This short introduction to dispersive waves provides an important prerequisite for moving from the dynamics of individual particle orbits to the dynamics of collective effects. As such, the ideas summarized here have a large impact on this book, especially on Part III, in which they are seen at work in the description of density and bending waves. The basic concepts (waves, wave packets, wave trains, group propagation, etc.) define a framework that has found wide and successful applications in hydrodynamics, geophysical fluid dynamics, and plasma physics.

In contrast with the case of hyperbolic waves, for which the description is centered on the properties of one class of partial differential equations, the study of dispersive waves focuses on one characteristic property of the solutions of many different types of dynamical equations that is called the dispersion relation. This is a relation, very often of a rather trivial algebraic form, between space and time modulation of elementary components of the wave process. The relation incorporates the constraints set by the dynamical equations.

It is sometimes believed that the main goal of the dynamicist should be to derive the dispersion relation for a given process and for a given model; the derived expression then would mark the end of the investigation.