thatthe maximum of [h] occurs at x = ξm when

(M2 − 1) + c0MdM

dc0= 0, (7.3.15)

while, the minimum of the bore speed U occurs at x = ξn when

M√

2M2 − 1 +c0(4M

2 − 1)√2M2 − 1

dM

dc0= 0. (7.3.16)

Using the lowest order approximation (i.e., equation (7.3.11)) for dM/dc0,



we find that [h] attains maximum at M = 1.3148, while the minimum ofbore speed occurs at M = 1.3027. These values are found to be somewhatdifferent from those predicted by the characteristic rule, according to whichthe maximum of [h] occurs at M = 1.146 and the minimum bore speed isattained at M = 1.428 (see [90]); strangely enough, the characteristic rulepredicts the same values ofM no matter what the initial bore strength and theundisturbed water depth are. The method presented here yields in a naturalmanner the critical values of the bore strength to a reasonably good accuracyby taking into account only the first few approximations (see below).

7.3.3 Higher order approximations

Following the truncation criterion discussed above, the evolution of boreto the first order approximation is described by the evolutionary equations(7.3.5) and (7.3.7) by setting Π2 = 0. Then the truncated system of transportequations forms a closed system consisting of a pair of equations involving twounknowns M and Π1; however, if we eliminate Π1 from these equations, weobtain the following second order ordinary differential equation, which whensupplemented by the initial conditions (7.3.10) with n = 1, can be integratednumerically to determine the evolutionary behavior of the bore to the firstorder

d2M

dx2+ T1

(dM

dx

)2

+T2

c0

dc0dx

dM

dx+T3

c0

d2c0dx2

+T4

c20

(dc0dx

)2

= 0, (7.3.17)

where T1, T2, T3 and T4, which are functions ofM only, are given in Appendix –B.

The evolution of a bore to the second order truncation approximation(n = 2) is described by the transport equations (7.3.5), (7.3.7) and (7.3.8)by setting Π3 = 0. This closed system, on eliminating Π1 and Π2, yields thefollowing third order differential equation

d3M

dx3+α1

c0

dc0dx

d2M

dx2+ α2

dM

dx

d2M

dx2+α3

c0

dc0dx

(dM

dx

)2

+α4

(dM

dx

)3

+α5

c0

d2c0dx2

dM

dx+α6

c0

d3c0dx3

+α7

c30

(dc0dx

)3

+α8

c20

dc0dx

d2c0dx2

+α9

c20

(dc0dx

)2dM

dx= 0 , (7.3.18)

where α1, α2, α3, α4, α5, α6, α7, α8 and α9, which are functions of M only, aregiven in Appendix – B. The implications of (7.3.17) and (7.3.18) concerningthe evolutionary behavior of a bore of arbitrary strength are discussed in thefollowing section.


7.3.4 Results and discussion

A transport equation (7.3.5) for the bore strength M , which is just oneof the infinite sequence of transport equations that hold on the bore front, isfound to be identical with equation (53) of Ho and Meyer (see Appendix – C).The coupling term Π1 in equation (7.3.5), is related to the wave strength ra-tio Γb of Ho and Meyer (see Appendix – C); in Ho and Meyer [72], since thequantity Γb remains unknown, the implications of their equation (53) are un-able to reveal the evolutionary behavior of a bore of arbitrary strength. Theinfinite hierarchial system for the bore strength M and the coupling termsΠk, derived in this chapter, enable us to investigate the dynamical couplingbetween the bore of arbitrary strength and the rearward flow. A closure onthe infinite hierarchy of equations is provided by setting Πk equal to zero forthe largest k retained. Here, we present the results of the first three successivetruncation approximations and compare them with the approximate resultsobtained by using the characteristic rule ([210] and [90]). It is quite inter-esting to note that the lowest order truncation approximation of the infinitehierarchical system of transport equations leads to the bore strength equa-tion (7.3.11), which bears a close structural resemblance with the equationobtained by using the characteristic rule (see equation (7.3.14)); indeed boththe equations lead to the same asymptotic behavior in the weak or strong borelimit as discussed in section - 3. It is shown in Fig. 7.3.1, that except for asmall range of M , i.e 1 < M < 5, the evolutionary behavior of the bores de-scribed by the lowest order truncation approximation is almost identical withthat predicted by the characteristic rule. Equations (7.3.15) and (7.3.16), inview of the lowest order approximation — (7.3.11) (respectively, the charac-teristic rule — (14)), determine that the bore height attains a maximum atM = M∗ = 1.3148 (respectively, M = M∗ = 1.146), while the bore speedattains a minimum at M = M∗ = 1.3026 (respectively, M = M∗ = 1.428). Itmay be noticed that the values of M∗ and M∗ determined by using the low-est order approximation are not influenced by M0 or the undisturbed waterdepth as it happens with the characteristic rule. To facilitate the comparisonbetween the results obtained by using the truncation approximation and thecharacteristic rule, we follow the same notation as in [90] and render the equa-tions (7.3.11), (7.3.14), (7.3.17) and (7.3.18) dimensionless by introducing thedimensionless quantities

