Chapter 1 provides the background, both the model equations and some of the mathematical transformations, needed to understand linear elastic waves. Only the basic equations are summarized, without derivation. Both Fourier and Laplace transforms and their inverses are introduced and important sign conventions settled. The Poisson summation formula is also introduced and used to distinguish between a propagating wave and vibration of a bounded body. A general survey of books and collections of papers that bear on the contents of the book are discussed at the end of the chapter.

A linear wave carries information at a particular velocity, the group velocity, which is characteristic of the propagation structure or environment. It is this transmission of information that gives linear waves their special importance. In order to introduce this aspect of wave propagation, we discuss propagation in one-dimensional periodic structures. Such structures are dispersive and therefore transmit information at a speed different from the wavespeed of their individual components.

Model equations

The equations of linear elasticity consist of:

1. (1) the conservation of linear and angular momentum; and

2. (2) a constitutive relation relating force and deformation.

In the linear approximation the density ρ is constant. The conservation of mechanical energy follows from (1) and (2). The most important feature of the model is that the force exerted across a surface, oriented by the unit normal nj, by one part of a material on the other is given by the traction ti = τjinj, where τji is the stress tensor.

This volume is dedicated to the memory of John G. Harris, whose life ended prematurely on the 6th of May, 2006. John's friendship and research impacted many people – he was a dedicated and loving husband, an accomplished scientist and applied mathematician, a passionate teacher, and an important mentor to many young scientists. This book was originally intended to be John's second book on elastic wave theory and diffraction. It grew from four lectures that were given at the Department of Mathematics and Mechanics, within IIMAS, at the National Autonomous University of Mexico, in January 2004. After John's passing, several of his colleagues, inspired by his wife Beatriz, began to convert these unfinished notes into a form suitable for publication. We have worked to combine the existing chapters with additional, contributed chapters from experts in the field of elastic wave theory.

Born and raised in Toronto, John entered McGill University as a mature student and graduated with a Bachelors in Electrical Engineering (Honours). After receiving a Masters of Science in Applied Physics from Stanford University, John traveled to Northwestern University to work toward a doctorate in Applied Mathematics with Jan Achenbach, which he completed in 1979. J. D. Achenbach had a lasting impact on John's work in elastic wave scattering, which formed the basis of much of John's research as a professor at the Department of Theoretical and Applied Mechanics at the University of Illinois at Urbana–Champaign between 1979 and 2005.

This chapter is concerned with cracks. Real cracks in solids are complicated: they are thin cavities, their two faces may touch, and the faces may be rough. We consider ideal cracks. By definition, such a crack is modeled by a smooth open surface Ω (such as a disc or a spherical cap); the elastic displacement is discontinuous across Ω, and the traction vanishes on both sides of Ω (so that the crack is seen as a cavity of zero volume). We suppose that we have one crack with a smooth edge, ∂Ω, embedded in an infinite, unbounded, three-dimensional solid. Extensions to multiple cracks, to cracks in two dimensions, to cracks in halfspaces or in bounded domains, or to cracks with less smoothness may be made, with varying degrees of difficulty. For a variety of applications, see the book by Zhang and Gross (1998).

After formulating our scattering problem, we give the governing hypersingular integral equation in §5.2. This equation is solved approximately for long waves (low-frequency scattering) in §5.3. The approach used is elevated to a well-known ‘strategy’ in §5.4 prior to further applications. For flat cracks and screens, we can simplify the governing hypersingular integral equation. This is done in §5.5. Alternatively, we can use a direct approach, using Fourier transforms; see §5.6. Methods for solving the resulting equations are discussed in §5.7. The final section is concerned with curved cracks and screens.

This book is concerned with the theory of gravity–capillary free-surface flows. Free-surface flows are flows bounded by surfaces that have to be found as part of the solution. A canonical example is that of waves propagating on a water surface, the latter in this case being the free surface.

Many other examples of free-surface flows are considered in the book (cavitating flows, free-surface flows generated by moving disturbances, rising bubbles etc.). I hope to convince the reader of the beauty of such problems and to elucidate some mathematical challenges faced when solving them. Both analytical and numerical methods are presented. Owing to space limitations, some topics could not be covered. These include interfacial flows and the effects of viscosity, compressibility and surfactants. Some further developments of the theories described in the book can be found in the list of references.

Many results presented in the book have grown out of my research over the last 35 years and, of course, out of the research of the whole fluid mechanics community. References to the original papers are given. For this book, I have repeated the older numerical calculations with larger numbers of grid points than was possible at the time. I am pleased to report that the new results are in agreement with the earlier ones!