UK Nonlinear News, August 1999

Introduction to Wave Propagation in Nonlinear Fluids and Solids

By D.S. Drumheller

Reviewed by Dr Penny Davies (Department of Mathematics, University of Strathclyde)

Published by Cambridge University Press, 1998
Paperback: ISBN 0 521 58746 8 (cost: £29.95)
Hardback: ISBN 0 521 58313 6 (cost: £80.00)

This book is described as a comprehensive introductory text on stress waves in nonlinear materials (both fluids and solids), and is aimed at advanced undergraduate or graduate students (and also as a reference for engineers and applied physicists). It contains five chapters and two appendices. Chapter 0 gives an overview of the rest of the book, and also describes two classical one dimensional (1D) experiments - the shock tube and flyer plate - that introduce the concepts of shock, rarefaction and structured waves. These experiments are revisited at points throughout the book.

Chapter 1 is devoted to kinematics and balance laws. Both the material (Lagrangian) and spatial (Eulerian) descriptions are used to derive the conservation laws of mass and momentum, firstly in 1D and then in 3D. In Chapter 2 a purely mechanical theory of waves in elastic materials is developed. The chapter opens with definitions and properties of elastic materials, and then goes on to discuss 1D nonlinear elastic wave equations (briefly) and 1D linear elastic waves (in detail). The theory of Riemann invariants is introduced to solve the linear wave equations (method of characteristics), and this is then generalised to Riemann integral solutions of the nonlinear equations. The chapter continues with sections on 1D structured (continuous) and shock (discontinuous) acoustic waves in nonlinear materials and wave-wave interactions. It concludes with a discussion of the book's key concept, namely that although smooth compressive acoustic waves in nonlinear elastic materials (mathematically) develop shocks, in fact shock waves do not exist in real materials. This apparent paradox is overcome by using thermoelastic rather than purely elastic constitutive models, and the next chapter is devoted to developing the theory of thermodynamics.

At the start of Chapter 3 Drumheller explains that when accurate experimental measurements are taken, waves in real elastic materials that appear to be shocks are found in fact to be continuous. This is because the waves cause changes in temperature as they pass through the material (as well as altering the stress) and some of the wave energy is absorbed by the material. This chapter mainly considers 1D deformations (although the 3D energy balance equation is also derived), and contains derivations of 1D constitutive laws for viscous, thermoelastic and thermoviscous materials using the second law of thermodynamics or Clausius-Duhem inequality. Types of solutions of the resulting equations that preserve various quantities (entropy, temperature, pressure, etc.) are discussed in detail.

The final chapter considers some more specific constitutive models (ideal gases and Mie-Grueneisen solids), and also considers elasto-plastic materials, porous solids, detonation and phase transformations. The appendices are devoted to numerical methods for solving linear and nonlinear wave equations (A), and tables of material constants (B).

As a non-thermodynamicist who has taught an honours course on continuum mechanics, I found aspects of this book interesting. I had not appreciated that shock waves are a mathematical approximation to what happens in practice, caused by using an incomplete set of constitutive equations. (Or equivalently that elastic waves cause oscillations in temperature as well as stress.) I also enjoyed some of the author's historical observations, and applaud his decision to include a modern derivation of various constitutive laws in detail using the second law of thermodynamics. I would however have some reservations (detailed below) both in using this book as a basis for teaching a course or in recommending it as a professional reference.


  1. M.E. Gurtin, An Introduction to Continuum Mechanics (Academic Press, 1981).
  2. P.D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (SIAM, 1973).
  3. R.J. Leveque, Numerical methods for conservation laws (2nd ed., Birkhauser, 1992).
  4. A.R. Mitchell & D.F. Griffiths, The Finite Difference Method in Partial Differential Equations (Wiley, 1980).
  5. K.W. Morton & D.F. Mayers, Numerical solution of partial differential equations (CUP, 1994).

UK Nonlinear News thanks Cambridge University Press for providing a copy of this book for review.

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15/06/99 (PJD)
Last Updated: 16th June 1999.