x = x/x0, t = tc0(0)/x0, u(x, t) = u(x, t)/c0(0),

H(x, t) = h(x, t)/h0(0), V = U/c0(0), N = [h]/h0(0),

where x0, h0(0) and c0(0) are known constants.

We integrate numerically the dimensionless forms of the equations (7.3.11),(7.3.14), (7.3.17), and (7.3.18) for different values of the initial bore strengthM0, and for two different dimensionless forms of the undisturbed water depth


H0(x), given by

H0(x) =

1, if x ≤ 0,1 − x, if 0 < x ≤ 1,

(7.3.19)

and

H0(x) =

1, if x ≤ 0,1 − x

2, if 0 < x ≤ 1.

(7.3.20)

Figure 7.3.2(a): Evolutionary behavior of a bore of moderate strength described

by the first and second truncation approximations and the characteristic rule. The

variations of bore height N and bore velocity V versus X are displayed for initial

bore strength M0 = (1.125)1/2

, which is less than the critical values M∗ and M∗

.

The first order approximation described by equation (7.3.17) shows thatfor a given M0 and H0(x), the bore height (respectively, the bore speed)attains a maximum at M = Mm (respectively, a minimum at M = Mn) thatsatisfies equations (7.3.15) and (7.3.17) (respectively, equations (7.3.16) and(7.3.17)) identically. Similar is the behavior described by the second orderapproximation (7.3.18); the computed results are shown in Tables 1 – 4. Itmay be noticed that in contrast to the characteristic rule, the higher orderapproximations exhibit that the bore strengthsMm andMn for which the boreheight and the bore speed attain respectively a maximum and a minimum dodepend on the initial bore strengthM0 and the undisturbed water depthH0(x)as one would anticipate (see Tables 1 – 4); indeed, a feature relating to thedependence of Mn on M0 is in agreement with the numerical results of [90],which show that the values of U/

√gh0(x) at the points where V = U/

√gh0(0)

is minimum, are 2.462, 2.539 and 2.674 for the initial bore height N(0) = 0.25,


0.225 and 0.125 respectively. In order to provide a comparison between ourresults and the numerical results of Keller et al. [90], we have carried out thecomputations of η/h0, U/√gh

0 and Mn up to second order approximationusing the same values of the initial bore height N(0) and undisturbed waterdepth H0(x), which were used in [90]; a comparison between our results andthe numerical results of Keller et al. [90] is shown in Table 5. It is interestingto note that for H0(x) given by (7.3.19), which corresponds to a beach ofuniform slope, the lowest order approximation yields values of M ∗ and M∗correct to six decimal digits as the first and second order approximationsdo not contribute any significant corrections to them (see Tables 1 and 2).However, if H0(x) is specified by equation (7.3.20), which corresponds to abeach of varying slope, the values of M ∗ and M∗ determined by the lowestorder approximation get modified by the first and second order approximations(see Tables 3 and 4); indeed, the values correct to four decimal digits are M ∗

= 1.1207 and M∗ = 1.4987, which differ quantitatively from those obtainedby using the characteristic rule. Further, if the initial bore strength M0 is suchthat M0 < M∗ (respectively,M0 < M∗) then the bore height (respectively, thebore speed) exhibits a maximum (respectively, minimum) lying in the interval0 ≤ x ≤ 1 ; however for M0 > M∗(respectively, M0 > M∗), the bore height(respectively, the bore speed) exhibits a monotonic decreasing (respectively,increasing) behavior.

