The new science of sound focuses on aspects of wave phenomena that have not been emphasized in traditional instruction. To elucidate these new aspects of wave phenomena, particularly the phase, in a clear exposition, we will rely on a number of simple models. We first define phase and group velocities using the one-dimensional monatomic harmonic crystal. Then, we advance to the diatomic one-dimension harmonic crystal and the one-dimensional harmonic crystal with alternating stiffness. We introduce the Green’s function approach to solving the wave equation, which will prove to be an indispensable tool in exploring phase related wave behavior. Three simple systems provide the basics of the Green’s function formalism, monatomic harmonic crystals with (1) a single mass defect, (2) a general perturbing potential, and (3) locally resonant structures. The introduction concludes with the introduction of Interface Response Theory (IRT) where we present the fundamental equations, introduce the cleavage operator, and demonstrate its use in a few examples. The Appendix 1 includes a Fortran77 code that will allow the reader to further explore the concepts presented here and in other chapters in greater detail.