This chapter begins by identifying four possible states or situations: continuous systems (state 1) modeled in discrete form (state 2), and naturally discrete systems (state 3) modeled in continuous form (state 4). We refer to state 2 as a discretized state whereas state 4 is referred to as a homogenized state. The analyses to follow are divided along these lines but the chapter begins with the discrete state because that is the new aspect of waves considered here.