Figure 7.3.2(b): Evolutionary behavior of a bore of moderate strength described

by the first and second truncation approximations and the characteristic rule. The

variations of bore height N and bore velocity V versus X are displayed for initial

bore strength M0 = (6.0)1/2

, which is greater than the critical values M∗ and M∗

.

Results for the dimensionless bore height N and bore speed V , which de-pend on M and x, through equations (7.3.2), are exhibited in Figs. 7.3.2 and


7.3.3 for typical values of M0; indeed, for M0 < M∗ (respectively, M0 < M∗),the bore height N (respectively, bore speed V ) exhibits a maximum (respec-tively, minimum) at M = Mm (respectively, M = Mn) at x = ξm (respec-tively, x = ξn) lying in the interval 0 ≤ x ≤ 1 (see Figures 7.3.2(a) and7.3.3(a)); however, for M0 ≥ M∗ (respectively, M0 ≥ M∗) the bore height(respectively, bore speed) exhibits a monotonic decreasing (respectively, in-creasing) behavior over the interval 0 ≤ x ≤ 1 (see Figures 7.3.2(b) and7.3.3(b)).

Figure 7.3.3(a): Evolutionary behavior of a bore described by the successive ap-proximations and the characteristic rule for the initial bore strength M0 = 1.005.

It is observed from Tables 1 – 4 that as M0 increases from 1 to M∗ (respec-tively, M∗), the location ξm (respectively, ξn) of the maximum (respectively,minimum) bore height (respectively, bore speed) is shifted towards x = 0.It may be remarked that only the first three approximations are sufficientto obtain a reasonably good accuracy in the computed values of ξn, ξm,Mm

and Mn. Further, it may be remarked that the lowest order approximationdescribes the evolutionary behavior of a bore to a good accuracy except whenthe initial bore is of moderate strength, i.e., 1 < M0 < 2; for such a bore,only the first few approximations are sufficient to describe the evolutionarybehavior to a reasonably good accuracy.


Figure 7.3.3(b): Variation of the bore height N and bore velocity V described bythe successive truncation approximations and the characteristic rule for the initialbore strength M0 = 4.0.

Table 1 : For a given initial bore strength M0, and the undisturbedwater depth H0(x) given by (7.3.19), the maximum bore height

is attained at ξ = ξm and M = Mm.

M0 ξm Mm

lowest first second lowest first secondorder order order order order order

solution solution solution solution solution solution1.2500 0.2242 0.1549 0.1572 1.3148 1.2914 1.29221.3000 0.0530 0.0352 0.0353 1.3148 1.3097 1.30971.3027 0.0436 0.0289 0.0290 1.3148 1.3106 1.31061.3148 0.0000 0.0000 0.0000 1.3148 1.3148 1.31481.3200 0.0000 0.0000 0.0000 1.3200 1.3200 1.3200

Indeed, the improvement takes place only on a subinterval (x1, x2), whoselength goes on diminishing as M0 moves closure to 1 or to 2 (see Figures7.3.2(a) and 7.3.3(a)). In particular, for moderate strength bores with initialvaluesM0 =

√1.125 andM0 =

√6, which correspond to the initial bore height

N(0) equal to 0.25 and 10, respectively, the flow variables are depicted in Figs.7.3.2(a) and 7.3.2(b), and the values of N and V in the interval (x1, x2) areimproved by the higher order truncation approximations; indeed only the firstthree approximations are sufficient to obtain the values of N and V correctto at least three decimal digits (see Figures 7.3.2(b) and 7.3.3(b)).


Table 2 : For a given initial bore strength M0, and the undisturbedwater depth H0(x) given by (7.3.19), the minimum bore speed

is attained at ξ = ξn and M = Mn.

M0 ξn Mn


solution solution solution solution solution solution1.2500 0.1889 0.2462 0.2406 1.3027 1.3220 1.32001.3000 0.0099 0.0131 0.0131 1.3027 1.3035 1.30351.3027 0.0000 0.0001 0.0001 1.3027 1.3027 1.30271.3148 0.0000 0.0000 0.0000 1.3148 1.3148 1.31481.3200 0.0000 0.0000 0.0000 1.3200 1.3200 1.3200

Table 3 : For a given initial bore strength M0, and the undisturbedwater depth H0(x) given by (7.3.20), the maximum bore height

is attained at ξ = ξm and M = Mm.

