### Table of Contents

**Chapter headings and selected contents:** **Fundamentals.** What is homogeneity? What is anisotropy? What is dispersion? What causes anisotropy of wave propagation. Appendix 1A: Analytical derivation of the relation between anisotropy and dispersion. **Tools for the Description of Wave Propagation under Piecewise Homogeneous Anisotropic Conditions.** Ray velocity and normal velocity. The ray-slowness surface; slowness and wave surface as polar reciprocals. Snell's Law. Appendix 2A: Formal description of the transformations used in this chapter. Analytic expression for inversion (reflection in a circle). Analytic derivation of the tangent curve from the footpoint curve. Analytic description of polar reciprocity. **Elasticity.** Tensors and vectors. Infinitesimal strain. Basic symmetries of the elastic tensor and the contracted notation. The elastic constants and material symmetry. Appendix 3A: The relation between elastic constants and rotational symmetry. Reduction of an arbitrary rotation to a sequence of rotations about one axis each. Tensors of rank two under rotation of the coordinate system. Appendix 3B: Invariants of the elastic tensor. Contraction of the elastic tensor on itself. Appendix 3C: FORTRAN subroutines for operations on elastic tensors in four- and two- subscript notation. **Elastic Waves - The Dispersion Relation and some Generalities about Slowness and Wave Surfaces.** The wave equation. Elements of inflection of the slowness surface. Slowness, polarization and symmetry. Singular directions. Appendix 4A: Orthogonality of polarization vectors. Appendix 4B: Explicit versions of the characteristic equation. Appendix 4C: Subroutines for the Kelvin-Christoffel matrix. **Stability Constraints.** The general stability condition. Stability conditions for isotropic, hexagonal, cubic, strong tetragonal and orthorhombic media. **One-Parameter Expressions for the Slowness Surfaces of Transversely Isotropic Media and the Slowness Curves in the Planes of Symmetry of Orthorhombic Media.** Decoupling of the across-plane polarization and a parameter expression for the "coupled" slowness curves. The "basic curves", the separating ellipse and the general shape of the "coupled" slowness curves. The associate parameter, the representing point, and a geometric construction for the polarization. A nomogram for the in-plane polarization. Appendix 6A: Closed explicit expressions for the coupled slowness curves in the symmetry planes of orthorhombic media. **One-Parameter Expressions for the Wave Curves in the Symmetry Planes of Orthorhombic Media.** Expressions for the wave surface in Cartesian and polar coordinates. Cusps. A measure of anisotropy. **Squared Slowness Surfaces and Squared Slowness Curves.** Phenomenology of slowness surfaces in the squared domain. The "framework" in the planes of symmetry. Deviations from the "framework" in the planes of symmetry. Appendix 8A: Determination of the type of the squared slowness curve. Appendix 8B: Properties of the square transformation. Coordinate grids. Straight lines. Symmetrically centered conics with aligned axes. General centered conics. Appendix 8C: Geometric tools for the conversion of a squared slowness curve to the ordinary domain. Geometric determination of curvature. **Causes of Anisotropy: Periodic Fine Layering.** Simple quasi-static strain modelling. Constraints on layer-induced anisotropy. Inversion of the compound stiffnesses to constituent stiffnesses. A nomogram for the determination of layer parameters. Appendix 9A: Generalized averages. **Anisotropy and Seismic Exploration.** Elliptical anisotropy. An equivalence theorem for surface-to-surface seismics. Some aspects of reflection seismics. **Eigentensors of the Elastic Tensor and their Relationship with Material Symmetry.** Rudimentary definition of a tensor space. Strain tensors and wave propagation. Coordinate-free representation of "wave-compatibility". Eigentensors and symmetry. Eigensystems of specific symmetries. Determination of the symmetry class. Appendix 11A: Construction of media with particular eigentensors. Reconstruction of an elastic tensor from its eigensystem. Construction of an eigensystem with particular eigenvectors. References. Index.

220 line drawings, 195 lit. refs.