In this chapter we introduce some basic concepts of the mathematics of wave propagation for acoustic and elastic waves. We shall often use heuristic or intuitive arguments rather than formal proofs, since our primary aim is to provide the necessary minimum background to readers not familiar with the fundamentals of continuum mechanics. Readers eager to educate themselves more extensively on the topics that we touch upon only briefly should consult the advanced seismology textbooks by Aki and Richards, Kennett or Dahlen and Tromp. Ĉervený and Chapman have written more specialized books on ray theory and its extensions.

We limit ourselves to wave propagation in isotropic media. This means that the elastic properties of the medium do not depend on its orientation in space: to shear a cube in the x-direction requires the same force as in the y- or z-directions. Even if the real Earth is locally anisotropic in regions of fine layering or of crystal alignment because of solid state flow, the background model – the model with respect to which heterogeneity is defined – is usually defined to be isotropic (and spherically symmetric). As we shall see in Chapter 16, anomalies with respect to the background model can be anisotropic. In the Sun, the magnetic field introduces anisotropy, but here too the background model is assumed to be isotropic (a gas).