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.

The problem of stability in the context of elliptical galaxies is completely different from the studies of stability described in Part III. For disks, there are several specific morphological properties, in particular, spiral structures, bars, warps, and corrugations, that we may match in terms of appropriate modes over more symmetric equilibrium configurations. In the case of elliptical galaxies, there are no outstanding morphological features to be addressed in a similar manner. Some features do exist, for example, isophotal twisting, shells, and peculiar kinematics, that might in principle be considered, but their three-dimensional (3D) spatial structure is not known, and it is not clear whether their origin can be traced to simple low-amplitude regular patterns. Linear stability analyses also would be of little use for addressing one obvious question related to the departures from spherical symmetry: What intrinsic shapes are realized? In fact, natural modes to be considered (and in some cases found to be unstable) are those with l= 2, but they are a degenerate class that includes oblate, prolate, and triaxial perturbations. Thus one general motivation at the basis of many modal analyses (see Chapter 9) is simply not present in the case of elliptical galaxies.

There remain two major physical reasons to carry out stability investigations. One is the need for a good knowledge of the intrinsic modes of a dynamical system as a prerequisite to the proper description of the driven problem (e.g., for the study of tidal interactions).

Normal galaxies have a variety of sizes and morphological categories. In this chapter their main empirically established structural characteristics are summarized schematically, and the problem of dark matter in the general cosmological context is briefly introduced. The problem of dark matter in galaxies will be better formulated in Chapter 5, among other broad issues addressed in this book, and will be discussed separately for spiral galaxies (Chapter 20) and for elliptical galaxies (Chapter 24) after the relevant dynamical tools have been properly developed.

Our world of galaxies is biased in favor of bright objects and against objects with low surface brightness. However, the tail of the galaxy luminosity function is widely populated by low-surface-brightness galaxies. We should thus be aware that our knowledge of the structure of galaxies and the discussion of the dynamics of normal galaxies that have been developed so far have focused mainly on bright galaxies; much remains to be done for fainter systems.

From astronomical observations, we ideally would wish to extract information on mass distributions and overall kinematics (mean flow motions and velocity dispersions). In practice, we must interpret images and spectroscopic information, gathered through various observational windows, and we must mediate the process by simplified models. Obviously, a major source of uncertainty is the a priori unknown three-dimensional structure of the object under investigation because the data give us quantities projected along the line of sight.