Section V consists of three chapters, of which the first two present the theoretical framework and equations needed to obtain the practical numerical methods and formulations included in the third. Thus, Chapters 8 and 9 are for the most part meant as a reference to Chapter 10, and less so for their own utility. Nonetheless, we decided to include this material herein in full length because no readily available reference exists containing the detailed derivation of the powerful stiffness matrix method for all the three major coordinate systems. Thus, the reader may wish to either browse lightly Chapters 8 and 9 – especially the example problems in Chapter 9 – or skip these altogether at first and return later only as may be needed to clarify matters.

Chapter 8 begins with a summary of the solutions to the scalar and vector Helmholtz equations in three-dimensional space, and then proceeds to give full derivations to these equations and to the wave equation in all three coordinate systems. Unlike most books on the theory of elasticity, we use matrix algebra throughout, and manage to express the final results as products of matrices, each of which depends on one coordinate only (whether or not the systems are layered). This greatly simplifies applications to layered media.

Chapter 9 makes a compact introduction to the integral transform methods commonly used to analyze stratified media, such as finite and infinite plates, rods, or spheres. Examples are also given to illustrate the application of these concepts.