M0 ξm Mm


solution solution solution solution solution solution1.1200 0.791 0.077 0.88 1.3148 1.1207 1.12091.1206 0.789 0.027 0.031 1.3148 1.1207 1.12071.1207 0.798 0.006 0.007 1.3148 1.1207 1.12071.1207 0.789 0.002 0.002 1.3148 1.1207 1.12071.1207 0.789 0.000 0.000 1.3148 1.1207 1.12071.2000 0.620 0.000 0.000 1.3148 1.2000 1.2000

Table 4 : For a given initial bore strength M0, and the undisturbedwater depth H0(x) given by (7.3.20), the minimum bore speed

is attained at ξ = ξn and M = Mn.

M0 ξn Mn


solution solution solution solution solution solution1.4500 0.0000 0.3763 0.3390 1.4500 1.4982 1.48841.4900 0.0000 0.1610 0.1440 1.4900 1.4986 1.49681.4950 0.0000 0.1060 0.0940 1.4950 1.4987 1.49791.4987 0.0000 0.0010 0.0010 1.4987 1.4987 1.49871.4987 0.0000 0.0000 0.0000 1.4987 1.4987 1.49871.5000 0.0000 0.0000 0.0000 1.5000 1.5000 1.5000


Table 5 : Comparison of our results, obtained using second orderapproximation, with the numerical results of Keller [90]; here

η/h0, U/√

gh0 and Mn are the values at points

where V is minimum.

M0 Method η/h0 U/√gh0 Mn√

1.1125, i.e., Present method 1.807 2.312 1.379N(0) = 0.225 Numerical method 2.13 2.539 1.436√

1.125, i.e., Present method 1.8024 2.308 1.378N(0) = 0.25 Numerical method 2.017 2.462 1.417√1.0625, i.e., Present method 1.8595 2.3490 1.389

N(0) = 0.125 Numerical method 2.316 2.674 1.469

However, outside the interval (x1, x2), the values of N and V obtainedby using the higher order approximations are indistinguishable from thoseobtained by using the lowest order truncation approximation (see Figure7.3.3(a)).

7.3.5 Appendices

Appendix – A

This Appendix contains the derivation of the transport equations (7.3.5)and (7.3.11). In view of bore conditions (7.3.2), the jump quantities[V], [A], [P] and [B] can be expressed in terms of the bore strength M and

the upstream flow quantities V0(x) = (c0(x), u0(x))T; indeed,

d[V]

dx=∂[V]

∂M

dM

dx+ (∇V0 [V])

dV0

dx.

As a result, equation (7.3.4) becomes

PU

∂[V]

∂M

dM

dx+ (∇V0 [V])

dV0

dx

+ P[A] (∂xV)0 + P[B] + [∂xV] = 0.

Pre-multiplication of this last result by the left eigenvector L1

gives the fol-lowing transport equation for the bore strength

α1(x,M)dM

dx+ β1(x,M) + Π1 = 0,

where

α1 = UL1P∂[V]

∂M,

β1 = L1UP (∇V0 [V]) + P[A]

dV0

dx+ L

1P[B],

Π1 = L1[∂xV] .


Equations (7.3.1) and (7.3.2) allow the calculation of α1 and α2 in terms ofM and c0(x) as follows. Since

[A] =c0√

2M2 − 1

2M(M2 − 1)(2M2 − 1) −

√2M2 − 1

)/2

2(2M2−1−

√2M2−1

)2M(M2−1)

,

[B] = 0, P =

√2M2 − 1

(4M2−1)(M2−1)c0

M(2M2 − 1

)/2

2(2M2 − 1

)M

,

∂[V]

∂M=

∂[c]

∂M

∂[u]

∂M

=

2c0√2M2 − 1

M

4M4 − 3M2 + 1

2M2 − 1

,

∇V0 [V] =

∂[c]

∂c0

∂[c]

∂u0

∂[u]

∂c0

∂[u]

∂u0

=

√2M2 − 1 − 1 0

2M(M2 − 1)√2M2 − 1

0

,

dV0

dx=

dc0dx

du0

dx

, L1 =

(1,

1

2

), and U = Mc0

√2M2 − 1,

it follows that

α1 = UL1P∂[V]

∂M=

(2M − 1)3(M + 1)2Mc0

(4M2 − 1)(M − 1)√

2M2 − 1,

β1 = L1 PU (∇V0 [V]) + P[A] dV0

dx+ L1P[B],

=

1 +

(M3 + 3M2 + 2M + 1

)√2M2 − 1

2M + 1

dc0dx

.

