ABSTRACT
This chapter constructs asymptotic solutions of elastodynamics equations for homogeneous anisotropic media in the form of ray ansatz. It shows that in this case the leading term of the ray ansatz is still a perturbed local plane wave, with its amplitude and phase smoothly varying from point to point. The ray theory requires solving first-order nonlinear partial differential equations (e.g., the classical eikonal equation in the case of isotropic media). A general approach to such equations can be based on the method of characteristics, which reduces the problem to specialized systems of ordinary differential equations. A ray satisfies the Fermat principle stating that the related travel time is extremal. The chapter provides a recipe for solving the Cauchy problem for the eikonal equation: it is sufficient to solve a characteristic system with proper initial data.
