UK Nonlinear News, March 2003

Selected Topics in Nonlinear Wave Mechanics

Edited by C.I.Christov and A.Guran

Reviewed by John Billingham


This book claims to 'give an overview of the current state of nonlinear wave mechanics, with emphasis on strong discontinuities (shock waves) and localized self-preserving shapes (solitons), in both elastic and fluid media', and also that 'the exposition is intentionally at a detailed mathematical and physical level ...'. Key features are claimed to include 'survey chapters written in an accessible style' and '(an) interdisciplinary approach integrating mathematical theory and physical applications ...'. The book is meant to be useful for 'applied mathematicians, physicists, mechanical, civil and aerospace engineers, as well as graduate students ...'.

I am not an expert in the fields that this book discusses, but I know the basics, and I am a practising applied mathematician, so the book should be suitable as an introduction to the field for me. It therefore seems reasonable that I judge its success by its interest and readability.

The book consists of seven self-contained chapters, each of which I shall discuss.

1) Elastic Surface Waves, Resonances and Inverse Acoustic Scattering, by A.Guran, A.Nagl and H.Uberall.

This chapter is concerned with the response of a fluid-loaded elastic plate or a sediment layer to incoming acoustic waves. The presentatonal difficulties of this chapter are pretty much encapsulated by the first paragraph, which launches into a discussion of the lowest and highest branches of the dispersion curve for a free plate, and heads off from there. This is not a chapter 'written in an accessible style'. The authors assume far too much background knowledge, and convey no sense of excitement in their topic. I found it all confusing and impenetrable. There are also problems with the figure labelling (figures transposed, indistinguishable labels) that are not really forgivable.

2) Amplitude Equation Models for the Interaction of Shocks with Nonlinear Dispersive Wave Envelopes, by P.K.Newton and R.M.Axel.

The interaction between shock waves and dispersive wave envelopes, in particular solitary waves, is the subject of this chapter, focussing on a model system based on the Zakharov equations from plasma physics. This is essentially Schrodinger's equation with a potential that satisfies a nonlinear conservation equation that can lead to shock formation. The authors go into a fair amount of detail for various analytical and asymptotic solutions, and show that several different types of interesting behaviour can occur, such as shock capture and speed change can occur. I would like to have seen some discussion of what this analytical work could represent in a real physical system, but generally, this is an interesting, well-written chapter.

3) Some Aspects of One-Dimensional Finite Amplitude Elastic Wave Propagation, by J.B.Haddow and R.J.Tait.

This chapter is concerned with initial/boundary value problems for elastic strings, membranes and shells. It begins with a survey of the theory of systems of hyperbolic partial differential equations in two independent variables. It would have been nice to see this before chapter 2, which makes extensive use of the theory, but gives less detail. The authors then move on to the specific case of finite elasticity with deformations functions of time and a single displacement variable. The discussion builds in complexity as it progresses, discussing both the appropriate constitutive laws and the mathematical structure of the resulting equations. A lot of detail is given without a huge amount of explanation, but the reader does get a feel for the mathematical structure of the equations, and extensive references are given to lengthier papers. I note that all of the figure captions are missing. This is a good introduction to the area.

4) Nonlinear Duality Between Elastic Waves and Quasi-particles, by G.A.Maugin and C.I.Christov.

In this chapter, the authors make a link between solitary waves in integrable or nearly integrable, systems and the properties of particles, namely mass, momentum and energy. The English is not good in this chapter, and the constant use of italics for almost every noun becomes irritating very quickly. I wasn't entirely clear what there was to get excited about here. I had it in mind that some sort of particle-particle interaction analogy for the interaction of two solitary wave was going to emerge, but only isolated solitary waves are discussed. The authors state that their ideas can be used to help improve the accuracy of numerical schemes by making sure that appropriate quantities are conserved, which is, of course, useful, and also that the presence of small forcing terms on the right hand sides of the conservation laws enables progress to be made in perturbation solutions of nearly integrable systems, although this is not illustrated. There are, however, many examples of interesting nonlinear partial differential equations, although I felt that too much prior knowledge of field theory was required of the reader (at least this reader!). Overall, the chapter is intriguingly titled, but ultimately rather disappointing.

5) Time-Harmonic Waves in Pre-Stressed Dissipative Materials, by A.Morro.

This author considers the propagation of small-amplitude time-harmonic waves in viscoelastic solids and fluids, with the emphasis on solids. The thermodynamics of the material is taken into account in an extremely detailed derivation of the governing equations. The possible forms of the wavenumber vector are then considered. This can be reduced to the solution of two cubic equations after suitable manipulations. The chapter concludes with a discussion of ray theory. The algebraic complexity of much of the analysis makes it rather hard to grasp the meaning of many of the equations, particularly as I am not expert in this field. The author demands rather a lot from his reader I think, but I felt that someone with a strong background in solid mechanics would probably get a lot out of this chapter.

6) Dissipative Effects on the Evolution of Internal Solitary Waves in a Sheared, Stably Stratified Fluid Layer, by W.B.Zimmerman and M.G.Velarde.

In this short chapter, the authors analyse the effect of viscous and thermal dissipation on solitary waves in sheared, stratified fluid layers. These dissipative effects are included in the analysis using perturbation theory, with inverse Reynolds number and wave amplitude as small parameters. The conclusion is that finite wavenumber effects are not very important, and that, unless the waves can draw energy from the external flow, the waves are dissipated. This is a well-written chapter that puts the problem in context and presents the subsequent analysis clearly.

7) Dissipative Nonlinear Strain Waves in Solids, by A.V.Porubov.

The final chapter deals with the propagation of solitary waves in free cylindrical elastic rods, cylindrical rods surrounded by a dissipative medium, and cylindrical rods with microstructure. The introduction and derivation of the equations are fairly clear. The author then describes many different nonlinear evolution equations for various strain waves. Although there are some linguistic peculiarities ('the longitudinal deformation process is similar to the beards movement on the thread'!) this is quite an interesting chapter. I did feel that there was rather too much algebraic detail.

Overall, the book is rather uneven in terms of readability, and could have done with more agressive editing to ensure that the standard of presentation was maintained throughout each chapter. I also felt that the title is misleading, as 5 of the 7 chapters are basically about solid mechanics, a bias that is not made clear in the introduction or description on the back of the book. However, the extensive lists of references make it a good place to start for anyone interested in current research in elastic wave propagation.

UK Nonlinear News would like to thank Birkhauser for providing a review copy of this book.

A listing of books reviewed in UK Nonlinear News is available.

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Last Updated: 1 March 2003.