This is the central chapter of the book. We emphasize that in this chapter we introduce the eikonal approximation without discussing the accuracy of the approximation, or how to deal with situations where it breaks down (for example, at caustics or in mode conversion regions). Those are matters for later chapters. The great advantage of eikonal methods is that they reduce the solution of systems of PDEs, or systems of integrodifferential equations, to the solution of a family of ODEs. This often results in a substantial increase in computational speed in applications. In addition, the ray trajectories themselves often provide useful physical insight. The outline of topics follows.

Eikonal theory for multicomponent wave equations is first developed in x-space where we derive the eikonal equation for the phase and the action conservation law (in the form of a nonlinear PDE). It should be noted that the dispersion function that arises from the variational principle is one of the eigenvalues of the dispersion matrix, restricted to its zero locus in phase space. The polarization is its associated eigenvector. We discuss the fact that the interpretation of these results and the method of solution of the eikonal and action transport equations are most natural when viewed in ray phase space.

We then discuss how to relate the x-space and phase space viewpoints, the key ideas being lifts and projections. The notion of a Lagrange manifold arises naturally in this context as a lifting of a local region of x-space into phase space. Singularities that appear under projection are related to caustics, which are dealt with in Chapter 5.