Nature produces many different kinds of waves and oscillations. In general, these periodic phenomena are very different from each other. However, certain physical and mathematical properties are common to small-amplitude waves almost irrespective of their nature. For instance, all small-amplitude waves can be characterized by dispersion relationhips and they transport physical quantities with the group velocity of the wave.

In this chapter we will examine some fundamental proprties of small-amplitude waves in neutral and conducting fluids. We will see that although neutral fluids exhibit a relatively small number of fundamental wave phenomena, conducting fluids (especially when they are magnetized) are a very fertile medium for the generation of a huge variety of plasma waves.

Here we shall concentrate on small-amplitude waves, when the wave equations can be linearized. This does not mean that nonlinear phenomena are unimportant — they are just too complicated for this introductory text. Also, we will limit our discussions to single species gases (or in the case of plasmas, to single ion plasmas). The results can be generalized to multispecies plasmas, when needed.

Linearized Fluid Equations

First of all, let us consider the linearized version of the ideal MHD equations. We choose to use the MHD equations, because mathematically the Euler equations represent a subset of these equations (one just has to set B to zero everywhere at all times). Let us assume that we have a solution of the full equation set, that is, ρm0, u0, p0, and B0 represent a steady-state solution of Eqs. (4.89).