The lowest order truncation approximation of the infinite system is obtainedby setting Π1 = 0 in equation (7.3.5), leading to the result

α1(x,M)dM

dx+ β1(x,M) = 0 , or

dM

dx+G(M)

c0

dc0dx

= 0 ,

where

G(M) = β1(x,M)c0

(α1(x,M)

dc0dx

)−1

=(M2−1)

(2M2−1)(2M4+5M3 +M2−1)+(4M2−1)

√2M2−1

M (8M6 + 12M5 − 6M4 − 11M3 + 3M2 + 3M − 1).


Appendix – B

This Appendix lists the coefficients in the transport equations (7.3.17) and(7.3.18).

T1 =2T11 + T21

2T10 + T20, T2 =

2T12 + T22

2T10 + T20, T3 =

2T13 + T23

2T10 + T20, T4 =

2T14 + T24

2T10 + T20,

ξ =M(4M4 −M2 + 1)

√2M2 − 1

(4M2 − 1)(M2 − 1), ξ∗ =

2M2(12M4 − 11M2 + 3)√2M2 − 1(4M2 − 1)(M2 − 1)

,

χ =(4M4 − 5M2 + 1)

M√

2M2 − 1, T10 =

−Mξ√

2M2 − 1

χ, T20 =

−Mξ∗√

2M2 − 1

χ,

T11 =80M11 + 8M9 − 3M7 − 21M5 + 7M3 +M

χ (4M4 − 5M2 + 1)2 ,

T12 =M2

(2M2 − 1

) (56M8 − 34M6 − 42M4 + 34M2 − 14

)

χ (4M4 − 5M2 + 1)2 ,

T13 = − 1

χ

M(2M2 − 1

)2 (M4 − 1

)

4M4 − 5M2 + 1+

M√2M2 − 1

,

T14 =15M3

(2M2 − 1

)2 (M2 + 1

)

χ (4M2 − 1)2 ,

T21 =2M2

(64M12 + 96M10 − 76M8 − 64M6 + 83M4 − 38M2 + 7

)

χ (2M2 − 1) (4M4 − 5M2 + 1)2 ,

T22 =4M

(16M12 + 36M10 − 72M8 + 8M6 + 17M4 − 4M2 − 1

)

χ (4M4 − 5M2 + 1)2 ,

T23 =1

χ

2√

2M2 − 1 − 2(10M6 − 5M4 + 4M2 − 1

)

4M2 − 1

,

T24 =2(8M10 + 104M8 − 60M6 − 17M4 + 12M2 − 2

)

