Dubious derivations resonate throughout a wavering work

Julian L. Davis has set out to describe wave motion in various media and to show how it can be analysed using a variety of mathematical tools. The aim is to produce a book that will interest researchers from different backgrounds with different levels of expertise. Thus the engineer might be attracted by investigating waves in elastic solids, the mathematician might be interested in water waves or in solutions by characteristics, and the physicist might wish to investigate gas dynamics or Hamiltonian systems.

When a student turns to a textbook of this kind, or when it is used by a researcher who is not a specialist in this field, he or she would like to find a clear, readable insightful description with careful derivations, helpful diagrams and exercises worth doing. In providing these essentials, an author has some freedom to cover however much of the subject they want, and to develop it from any standpoint. Davis is perfectly at liberty to tackle the one-dimensional wave equation from the standpoint of characteristics before looking at the same equation via separation of variables, which might well be thought to be a more fundamental approach. But care and clarity are lacking from this book; the errors are many, starting with the preface and running throughout.

Admittedly, many are typographical, but they are so numerous that they really do get in the way of the description, and often they occur where an undergraduate or other young researcher could easily become confused. These errors would not have been present if the author had read through the proofs; how can he expect others to read this book carefully when he seems not to have done so himself?

There are also other, more serious failings: the nonsense on page 12 that is supposed to be describing oscillations as either longitudinal or transverse; the missing figure 2.3 on page 62 (not to be confused with the unrelated figure 2.3 on page 78); the failure to perform separation of variables properly for a rectangular membrane (followed by an inability to turn it into a square membrane successfully), and several shortcomings in tackling the circular membrane; a derivation of the Navier-Stokes equations that, on several occasions, equates vectors to scalars, and where the pressure can change sign or disappear altogether. The book also suffers from important omissions from the index, such as "gas dynamics" and "supersonic".

There are a few things I like about the book. The illustration of shock fronts advancing or receding in terms of lines of cars crashing on a highway is a memorable image. But I cannot recommend this book. Lighthill's Waves in Fluids is a direct, highly preferable parallel to this work; it covers similar material, is full of insight but contains just one error, an amusing misspelling.

Mike Simon is lecturer in mathematics, University of Manchester.