In the earlier chapters we have described the mathematical background – and the mathematical details – of many classical linear and nonlinear water-wave phenomena. In addition, in the later chapters, we have presented many of the important and modern ideas that connect various aspects of soliton theory with the mathematical theory of water waves. However, much that is significant in the practical application of theories to real water waves – turbulence, random depth variations, wind shear, and much else – has been omitted. There are two reasons for this: first, most of these features are quite beyond the scope of an introductory text, and, second, the modelling of these types of phenomena follows a less systematic and well-understood path. Of course, that is not meant to imply that these approaches are unimportant; such studies have received much attention, and with good reason since they are essential in the design of man-made structures and in our endeavours to control nature.

What we have attempted here, in a manner that we hope makes the mathematical ideas transparent, is a description of some of the current approaches to the theory of water waves. To this end we have moved from the simplest models of wave propagation over stationary water of constant depth (sometimes including the effects of surface tension), to more involved problems (for example, with ‘shear’ or variable depth), but then only for gravity waves.