χ (4M2 − 1)2 ,

α1 =2T11 + T21

2T10 + T20

, α2 =2T12 + T22

2T10 + T20

, α3 =2T13 + T23

2T10 + T20

, α4 =2T14 + T24

2T10 + T20

,


α5 =2T15 + T25

2T10 + T20

, α6 =2T16 + T26

2T10 + T20

, α7 =2T17 + T27

2T10 + T20

, α8 =2T18 + T28

2T10 + T20

,

α9 =2T19 + T29

2T10 + T20

, T10 = −MT 10

√2M2 − 1, T20 = −MT 20

√2M2 − 1,

T11 = −M√

2M2 − 1(T 10 + T 12

)+ 2(ψ − 1)T 20 + 2.5ψ∗T 10,

T12 = −M√

2M2 − 1

(dT 10

dM+ 2T 11

)+ 2ξT 20 + 2.5ξ∗T 10,

T13 = −M√

2M2 − 1

(T 11 +

dT 12

dM

)+ 2(ψ − 1)T 21 + 2.5ψ∗T 11

+2ξT 22 + 2.5ξ∗T 12,

T14 = −M√

2M2 − 1dT 11

dM+ 2ξT 21 + 2.5ξ∗T 11,

T15 = −M√

2M2 − 1

(T 12 +

dT 13

dM

)+ 2ξT 23 + 2.5ξ∗

(T 13 − 1

),

T16 = −MT 13

√2M2 − 1 +

2M(M2 − 1)√2M2 − 1

,

T17 = MT 14

√2M2 − 1 + 2(ψ − 1)T 24 + 2.5ψ∗T 14,

T18 = −2MT14

√2M2 − 1 + 2(ψ − 1)T 23 + 2.5ψ∗

(T 13 − 1

),

T19 = −M√

2M2 − 1dT 14

dM+ 2ξT 24 + 2(ψ − 1)T 22 + 2.5ξ∗T 14 + 2.5ψ∗T 12,

T21 = −M√

2M2 − 1(T 20 + T 22

)+ 6(ψ − 1)T 10 + 3ψ∗T 20,

T22 = −M√

2M2 − 1

(dT 20

dM+ 2T 21

)+ 6ξT 10 + 3ξ∗T 20,

T23 = −M√

2M2 − 1

(T 21 +

dT 22

dM

)+ 6(ψ − 1)T 11 + 3ψ∗T 21

+6ξT 12 + 3ξ∗T 22,

T24 = −M√

2M2 − 1dT 21

dM+ 6ξT 11 + 3ξ∗T 21,


T25 = −M√

2M2 − 1

(T 22 +

dT 23

dM

)+ 6ξ

(T 13 − 1

)+ 3ξ∗T 23,

T26 = −MT 23

√2M2 − 1 + 2

√2M2 − 1 − 2,

T27 = MT 24

√2M2 − 1 + 6(ψ − 1)T 14 + 3ψ∗T 24,

T28 = −2MT24

√2M2 − 1 + 6(ψ − 1)T 13 + 3ψ∗T 23 − 6ψ,

T29 = −M√

2M2 − 1dT 24

dM+ 6ξT 14 + 6(ψ − 1)T 12 + 3ξ∗T 24 + 3ψ∗T 22,

ψ =(2M2 − 1)3/2(M2 + 1)

4M2 − 1+ 1, ψ∗ =

10M3√

2M2 − 1

(4M2 − 1),

T 10 = T10 +(2M2 − 1)T20

2MT 20 =

2(2M2 − 1)T10

M+ T20,

T 11 = T11 +(2M2 − 1)T21

2MT 21 =

2(2M2 − 1)T11

M+ T21,

T 12 = T12 +(2M2 − 1)T22

2MT 22 =

2(2M2 − 1)T12

M+ T22,

T 13 = T13 +(2M2 − 1)T23

2MT 23 =

2(2M2 − 1)T13

M+ T23,

T 14 = T14 +(2M2 − 1)T24

2MT 24 =

2(2M2 − 1)T14

M+ T24,

Appendix – C

This Appendix shows that the transport equations (5) and (53) in Ho andMeyer [72] are identical, and also that the coupling term in this article isrelated to the wave strength ratio Γ in the work of Ho and Meyer (see theirequation (54)).

Ho and Meyer derived an ordinary differential equation for bore strengthM by introducing a transformation from (x, t) plane to (α, β)− characteristicplane, where α = constant on advancing characteristic lines determined bydx/dt = u+ c, and β = constant on receding characteristic lines determinedby dx/dt = u − c. In view of the relationships u = (α − β)/2 − γt andc = (α + β)/4 used by Ho and Meyer [72], the coupling term in this article

Π1 = [cx + ux/2] can be written as Π1 = −[

1

4ctα

], since αx = −1/2ctα. In

addition, in the medium ahead of the bore α− β = 2γt, it follows that

Π1 =γ

2c0− 1

4cb(tα)b=

γ

2c0− γ

2(ub + cb)− γΓb

4cb,


where Γb =1

γ

1

(tα)b− 2γcbub + cb

and the suffixes b and 0 refer, respectively,

to the quantities just behind and ahead of the bore.Combining the expression for Π1 with h0 = −γx/g, cb = c0

√2M2 − 1

and ub = 2M(M2−1)c0/√

2M2 − 1, it can be easily seen that equation (7.3.5)coincides with equation (53) in Ho and Meyer [72].

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sharma v. quasi linear hyperbolic systems, compressible flows, and waves (crc, 2010)(isbn...

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